Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

In this talk, I will discuss the relation between Killing spinors, symmetry operators and conserved quantities. For the case of Maxwell test fields on the Kerr spacetime, the Killing spinor can be used to construct conserved currents not related to Noether currents, as well as higher order conserved tensors.

Abstract:

For any given integer N larger than 2, we show that every bounded measurable vector field on a bounded domain in Euclidean space is N-cyclically monotone up to a measure preserving N-involution. The proof involves the solution of a multidimensional symmetric Monge-Kantorovich problem, which we first study in the case of a general cost function. The proof exploits a remarkable duality between measure preserving transformations that are N-involutions and Hamiltonian functions that are N-cyclically antisymmetric.

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

Consider axisymmetric initial data for the Einstein equations, satisfying the dominant energy condition, and having two ends, one asymptotically flat and the other either asymptotically flat or asymptotically cylindrical.

Heuristic physical arguments lead to the following inequality m≥√|J| relating the total mass and angular momentum. Equality should be achieved if and only if the data arise from the exrteme Kerr spacetime. Dain established this inequality (along with the corresponding rigidity statement) when the data are maximal and vacuum, and subsequently several authors have improved upon and extended these results. Here we consider the general non-maximal case in which the matter fields satisfy the dominant energy condition, and introduce a natural deformation back to the maximal case which preserves all the relevant geometry. This procedure may then be used to establish the angular momentum-mass inequality (and rigidity

statement) in the general case, assuming that a solution exists to a canonical system of two elliptic equations. This is joint work with Ye Sle Cha.

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

The minimal surface equation for timelike surfaces of the Minkowski space is a quasi-linear wave equation. The nonlinear part of the equation exhibits a cancellation known as the null condition. We replicate the strategy of Smith and Tataru of constructing a wave-packet parametrix, which coupled with a space-time estimates for the null form allows us to lower the regularity compared to the general result for the quasilinear wave equation obtained by Smith and Tataru.

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

We discuss long time behavior of linear scalar waves on Kerr and Kerr-de Sitter black hole backgrounds and their stationary perturbations. The physical motivation comes from the analysis of gravitational waves emitted during the ringdown stage of a large scale event (such as merging with another black hole). The properties of these waves, and their frequencies, called quasi-normal modes or resonances, depend on the structure of the set of all trapped light rays.

In the considered Kerr(-de Sitter) case, the trapped set is r-normally hyperbolic and we can provide a detailed description of quasi-normal modes and long-time behavior of linear waves.

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

There are several recent deep results concerning future stability of solutions to the Einstein-Vlasov and Einstein-Euler-system with a cosmological constant. In this talk we will present some results concerning the case of a vanishing cosmological constant and assuming that the spacetime is homogeneous. Recent results will be highlighted with an outlook of how they can be generalized.

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

A considerable amount of research activity in recent years has been devoted to the study of nonlinear partial differential equations of Monge-Ampere type (MATEs) in connection with their applications to conformal geometry, optimal transportation and geometric optics. In this talk we plan to skim through a potpouri of recent results pertaining to  the underlying structural condition for regularity originating in a paper in 2005 by Ma, Wang and myself.

Abstract:

We consider the problem of unique continuation from infinity for linear waves on a curved background. We derive new Carleman estimates from infinity for wave operators on Minkowski, Schwarzschild and certain perturbations thereof, which allow us to conclude that solutions to a wave equation vanishing to infinite order on suitable portions of future and past null infinity imply that the solution itself vanishes in an open region of spacetime. Surprisingly, the result in Schwarzschild is stronger than the one in Minkowski spacetime. Moreover, we show that our results are sharp (in particular infinite order vanishing is necessary). These results are motivated by questions in General Relativity. This is joint work with Spyros Alexakis and Arick Shao.

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

I'll review optimal transport from a dynamical point of view, and consider some elementary examples of periodic orbits of probability measures on the circle, The definition of a rotation number for such orbits seems to be a natural object in this setting. I'll also consider minimal transport plans subjected to a prescribed rotation number.

Abstract:

I will review the present status on the existence, structure and stability of static and stationary solutions of the Einstein-Vlasov system. Both analytical and numerical results will be discussed.

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

In some recent papers, a connection between optimal transport and density functional theory has been noticed. The transport model involves a cost function of Coulomb type which decreases with distance. We present some initial results on this multimarginal transport problem. In particular, we introduce a natural notion of cyclical Monge problem and we show the equality between its infimum and the  minimum in the classical Kantorovich problem. Finally, we study the problem in dimension one, building an optimal map.

Abstract:

Asymptotically hyperbolic manifolds appear in many situations in General Relativity, such as isolated systems in a universe with negative cosmological constant and in the AdS/CFT correspondence theory as conformally compact spaces.

As for the ADM mass of asymptotically Euclidean manifolds, one can define mass-like asymptotic invariants. After a review of classical results about these quantities such as positive mass theorems by X. Wang and P.T. Chrusciel - M. Herzlich, we will focus on the mass-aspect tensor that appears in Wang's definition, when the boundary at infinity is assumed to be spherical.

We will give examples for Kottler-Schwarzschild-AdS manifolds as well as a new family which has interesting properties near infinity, such as being conformally flat and of constant scalar curvature. We will finally discuss how to look for new mass-like invariants constructed from the mass-aspect tensor.

This is partly based on a joint work with Mattias Dahl and Romain Gicquaud.

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

The current standard model of the universe is spatially homogeneous, isotropic and spatially flat. Furthermore, the matter content is described by two perfect fluids (dust and radiation) and there is a positive cosmological constant. Such a model can be well approximated by a solution to the Einstein-Vlasov equations with a positive cosmological constant. As a consequence, it is of interest to study stability properties of solutions in the Vlasov setting. The talk will contain a description of recent results on this topic. Moreover, the restriction on the global topology of the universe imposed by the data collected by observers will be discussed.

Abstract:

We will discuss the construction on new Type II compact ancient compact solutions to the  Yamabe flow.  Our solutions are rotationally symmetric and converge, as $t \to -\infty$,  to a tower of N spheres. Their curvature operator changes sign. Our technique may be viewed as a  parabolic analogue of  gluing  two  exact solutions to the rescaled equation, that is the spheres, with narrow cylindrical necks to obtain a new ancient solution to the Yamabe flow.

Abstract:

In recent years, there has been a resurgence in interest in the relationship between Newtonian gravity and General Relativity on cosmological scales, which is motivated by questions surrounding the physical interpretation of large scale cosmological simulations using Newtonian gravity and the role of Newtonian gravity in cosmological averaging. The overriding fundamental question is in what sense, if any, does Newtonian gravity provide an approximation to General Relativity on a cosmological scale. In this talk, I will describe a wide class of inhomogeneous relativistic solutions that are well approximated on cosmological scales by solutions of Newtonian gravity. The initial value formulation for these solutions will be presented and I will outline the estimates that lead to existence and guarantee convergence to solutions of Newtonian gravity.

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

This seminar is open to all faculty, postdocs and doctoral students involved in either program at MSRI in Fall 2013.  It is designed to bring the two programs together as both fields require an understanding of the convergence of manifolds and metric measure spaces.  We meet Wednesday afternoons 3-5pm in the Baker Boardroom.

For information about titles and speakers, please visit: https://sites.google.com/site/professorsormani/teaching/msri-reading-seminar

Abstract:

The speaker will be chosen among the members of the audience at the time of the event. One or more volunteers from the audience will present a current topic of research, possibly but not necessarily one on which they are currently working.

Abstract:

Hamilton-Jacobi equations (HJE) in the space of measures arise naturally when approximating the dynamics of many particle systems in classical mechanics. At first sight, these equations appear to be very difficult to analyze. We argue, however, that the space of measures has just enough regularity to admit a uniqueness theorem for solutions of these HJE.

Our hope is that the ideas presented in the talk will provide methods for obtaining properties of solutions and in turn information on many particles systems.

Abstract:

I describe work with with Stefan Hollands that establishes a new criterion for the dynamical stability of black holes in $D \geq 4$ spacetime dimensions in general relativity with respect to axisymmetric perturbations: Positivity of the canonical energy, $\mathcal E$, on a subspace of linearized solutions that have vanishing linearized ADM mass, momentum, and angular momentum at infinity and satisfy certain gauge conditions at the horizon implies mode stability. Conversely, failure of positivity of $\mathcal E$ on this subspace implies the existence of perturbations that cannot asymptotically approach a stationary perturbation. We further show that $\mathcal E$ is related to the second order variations of mass, angular momentum, and horizon area by $\mathcal E = \delta^2 M - \sum_i \Omega_i \delta^2 J_i - (\kappa/8\pi) \delta^2 A$, thereby establishing a close connection between dynamic stability and thermodynamic stability. Thermodynamic instability of a family of black holes need not imply dynamic instability because the perturbations towards other members of the family will not, in general, have vanishing linearized ADM mass and/or angular momentum. However, we prove that all black branes corresponding to thermodynamically unstable black holes are dynamically unstable. We also prove that positivity of $\mathcal E$ is equivalent to the satisfaction of a ``local Penrose inequality,'' thus showing that satisfaction of this local Penrose inequality is necessary and sufficient for dynamical stability.

Abstract:

Outline: Several examples of hidden convexity in nonlinear PDEs will be addressed. Most of them are related to the theory of “optimal transportation”, which is the theme of one of the two MSRI programs this fall: see http://www.msri.org/programs/277.

1) Fully nonlinear elliptic equations: The canonical example is the reduction of the Monge-Amp`ere equation (a fully nonlinear elliptic PDE related to the “Minkowski problem” in geometry) to a convex minimization problem, `a la Kantorovich, which can be solved by convex duality techniques. Connections with combinatorial optimization and “optimal transport” will be discussed.

2) Mathematical fluid mechanics:
i) Some solutions to the Euler equations can be obtained by using the least action principle (following V.I. Arnold’s geometric interpretation), leading to the problem of “optimal incompressible transport”; this problem has a hidden convex structure which provides existence, uniqueness and partial regularity for its solution.
ii) The hydrostatic and semi-geostrophic limits of the Euler and Navier-Stokes equations are examples of very singular limits that cannot be justified on standard linear functional spaces but can be addressed successfully on suitable convex functional cones.

3) Electromagnetism and magnetohydrodynamics:
i) The nonlinear theory of electromagnetism by Born and Infeld is described by a system of nonlinear conservation laws (which leads to the standard Maxwell equations for fields of low intensity), which has an interesting hidden structure, showing very easily its hyperbolic nature; the high field limit of the BI equations will be discussed and related to the “optimal transport of currents”.
ii) The magnetic relaxation equations proposed by Moffatt can be seen as a natural extension to divergence-free vector fields of the scalar heat equation (following the interpretation of Jordan-Kinderlehrer-Otto); they form a very degenerate parabolic systems of PDEs with a hidden convex structure that enables us to show the global existence of “dissipative solutions” (in the spirit of P.-L. Lions and Ambrosio-Gigli-Savar´e).

Abstract:

Thursday Sept 12:

A crash course on rational mechanics (with a personal viewpoint) will relate standard optimal transportation problems to Euler and ideal MHD equations.

Abstract:

Tuesday Sept 10

The Monge-Ampere second boundary value problem has been related to the Monge-Kantorovich transportation problem. Existence and uniqueness of maps with convex potential between given compactly supported probability measures on R^d are established.

Abstract:

In recent years there has been an explosion of interest in black holes in higher dimensional gravity.  This, in particular, has led to questions about the topology of black holes in higher dimensions. In this talk we review Hawking's classical theorem on the topology of black holes in 3+1 dimensions (and its connection to black hole uniqueness) and present a generalization of it to higher dimensions.  The latter is a geometric result which imposes restrictions on the topology of black holes in higher dimensions.  We shall also discuss recent work on the topology of space exterior to a black hole.  This is closely connected to the Principle of Topological Censorship, which roughly asserts that the topology of the region outside of all black holes (and white holes) should be simple.   All of the results to be discussed rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry.  This talk is based primarily on joint work with Rick Schoen and with Michael Eichmair and Dan Pollack.

Abstract:

The space $\mathbb X$ of all metric measure spaces $(X,d,m)$ plays an important r\^ole in image analysis, in the investigation of limits of Riemannnian manifolds and metric graphs as well as in the study of geometric flows that develop singularities. We show that the space $\mathbb X$ -- equipped with the $L^2$-distortion distance $\Delta\!\!\!\!\Delta$ -- is a challenging object of geometric interest in its own. In particular, we show that it has nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on $\mathbb X$ are presented.

Abstract:

In 2007, A. Ionescu and S. Klainerman introduced a new approach to considering the black hole uniqueness problem which is based on a characterisation (which originated in the works of M. Mars and W.

Simon) of the family of Kerr black holes among stationary vacuum solutions of Einstein's equations. In this talk I'll start by giving a review of the black hole uniqueness problem. Then I will review the algebraic properties of general stationary electro-vacuum solutions, which leads us to a characterisation of the Kerr-Newman family. I will conclude with a discussion of how to turn the characterisation into a smallness measure and how this can be used to demonstrate various "perturbative" uniqueness results for Kerr-Newman black holes.

Abstract:

2nd lecture: Thursday Sept 5:

-Monge-Ampere has been related to the Monge-Kantorovich optimization problem, existence and uniqueness of maps with convex potential between given measures has been stated; -A comprehensive proof has been started in the simple case of compactly supported measures, end of the proof due Tuesday Sept 10

Abstract:

1st lecture: Tuesday Sept 3:

-General introduction to the equations to be discussed:

Monge-Ampere, Euler, Born-Infeld, magnetic relaxation; -Tools we need from Convex analysis: Legendre-Fenchel transforms  and Rockaffellar's duality theorem for Banach spaces.

Abstract:

Swarming by Nature and by Design
Andrea Bertozzi, University of California, Los Angeles

The cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, such phenomena have attracted a renewed interest from the engineering community.

This talk will cover a survey of the speakers research and related work in this area ranging from aggregation models in nonlinear partial differential equations to control algorithms and robotic testbed experiments.

We conclude with a discussion of some interesting problems for the mathematics community.

Abstract:

A paper of Cascini, Tanaka and Xu.

Abstract:

Cluster algebras were introduced by Fomin and Zelevinsky in 2001 and have become an important tool in representation theory, higher category theory, and algebraic/Poisson geometry.

The goal of my talk (based on a joint paper with V. Retakh) is to introduce totally noncommutative clusters and their mutations, which can be viewed as generalizations of both ``classical" and quantum cluster structures.

Each noncommutative cluster X is built on a torsion-free group G and a certain collection of its automorphisms. We assign to X a noncommutative algebra A(X) related to the group algebra of G, which is an analogue of the cluster algebra, and expect a Noncommutative Laurent Phenomenon to hold in the most of algebras A(X).

Our main examples of "cluster groups" G include principal noncommutative tori which we define for any initial exchange matrix B and noncommutative triangulated groups which we define for all oriented surfaces.

Abstract:

In this joint work with Jake Goodman we show that the crossed product of a twisted Calabi-Yau algebra by its modular (aka Nakayama) automorphism is a Calabi-Yau algebra as was shown before by several authors for various special classes of twisted Calabi-Yau algebras.

Abstract:

2:20 - 3:00 PM
F-signature of a Cartier module
Speaker: Kevin Tucker
 

Abstract:

In 2004 Derksen made a conjecture concerning bounds on the degrees of the generators of the syzygies of a ring of polynomial invariants, the resolution being over a polynomial ring that maps onto the invariants.

We will present a proof of an amended version of this conjecture that only uses elementary homological algebra.

This is joint work with Marc Chardin.

Abstract:

We prove some homological properties of the category of Lyubeznik's F-modules. In detail, we show that this category has finite global dimension. And we point out that the injective hull of the residue field, though injective in the category of R-module, is not injective in the category of F-modules.

Abstract:

Hartshorne's Connectedness Theorem asserts that, if a local ring (R,m) has depth at least 2, then Spec(R)\{m} is connected. On the other hand, Faltings' Connectedness Theorem says, if an ideal I of a d-dimensional complete local domain (R,m) can be generated by d-2 elements, then Spec(R/I)\{m} is connected.

Abstract:

A new method, currently under development, is brought forward to establish an upper bound on the growth of any finitely generated group, using a variant of monoid categories and analagous CW-complexes.

Abstract:

A real matrix is totally positive if all of its minors are greater than zero, and, more generally, is totally nonnegative if all of its minors are nonnegative.

There is a cell decomposition of the space of totally nonnegative matrices of a given size, and the set of totally positive matrices forms the big cell in this decomposition. There are well-known efficient criteria for recognising total positivity, but until recently not for the other cells.

In this talk, I'll introduce some of the main ideas concerning totally nonnegative matrices, and then discuss the cell recognition problem, and give a solution that was motivated by connections with the theory of torus invariant prime ideals in quantum matrices.

This is joint work with Ken Goodearl and Stephane Launois.

Abstract: We will review the definition and basic facts about Gorenstein liaison of varieties in projective space. Taking the case of codimension 2, which is well understood, as a paradigm, we will consider the questions that arise in higher codimensions. A major problem is whether every arithmetically Cohen-Macaulay variety is in the Gorenstein liaison class of a complete intersection (in acronyms: does ACM => glicci?). While this has been proved to be so in many particular cases, the problem is open in general. We will examine some known cases and two striking recent results in the hope of clarifying where the difficulty lies.

Abstract:

Let $k$ be an algebraically closed field of characteristic zero, $A$  be an Artin-Schelter regular algebra, and $G$ be a group of graded automorphisms of $A$.  J{\o}rgensen and Zhang proved that if all elements of $G$ have homological determinant 1, then $A^G$ is Artin-Schelter Gorenstein.  For a family of AS regular algebras of dimension 3 (the Noetherian graded down-up algebras) we determine when $A^G$ is a ``complete intersection" (in a sense to be defined), and we relate this condition to the form of the Hilbert series of $A^G$, and to generators of $G$.  For this class of algebras, we obtain an extension to $A$ of a theorem for $k[x_1, \cdots, x_n]$ due to Kac-Watanabe and Gordeev.

(This is joint work with James Kuzmanovich and James Zhang).

Abstract:

It is a long standing conjecture that F-injectivity deforms. In this talk, we discuss some simple criteria on local cohomology sufficient to give that F-injectivity deforms. These will apply to show that F-injectivity deforms when the non-CM locus of the special fiber is isolated and that F-purity deforms to F-injectivity.. This is joint work with Kazuma Shimomoto and Jun Horiuchi. 

Abstract: Commutative Algebra and Algebraic Geometry Seminar
Tuesdays, 3:45-6:00
939 Evans Hall

Organized by David Eisenbud
More information at http://hosted.msri.org/alg/

Abstract:

An "elliptic Weyl algebra" is obtained by localising a (generic) Sklyanin algebra at the central element g.  It is a hereditary domain of GK-dimension 2 that has the surprising and beautiful property that all of its subalgebras are finitely generated and noetherian.  We present the classification of maximal orders in elliptic Weyl algebras and give some consequences, focusing on the ideal structure.  We show that any order in an elliptic Weyl algebra has DCC on ideals, and deduce that any graded order in (the 3rd Veronese of) a Sklyanin algebra must be noetherian.  This is joint work with Dan Rogalski and Toby Stafford.

Abstract:

Orlov showed in 1997 that all exact, fully faithful functors between the bounded derived categories of two smooth projective varieties are isomorphic to a Fourier-Mukai transform. In this talk we will discuss what happens when removing the fullness and faithfulness hypotheses.

Abstract:

The Lyubeznik numbers are a family of invariants of a local ring containing a field that detect many ring-theoretic properties, and also have topological and geometric interpretations.  Motivated by their utility, in joint work with Luis Núñez-Betancourt, we define a new family of invariants of any local ring with characteristic p>0 residue field; in particular, they are defined for local rings of mixed characteristic.  For local rings of equal characteristic p>0, both the Lyubeznik numbers and these new invariants are defined.  In some cases, they agree, but we give an example where they differ.

Abstract:

I compute the coinvariants of functions under Hamiltonian flow for complete intersections with isolated singularities. For surfaces the Hamiltonian flow is with respect to the Jacobian Poisson structure; in higher dimensions the Hamiltonian flow can be generalized using the natural top polyvector field which can be thought of as a degenerate Calabi-Yau structure. In particular the dimension of the coinvariant space equals the sum of the dimension of the top cohomology of the singular variety with the sum of the Milnor numbers of the singularities. In other words this equals the top cohomology of the smoothing of the variety. These results follow from more general ones computing the D-module which represents invariants under the Hamiltonian flow. The structure is interesting: it turns out to have a summand which is the maximal extension on the bottom, but not on the top, of the intersection cohomology D-module. This is joint work with Pavel Etingof.

Abstract:

Mulmuley and Sohoni came up with the approach to the Valiant version of the P versus NP problem using invariant theory. This approach leads to interesting questions related to the orbit closures of the algebraic group actions. It also shows that the P versus NP problem is closely related to two classical problems in representation theory of the general linear group. In this talk, I will discuss these connections and point out some easier problems that might be of interest to commutative algebraists.

Abstract:

Commutative Algebra and Algebraic Geometry Seminar
Tuesdays, 3:45-6:00
939 Evans Hall

Organized by David Eisenbud
More information at http://hosted.msri.org/alg/ 3:45 PM Quantitative Cocycles Speaker: Persi Diaconis Abstract: Choosing sections for natural maps is a basic mathematical activity. In joint work with Soundarajan and Shao we study the problem of choosing "nice" sections. Here is a typical theorem: Let $G$ be a finite group, $H$ a normal subgroup. Let $X$ be coset representatives for $H$ in $G$. suppose the proportion of $x,y$ in $X$ with $xy \in X$ is more than $1-1/60$. Then the extension splits. There are many variations, many open problems and applications to basic arithmetic and computer science. 5:00 PM Clifford Algebras and the Ranks of Modules in Free Resolutions Speaker: David Eisenbud Abstract: A conjecture of Horrocks, and independently of Buchsbaum and myself, asserts that the sum of the ranks of the modules in the free resolution of a module annihilated by a regular sequence of length c is at least $2^{c}$. I'll describe a related new conjecture by Irena Peeva and myself and show how we proved a special case, in our work on complete intersections, using results on Clfford algebras and enveloping algebras.

Abstract:

I am going to describe Hochschild-Witt homology, a new homology theory for associative algebras and DG algebras over a finite field that extends crystalline cohomology to the non-commutative setting. This is based on reexamination and generalization of the classical notion of Witt vectors. Using this, we introduce what we call a Hochschild-Witt complex, a gadget refining the usual Hochschild homology complex of an algebra in exactly the same way as the de Rham-Witt complex of Deligne and Illusie refines the usual de Rham complex of a smooth algebraic variety over a finite field.

Abstract: The use of motivic techniques in Quantum Field Theory has been widely explored in the past ten years, in relation to the occurrence of periods in the computation of Feynman integrals. In this lecture, based on joint work with Aluffi, I will show how some of these techniques can be extended to a motivic analysis of the partition function of Potts models in statistical mechanics. An estimate of the complexity of the locus of zeros of the partition function, can be obtained in terms of the classes in the Grothendieck ring of the affine algebraic varieties defined by the vanishing of the multivariate Tutte polynomial, based on a deletion-contraction formula for the Grothendieck classes.

Abstract:

Let (R, m) be a local ring of characteristic p with perfect residue field. Let I be an ideal of R. In this talk I will describe some recent results, joint with Ilya Smirnov and Keiichi Watanabe, on the asymptotic behavior of length of the zeroth local cohomology module of R/I^[q]. This is an obvious generalization of the classical concepts of Hilbert-Kunz function and multiplicity.

Abstract: Hecke algebras associated with Weyl groups appear naturally as endomorphism rings in the study of finite reductive groups. Broué, Malle and Rouquier have generalized their definition to all complex reflection groups. Hecke algebras depend on a parameter q and their representation theory becomes very interesting when q specializes to a root of unity. The existence of canonical basic sets and the cellular algebra structure have been key elements in the understanding of the representation theory of Hecke algebras in that case. An open question is whether these structures exist for any choice of parameters and for all complex reflection groups. In this talk will see how the connections with the representation theory of rational Cherednik algebras, another type of algebras associated with complex reflection groups, could provide an answer to this problem.

Abstract: In this talk, we will discuss certain invariants of hypersurfaces defined via the Frobenius endomorphism in characteristic p. In addition to presenting lots of examples, we will also recall the relationship between these invariants and ones arising in the characteristic zero setting (e.g., log canonical thresholds and Bernstein-Sato polynomials). *Warning*: This talk will be more of an elementary, colloquium-style overview than a serious research talk, and will not be aimed towards experts. In particular, we hope to make it accessible to postdocs from both programs. Though we will discuss some recent results, we will probably not prove anything.

Abstract: This talk will present an overview of the Theory of Combinatorial Games (CG)

In 1901, a Harvard math professor named Bouton published his mathematical solution to the then-popular barroom game called Nim.
This marked the birth of a subject call combinatorial game theory, which seeks to determine optimal strategies for positions in board games. The theory has been most successful for games whose positions tend to breakup into disjoint regions, which are treated as sums.
Examples of such games include classics such as Go, Dots-and-Boxes, Fox-and-Geese, and Konane (also known as Hawaiian checkers), as well as more modern games such as Domineering, Hackenbush, Amazons, and Clobber.
Combinatorial Game Theory yields insights into endgame positions in all of these games, as well as a few special positions in chess and Anglo-American checkers.

The mathematical emphasis of the theory is on a partial order and the behavior of equivalence classes of combinatorial game positions under the operation of addition. This enables CG to be treated as a large commutative monoid, which contains some well-known groups. The dyadic rationals appear immediately, and provide numerical bounds on all other CG. Among the most interesting CG are the infinitesimals. One common class of infinitesimals is the nimbers, a countably infinite group in which every element has order two. There is also a first order of infinitesimals, and many higher orders, none of which can be viewed as the "second" order. By allowing games with infinitely many positions while imposing other severe restrictions, one obtains Conway\'s elegant constructions of the real, transfinite, and surreal numbers.

The set of finite CG also naturally contains several idempotents with useful properties. And more: There are some powerful theorems which can be viewed as properties of other "contrived" idempotents.

This talk will explain some of these results. No prior background is assumed.

Abstract: This talk will focus in part on joint work with T. Cassidy regarding a definition of complete intersection using point modules in the context of skew polynomial rings.
The hurdles involved in extending the definition to some other types of algebras will then be explored.

Abstract: For any field, we produce an explicit series 4-dimensional domains which are homogeneous graded algebras over the field and whose normalizations (i) have the same degree 1 components as the original algebras, (ii) are generated as algebras in degrees at most 2, yet (iii) admit no uniform upper bound for the discrepancy between the Hilbert functions and the Hilbert polynomials, when viewed as modules over the original homogeneous algebras. The natural habitat of this sort of phenomena is very ample line bundles on projective toric varieties. In the talk, we will overview various related uniform bounds in the homological and combinatorial theory of the rings of sections of such line bundles, some known to exist and some conjectured. Most of these results are joint works with Bruns, Trung, Beck, and Delgado.

Abstract: Commutative Algebra and Algebraic Geometry Seminar
Tuesdays, 3:45-6:00
939 Evans Hall

Organized by David Eisenbud
More information at http://hosted.msri.org/alg/ 3:45 PM Finite criteria for a ring to be Golod Speaker: Luchezar Avramov Abstract: Golod rings can be characterized by a numerical condition on the Betti numbers of the residue class field, or by a homological condition on the products in Koszul homology. The original proof of the equivalence shows that each condition can be checked in a known number of steps, but many subsequent proofs do not provide such information. The talk will explain a proof that does. 5 PM Sharp upper bounds for higher linear syzygies of projective varieties Speaker: Kangjin Han Abstract: In this talk, we are going to consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first remind `Partial Elimination Ideals (PEIs)' theory and introduce a new framework in which one can study the syzygies of embedded projective schemes well using PEIs theory and the reduction method via inner projections. Next we establish quite useful inqualities which govern the relations between the graded Betti numbers in the first linear strand of an algebraic set $X\subset\mathbb{P}^{N}$ and the ones of its inner projection $X_q\subset\mathbb{P}^{N-1}$. Using these results, we obtain some natural sharp upper bounds for higher linear syzygies of any nondegenerate projective variety in terms of the codimension with respect to its own embedding and classify what the extremal case and next to the extremal case are. This gives us interesting generalizations of classical characterizations on varieties of small degree by Castelnuovo and Fano from the viewpoint of `syzygies'. Note that our method could be also applied to get similar results for more general categories (e.g. connected in codimension one algebraic sets). This is a joint work with Sijong Kwak.

Abstract: I will explain why, when studying derived autoequivalences of 3-folds, it is necessary to understand noncommutative deformations of curves. In the talk I will give a construction of a certain "noncommutative twist" associated to any floppable curve that recovers the flop-flop functor on the level of the derived category. The idea is that the commutative deformation base is too small for the homological algebra to work, so we need to fatten it by considering noncommutative directions. Our construction generalizes work of Seidel--Thomas and Toda who considered the special case when the curve deforms in only one direction. I will try to explain why considering noncommutative deformations is strictly necessary, as I will show that considering only the commutative deformations does not give a derived autoequivalence as one might hope. This all sounds very fancy, but the talk will be based around one basic example, where the birational geometry of a certain 3-fold is controlled by the cusp in the quantum plane, which is a 9-dimensional self-injective algebra. This is all based on joint work with Will Donovan.

Abstract: Recently there has been an advance in understanding the relationship between the derived categories of the stack quotient of a variety X by a group G and its GIT quotients. I will explain the connection. Then I will present joint work with Dan Halpern-Leistner on how a variation of GIT can be used to produce autoequivalences, which turn out to be special. Along the way, we will see a tight connection between spherical functors and mutations of semiorthogonal decompositions. If time and courage permit, I will remark on how our results confirm a prediction from mirror symmetry.

Abstract: Let $R=k[x_1,\cdots, x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0.$ If $I$ is an ideal of $R,$ we denote $H^i_I(R)$ the $i$-th local cohomology module of $R$ with support in $I.$ After some introductory material on local cohomology, we will give a lower bound of the dimension of associated primes $P$ of $H^i_I(R)$ in terms of the degrees of the generators of $I.$ Let $\m=(x_1,\cdots, x_n)$ be the maximal ideal generated by the variables and let $I_1,\cdots, I_s$ be homogeneous ideals of $R.$ We will describe the grading of $H^i_{\mathfrak{m}}(H^{j_1}_{I_1}\circ\cdots \circ H^{j_s}_{I_s}(R))$ and also give two algorithms to calculate it.

Abstract: For more information visit http://math.berkeley.edu/about/events/lectures/chern

Abstract: 10:00 - 11:00 AM Cohomological and homogeneity conditions on the Jacobian of a polynomial Speaker: Uli Walther Abstract: The free resolution of the Jacobian ideal of a polynomial contains a lot of geometric information on the hypersurface. "Freeness" arises if the resolution is as short as it can be (barring smoothness). We discuss natural sources, weakenings, and implications of freeness under varying homogeneity conditions. 11:30 AM - 12:30 PM Associated primes of local cohomology modules Speaker: Gennady Lyubeznik Abstract: : I will give a survey of work on associated primes of local cohomology modules, including the very latest results. 2:00 - 3:00 PM Regularity of group cohomology Speaker: Peter Symonds Abstract: We will survey what is known about local cohomology and regularity of cohomology rings of groups and invariants.

Abstract: There are two fundamental classes of objects in representation theory: path algebras of quivers and Geigle-Lenzing's weighted projective lines (or derived equivalently, Ringel's canonical algebras). Their importance comes from the fact that they give rise to abelian categories of global dimension 1. The aim of this talk is to consider higher dimensional analogs of these objects.

Abstract: There appeared not long ago a Reduction Formula for derived Hochschild (co)homology, that has been useful e.g., in the study of Gorenstein maps and of rigidity w.r.t. semidualizing complexes. The formula involves the relative dualizing complex of a map, and so brings out a connection between Hochschild homology and Grothendieck duality. The proof, somewhat ad hoc, uses homotopical considerations via numerous non canonical resolutions, both projective and injective, of differential graded objects. Recent efforts are producing more intrinsic approaches, which one hopes to upgrade to "higher" contexts, for example bimodules over algebras in infinity categories. This would lead to wider applicability, for example, to ring spectra; and the methods might be globalizable, revealing some homotopical generalizations of Grothendieck duality. (The original formula has a geometric version, proved by completely different methods coming from duality theory.) After reviewing the formula, I will just try to explain a few elementary things about infinity categories, and how one can express therein the basic relation (adjoint associativity) between Hom and tensor products that underlies any proof.

Abstract: The 2013 Chern Lectures 141 McCone Hall For more information visit http://math.berkeley.edu/about/events/lectures/chern

Abstract: Commutative Algebra and Algebraic Geometry Seminar
Tuesdays, 3:45-4:45
939 Evans Hall

Organized by David Eisenbud
More information at http://hosted.msri.org/alg/

3:45 PM
Speaker: Vu Thanh
Koszul algebras and the Frobenius endomorphisms
Abstract: Let $R$ be a standard graded algebra over a field of characteristic $p > 0$. Let $\phi:R\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let $^{\phi}M$ be the abelian group $M$ with an $R$-module structure induced by the Frobenius endomorphism. The $R$-module $^{\phi}M$ has a natural grading given by $\deg x=j$ if $x\in M_{jp+i}$ for some $0\le i \le p-1$. In this talk, I\\'ll discuss our new characterization of Koszul algebras saying that $R$ is Koszul if and only if there exists a non-zero finitely generated graded $R$-module $M$ such that $\reg_R \up{\phi}M <\infty$. This is a joint work with Hop Nguyen.

Abstract: Question: for A in SL(4), when does CC^4/A have a crepant resolution? If you expect Calabi-Yau 4-fold geometry to be analogous to the 3-fold case you are probably in for a disappointment. Even in cases when a crepant resolution exists, A-Hilb CC^4 may be bad, for example containing exuberant components. In the longer term, this may be a test case for derived geometry. I will give some algorithms and examples, and suggest some questions on relations between A-Hilb CC^4 computed in toric geometry and Hilb^n CC^3. This is speculative material, so don't expect any of it to be correct. For related material, see our website www.warwick.ac.uk/staff/T.Logvinenko/Traps

Abstract: An organizational meeting to discuss working groups for upcoming focus periods

Abstract: I will report on joint work with Claudiu Raicu and Emily Witt on computing the GL-equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix. I will explain the main tools that we employ in our study, namely (1) the BGG correspondence between the homology groups of linear complexes over a polynomial ring and minimal free resolutions over the exterior algebra, and (2) the exterior algebra analogue of the geometric technique for computing syzygies. If time permits, I will mention how the same techniques apply to give the description of the local cohomology modules with support in the ideal of 2n x 2n Pfaffians of a (2n+1) x (2n+1) generic skew-symmetric matrix, and mention a possible approach to extend our results to the case of non-maximal minors.

Abstract: For more information visit http://math.berkeley.edu/about/events/lectures/chern

Abstract: I describe the projective resolution of a codimension 4 Gorenstein ideal, aiming to extend Buchsbaum and Eisenbud's famous result in codimension 3. The main result is a structure theorem stating that the ideal is determined by its (k+1) x 2k matrix of first syzygies, viewed as a morphism from the ambient regular space to the Spin-Hom variety SpH_k in Mat(k+1, 2k). This is a general result encapsulating some theoretical aspects of the problem, but, as it stands, is still some way from tractable applications.

Abstract: Birge Hall, Room 50 For more information, visit http://math.berkeley.edu/about/events/lectures/chern

Abstract: Commutative Algebra and Algebraic Geometry Seminar
Tuesdays, 3:45-6:00
939 Evans Hall

Organized by David Eisenbud
More information at http://hosted.msri.org/alg/ Differential Graded Categories and Milnor's Theorem Oren Ben-Basset (Oxford) In his book on Algebraic K-theory, Milnor gave a technique for gluing projective modules. This technique constructs a projective module on a ring A out of a projective modules on rings B and C and a certain isomorphism. In joint work with Jonathan Block, we generalized this construction to a gluing (involving a homotopy fiber product) of certain differential graded categories. In this talk, I will review differential graded categories and introduce the construction of Jonathan Block which assigns a differential graded category to each differential graded algebra. This construction is fundamentally different than a well known construction involving differential graded modules. If time permits I will give some geometric consequences relating to compact complex manifolds. One of these consequences relates to forming vector bundles on a complex manifold X out of vector bundles "near" a submanifold Z and vector bundles on X-Z. I will also mention a related gluing construction involving gluing categories of coherent sheaves over a punctured formal neighborhood of a subvariety Z inside a variety X over any field k. The punctured formal neighborhood is described using Berkovich analytic geometry. That construction is joint work with Michael Temkin.

Abstract: We discuss the asymptotic regularity of high powers of an ideal, and various saturations of powers of an ideal. We give some illustrative examples, say something about what is known, and speculate on what is not known.

Abstract: I will talk about a recent classification of abelian hereditary numerically finite categories with Serre duality, up to derived equivalence. In that classification, one distinguished three main types. The focus of the talk will lie on giving examples of each of these three types, and new examples that occur when one removes the condition that these categories need to be numerically finite.

Abstract: Abstract: I'll describe a new perspective on the category O over symplectic reflections algebras for the groups $Z/\ell Z \wr S_n$, using ideas from the theory of categorical actions of Lie algebras. This builds on earlier work of Rouquier, Shan, Vasserot, Losev and others, but takes a more explicit diagrammatic perspective. In particular, I'll give an explicitly presented finite dimensional algebra (generalizing KLR algebras) whose representation category is the same as a block of the SRA category O mentioned above.

This algebra is quite interesting in its own right (for example, it is cellular), but it also allows hands-on proofs of several facts which are hard to see from the SRA perspective:

* it is manifestly graded, and thus provides a grading on category O

* the derived equivalences between blocks predicted by Rouquier are
given by easily guessed bimodules

* the identification of decomposition numbers with coefficients of a
canonical basis follows from the application of well-established
techniques from geometric representation theory, for example, from
Varagnolo and Vasserot's proof that projective modules over KLR
algebras give the canonical basis.

I'll try to convince you that this is a fruitful perspective and indicate avenues for better understanding its ties to other approaches to these categories.

Abstract: Bionic symplectic manifolds are smooth symplectic varieties equipped with a particularly good action of a 2-torus.
These spaces arise naturally in geometric representation theory, for instance when studying rational Cherednik algebras or W-algebras. In this talk I will describe how one can use the bionic structure to learn a great deal about the categories of deformation-quantization modules on these spaces. For instance, one can calculate the K-theory and Hochschild (co)-homology of these categories. The talk is based on joint work in progress with C. Dodd, K. McGerty and T. Nevins.

Abstract: Commutative Algebra and Algebraic Geometry Seminar
Tuesdays, 3:45-6:00
939 Evans Hall

Organized by David Eisenbud
More information at http://hosted.msri.org/alg/ 3:45 PM: Links and resolutions of rational and Cohen-Macaulay singularities J'anos Koll'ar We study the connection between algebraic properties of singularities and the topology of their links and their resolutions. 5:00PM: Tropicalization of classical moduli spaces Qingchun Ren The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves. (Work with Steven Sam and Bernd Sturmfels)

Abstract: Let  be a finite dimensional algebra over an algebraically closed field, and Rep(, d) the standard affine variety parametrizing the isomorphism classes of -modules with dimension vector d. We will address the problem of determining the irreducible componentsof the Rep(, d) in terms of representation-theoretic invariants, and of exploring the generic structure of the modules these components encode. After giving an overview of existing theory, stemming from work of Kac, Schofield, Schr¨oer, Crawley-Boevey, Riedtmann, among others, we will present new results. We will conclude with a conjecture addressing the next plausible step in trying to push beyond the status quo.

Abstract: Hochster noted that the eventual 2-periodicity of projective resolutions over hypersurface rings can be used to define a stable intersection pairing.
We will discuss this notion and show how it links to various geometric and topological incarnations.

Abstract: Ten years ago, G.Bergman asked: "Can one factor the classical adjoint of a generic matrix?"
With G.Leuschke we showed that, yes, sometimes you can. We arrived at this by translating the question into one on morphisms between matrix factorizations of the determinant.

Understanding all such morphisms lead through joint work with Leuschke and van den Bergh to a description of a noncommutative desingularization of determinantal varieties in a characteristic-free way and we are just about to put the finishing touches on this work here at MSRI.

I will desribe the highlights of this journey and mention some resulting, and remaining open questions.

Abstract: In this talk, I will first present the definition and some of the classical properties of Castelnuovo-Mumford regularity. Then we will turn to some known results and open questions on estimates for this invariant, when the base ring is a field. Next, we present a case where working over a more general base ring is very useful. Finally, if time permits, extensions of this notion to more general ones will be motivated and briefly described.

Abstract: We'll prove that the category O of some class of Cherednik algebras is equivalent to a highest weight subcategory of the (parabolic) category O of affine gl_n. This implies, in particular, some conjectural Kazhdan-Lusztig type character formulas for the simple modules. The proof uses categorifications and highest weight theory. This is a joint work with R. Rouier, P. Shan and M. Varagnolo.

Abstract: In joint work with Gebhard Böckle, we systematically study modules with a right action of Frobenius.
Up to nilpotent actions, the resulting category of Cartier Crystals satisfies some surprising and very strong finiteness conditions. The theory has connections to questions in commutative algebra (finiteness of local cohomology), birational geometry (test ideals) and number theory (constructible p-torsion sheaves, L-functions). In my talk I will explain the key features of the theory via simple examples, elaborate its connection with Grothendieck-Serre duality and discuss some of the aforementioned applications.

Abstract: Reverse homological algebra deals with questions like the following ones, concerning a ring $R$ and a (left) $R$-module $k$: What $Ext_R(k,k)$-modules have the form $\mathrm{Ext}_R(M,k)$ for some $R$-module $M$? What are the essential images of the functor $\mathrm{RHom}_R(?,k)$ from various subcategories of the derived category of $R$-modules to the derived category of DG modules over $\mathrm{RHom}_R(k,k)$?

Some answers to the second question will be presented when $R$ is commutative, noetherian and local and $k$ is its residue field.
Under an additional hypothesis on $R$, which holds for complete intersections and for Golod rings, the first question will be answered "up to truncations." A crucial step of the proof involves a contravariant Koszul duality for (not necessarily commutative) connected DG algebras.

Part of the talk is based on joint work with David Jorgensen.

Abstract: The talk is based on my joint work with Robert Fisette.
We consider the A-infinity algebra associated with a certain generator of the derived category of coherent sheaves on a smooth projective curve. In the case of an elliptic curve this A-infinity algebra can be computed explicitly, and in particular, one can recover the j-invariant by looking at the higher products m_6 and m_8. In the higher genus curve we prove that the A-infinity structure can be uniquely recovered up to homotopy from the higher products up to m_6. We also study the map from the moduli space of curves with marked points given by the products m_3.

Abstract: Let H be a semisimple (so, finite dimensional) Hopf algebra over an algebraically closed field k of characteristic zero and let A be a commutative domain over k. We show that if H acts on A \\'inner-faithfully\\', then H must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen. This result also extends to working over k of positive characteristic. We also discuss results on Hopf actions on Weyl algebras as a consequence of the main theorem. This is joint work with Pavel Etingof.

Abstract: Test ideals are a measure of singularities in positive characteristic, and are analogs of multiplier ideals from characteristic zero. In this talk, I will describe some recent joint work with Karl Schwede on the test ideals of non-principal ideals. In particular, time permitting I will discuss a description of test ideals using regular alterations, as well as positive characteristic global division theorem for test ideals.

Abstract: We present some recent progress on the LG/CFT correspondence which relates certain representations of the bosonic part of the vertex operator superalgebra for the N=2 minimal model and subcategories of permutation type matrix factorizations. Joint work with Alexei Davydov and Ingo Runkel.

Abstract: I begin by reviewing the original construction of Khovanov-Rozansky knot invariant based on tensor products of matrix factorizations. I will then explain how techniques from homotopical algebra apply to show that it can also be described in terms of stable Hochschild homology of Soergel bimodules; the latter are prominent in representation theory and also occur in the construction of other knot invariants. The description and techniques also allow for an alternative proof of the fact that one gets a knot invariant.

Abstract: Motivated by topological quantum field theory, one can start from any pivotal bicategory B and construct its "orbifold completion" B_orb in terms of certain Frobenius algebras. The completion satisfies the natural properties B \subset B_orb and (B_orb)_orb = B_orb, and it gives rise to various new equivalences and nondegeneracy results. I will explain this construction (which is joint work with Ingo Runkel) and apply it to the bicategory of isolated singularities and matrix factorisations. Applications include a unified perspective on ordinary equivariant matrix factorisations, a one-line proof of Knörrer periodicity, and new equivalences for simple singularities.

Abstract: One can view the affine Hecke category of equivariant coherent sheaves on the Steinberg variety as a "matrix algebra" over the adjoint quotient of a reductive group. Thus one expects its center and abelianization (Hochschild cohomology and homology categories) to involve equivariant coherent sheaves on the commuting variety. We will present precise descriptions involving coherent sheaves with prescribed singular support. This is joint work with D. Ben-Zvi (Texas) and A. Preygel (Berkeley).

Abstract: In a 2008 paper, Avramov and Iyengar studied Cohen-Macaulay and Gorenstein morphisms and characterized them in terms of invariants from Hochschild(co)homology.
The invariants in question turned out to be related to Grothendieck duality, and in a 2010 paper Avramov, Iyengar, Lipman and Nayak pursued the ideas to prove reduction formulas for the Hochschild (co)homology groups in question. In this talk, we will discuss a new approach to the problem, based on Brown representability.

Abstract: There is an old conjecture by Stanley which gives a numerical
criterion for a module of covariants to be Cohen-Macaulay.
To prove this conjecture one has to be able to compute, or
at least estimate the local cohomology with support in the nullcone,
a question that has regained interest recently. For this reason I will
discuss my old work on this in which I give a precise (but complicated)
formula for the local cohomology with support in the nullcone as a
G-equivariant D-module, under some reasonably generic conditions.

Abstract: Commutative Algebra and Algebraic Geometry Seminar Tuesdays, 3:45-6:00 939 Evans Hall Organizer: David Eisenbud http://hosted.msri.org/alg/ 3:45 PM
F-singularities in families
Speaker: Wenliang Zhang

F-singularities are classes of singularities defined via the Frobenius endomorphism in characteristic p. One prominent method to measure these singularities is to introduce ideals that define the loci, such as the test ideal (a characteristic p analog of the multiplier ideal). However, these ideals may behave quite differently from their characteristic 0 counterparts. For instance, the test ideal may fail to satisfy the generic restriction theorem that holds for the multiplier ideal. This prompts a natural question, how do we study F-singularities in families?

In this talk, I will discuss recent joint work with Zsolt Patakfalvi and Karl Schwede on the behavior of F-singularities in families. In particular, I will discuss a relative version of the ideals mentioned in the previous paragraph and some restriction theorems for them. If time permits, some global geometric consequences will also be discussed.

5:00 PM
Periodicity of betti numbers of monomial curves.
Speaker: Vu Thanh

Let $k$ be an arbitrary field. Let $S = (a_1< ...

Updated on Mar 18, 2013 03:40 AM PDT

Abstract: Riemann surfaces, which we now think of abstractly as smooth algebraic
curves over the complex numbers, were described by Riemann as graphs of
multivalued holomorphic functions-in other words, branched covers of the
Riemann sphere $\P^1$. The Hurwitz spaces, varieties parametrizing the
set of branched covers of $\P^1$ of given degree and genus, are still
central objects in the study of curves and their moduli. In this talk, we'll
describe the geometry of Hurwitz spaces and their compactifications, leading
up to recent work of Anand Patel and others.

Abstract: Let X be a holomorphic symplectic variety and F a coherent sheaf on X. Then a natural question to ask is if we take a deformation quantization of X can we also quantize the coherent sheaf F? We give an answer to this question, up to order 2, when the coherent sheaf is a direct image of a vector bundle supported on a smooth subvariety. If time permits we will also construct BV differentials on Tor and Ext groups between modules which admit quantizations. This is joint work with V.Baranovsky and V. Ginzburg.

Abstract: Free divisors are certain non-normal hypersurfaces in a complex manifold, whose singular loci have nice algebraic properties: the Jacobian ideals are Maximal Cohen Macaulay modules for the hypersurface rings.
In this talk we give a short introduction to free divisors, also to their original definition (due to K. Saito) via logarithmic derivations and logarithmic differential forms, and indicate in which areas of mathematics they appear.
Then we talk about a conjecture about the singularities of some well-known free divisors, namely, normal crossing divisors. Finally we move on to logarithmic residues, a recent topic of interest, and to a conjecture about relationship between the logarithmic residue and the normalization of a free divisor.

Abstract: Let (A,m) be a Noetherian Gorenstein local ring of dimension d. An m-primary ideal of A is called "good" if G(I):=\oplus I^n/I^{n+1} is Gorenstein with a(G) =1-d. Equivalently, if Q is a minimal reduction of I, I is good if and only if I^2=QI and Q:I = I.

In a paper Goto-Iai-Watanabe, Good ideals in Gorenstein local rings, T.A.M.S. 353 (2000), we studied good ideals in 2 dimensional rational singularities. In this talk, we show existence of good ideals for non-rational surface singularities using cohomology groups of anti-nef cycles on a resolution of the singularity.

This is a joint work with Tomohiro Okuma and Ken-ichi Yoshida.

Abstract: The Poincare series of a finitely generated module over a commutative local ring is defined to be the generating series of its Betti numbers. Although examples of rings with transcendental Poincare series exist, there seems to be an abundance of rings for which the Poincare series of all finitely generated modules are rational, sharing a common denominator. I will present recent work with Marilina Rossi, in which we prove that this property holds for generic Gorenstein Artinian algebras. When the socle degree is different than 3, this property holds more generally, for all local Artinian rings whose associated graded algebra is Gorenstein and has extremal (compressed) Hilbert series.

Abstract: I will speak on the work of Enright-Hunziker and Pruett on resolutions of determinantal varieties.

Abstract: Commutative Algebra and Algebraic Geometry Tuesdays, 3:45-6pm in Evans 939 organizer: David Eisenbud http://hosted.msri.org/alg 3:45 PM Algebra structures on short free resolutions Speaker: Lars Christensen Let $R$ be a quotient of a regular local ring $Q$. If the free resolution of $R$ over $Q$ has length $2$, then its structure is given by the Hilbert--Burch Theorem. For longer resolutions the structure is significantly harder to describe, but for resolutions of length $3$ a partial classification is available. I will discuss some recent progress, achieved in collaboration with Oana Veliche and Jerzy Weyman, towards completion of this classification. 5:00 PM The Golod Property for Monomial Ideals Speaker: David Berlekamp The Golod property for monomial ideals is implied by a simple criterion on GCD's. This can be used to prove that the product of any two monomial ideals is Golod.

Abstract: To a finite acyclic quiver we can associate a path algebra kQ (k a field), a preprojective algebra and a Coxeter group. Using a natural map from the Coxeter group to ideals in the preprojective algebra,investigated in work with Buan, Iyama and Scott, we establish a 1-1 correspondence between elements in the Coxeter group and cofinite quotient closed subcategories of the finite dimensional kQ-modules. This is based on work with Oppermann and Thomas.

Abstract: I shall begin with a gentle introduction to modular representation theory of finite groups. Many questions about these reduce to the case of an elementary abelian p-group, so I shall spend most of the talk on these. In particular, I shall talk about modules of constant Jordan type, and what they have to do with algebraic vector bundles on projective space.

Abstract: The noncommutative minimal model program (ncMMP) uses the techniques of Mori theory to classify noncommutative surfaces, which we assume to be finite over their centres. The main result is a very pleasing analogy with the commutative theory - a noncommutative surface is either birational to a unique minimal model, or a noncommutative ruled surface. In this talk, I will explain how to associate a Brauer pair to a noncommutative surface, and how to run the ncMMP for these Brauer pairs. Although this is similar in spirit to the log MMP, there are some differences. Noncommutative ruled surfaces arise naturally in this context, and we will conclude with some structural results about them obtained by moduli theory.

Abstract: The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. Rather than considering the whole ring of invariants, one considers a separating set, that is, a set of invariants whose elements separate any two points which can be separated by invariants. In this talk, we focus on representations of finite groups. We show that if there exists a polynomial separating algebra, the the group action must be generated by (pseudo-)reflections. This produces a new, simpler proof of the classical result of Serre that if the ring of invariants is polynomial then the group action must be generated by (pseudo-)reflections.

Abstract: Under very general conditions, asymptotic limits of graded families of ideals can be be computed by associating a cone generated by a corresponding integral semigroup and then taking the volume of an appropriate slice. We discuss this method, its origins, and its application to problems in commutative algebra.

Abstract: A skew Calabi-Yau algebra A is a generalization of a Calabi-Yau algebra which allows for a non-trivial Nakayama automorphism. We develop formulas for the Nakayama automorphisms of other algebras derived from A by important constructions: (1) smash product algebras A#H for a Hopf algebras H acting on A, and (2) graded twists of A. We discuss applications of these formulas and open questions. This is joint work with Manny Reyes and James Zhang.

Abstract: The Bernstein-Sato polynomial of a hypersurface is related to several concepts in commutative algebra. After an introduction to the general properties of this polynomial, we will focus on the case of hyperplane arrangements. We will discuss relations to Milnor fibers, to the intersection lattice of the arrangement, and the Jacobian ideal of the arrangement. We will also discuss new results, in the arrangement case, on the conjectured connection between the topological zeta function and the Bernstein-Sato polynomial.

Abstract: I will conclude speaking on resolutions of length 3 and Kac-Moody Lie algebras.

Abstract: Commutative Algebra and Algebraic Geometry Seminar
Tuesdays, 3:45-6:00
939 Evans Hall

Organized by David Eisenbud
More information at http://hosted.msri.org/alg/ 3:45PM: Symmetric tensors, rank versus cactus rank. Kristian Ranestad The cactus rank of a form is the minimal length of an apolar subscheme to the form, and has therefore also been called the scheme length of the form. It is in general smaller than the rank, but its value for a general form is not known. We shall relate it to the dimension of the family of polynomials with given dimension for the space of partials of all orders. I shall report on recent results in work with Bernardi and Marques. 5:00PM: Maximal Cohen-Macaulay modules over cubic hypersurface rings of dimension 4 and 5 Marti Lahoz In this talk, I will give a new construction of stable Arithmetically Cohen-Macaulay (ACM) bundles on cubic hypersurfaces in $P^{4}$ and $P^{5}$ Our construction relies on the interpretation of MCM modules as objects in the derived category of modules over the even part of the Clifford algebra associated to a quadric fibration. If time permits, I will explain how choices in the threefold case induce different compactifications of the moduli space of Ulrich bundles of rank 2. This is a joint work with Emanuele Macrì and Paolo Stellari.

Abstract: We will introduce the theory of varieties for modules and tensor ideals in categories of modules of a Hopf algebra. We will describe a classification of (thick) tensor ideals for a class of Hopf algebras: the quantum elementary abelian groups. This is joint work with Julia Pevtsova.

Abstract: Categorization Seminar
1:30 - 3:30 PM
Commons Room
Organizer: Catharina Stroppel
Speaker: Michael Ehrig

Abstract: Rings of polynomial invariants of finite group actions are among the most classical objects in commutative algebra. There are many beautiful theorems ensuring that the invariant ring has good properties when the order of group is invertible, but if the order of the group is not a unit (i.e., is divisible by the characteristic of the ground field), many of these properties become more subtle. One key issue is that the ring of invariants may fail to be a direct summand of the polynomial ring. In this talk, we will review some of these subtleties, and provide a generalization of a result of Singh on this direct summand property for rings of modular invariants.

Abstract: Starting with quadrics I shall discuss old and more recent results on the variety of powersum decompositions of a form and related results and open problems on apolar subschemes of minimal length.

Abstract: I start the series of talks on the connection of resolutions of length 3 and Kac-Moody Lie algebras. In the first part I will discuss the background, definitions and older results.

Abstract: Commutative Algebra and Algebraic Geometry Tuesdays, 3:45-6pm in Evans 939 Organizer: David Eisenbud http://hosted.msri.org/alg Date: Feb 26 3:45 A model of the moduli space of marked cubic surfaces Speaker: Laurent Gruson It is well known that the restricted Picard group $Pic_0(S)$ of a cubic surface $S$ in $P_3$ is isomorphic to the root lattice $E_6$. The choice of such an isomorphism is a markng of the surface. The moduli space of marked cubic surfaces is thus acted on by the Weyl group $W(E_6)$. We seek a equivariant morphism of this moduli space into $P(E_6 \otimes C)$. Ellingsrud and Peskine have identified $Pic_0(S)\otimes C$ with the vector space W of dual quadrics ``apolar" to S. Our morphism transforms $S$ into the hyperplane of $W$ consisting of quadrics containing the faces of the Sylvester pentahedron of $S$. We give an explicit form of this parametrization. 5:00 Symmetric output feedback control and isotropic Schubert calculus Speaker: Frank Sottile One area of application of algebraic geometry has been in the theory of the control of linear systems. In a very precise way, a system of linear differential equations corresponds to a rational curve on a Grassmannian. Many fundamental questions about the output feedback control of such systems have been answered by appealing to the geometry of Grassmann manifolds. This includes work of Hermann, Martin, Brockett, and Byrnes. Helmke, Rosenthal, and Wang initiated the extension of this to linear systems with structure corresponding to symmetric matrices, showing that for static feedback it is the geometry of the Lagrangian Grassmannian which is relevant. In my talk, I will explain this relation between geometry and systems theory, and give an extension of the work of Helmke, et al. to linear systems with skew-symmetric structure. For static feedback, it is the geometry of spinor varieties which is relevant, and for dynamic feedback it is quantum cohomology and orbifold quantum cohomology of Lagrangian and orthogonal Grassmannians. This is joint work with Chris Hillar.

Abstract: Launois, Lenagan and Rigal proved that many generic quantized coordinate rings and quantum algebras are noncommutative UFDs in the sense introduced by Chatters and Jordan. Other ncUFDs and ``near UFDs" (in which all but finitely many height 1 prime ideals are principal) have appeared in recent joint work with K. Brown. We shall describe these results and relations with the problem of determining the Zariski topology on prime spectra of quantum algebras.

Abstract: In recent years a surprising number of insights and results in noncommutative algebra have been obtained by using the global techniques of projective algebraic geometry. Many of the most striking results arise by mimicking the commutative approach: classify curves, then surfaces, and we will use this approach here. As we will discuss, the noncommutative analogues of (commutative) curves are well understood while the study of noncommutative surfaces is on-going. In the study of these objects a number of intriguing examples and significant techniques have been developed that are very useful elsewhere. In this talk I will discuss several of them.

Abstract: We study the invariants of an algebraic group action on an affine variety via separating morphisms, that is, dominant G-invariant morphism to another affine variety such that points which are separated by some invariant have distinct image. This is a more geometric take on the study of separating invariants, a new trend in invariant theory initiated by Derksen and Kemper.
In this talk, I will discuss some results which indicate that the fact that the invariants are not always finitely generated is less significant than the fact that what we would want to call the quotient morphism is not always surjective.
(Joint work with Hanspeter Kraft)

Abstract: This will be a survey talk focused on providing important and understandable examples. We will investigate the relationships among differential operators, topology, and algebraic geometry/commutative algebra as the characteristic of the base field changes from 0 to p > 0.

Abstract: I'll explain a sufficient condition on a small category for its category of representations to be Gorenstein in an appropriate sense. I then want to discuss the problem of finding generators for singularity categories of such categories of representations. More specifically, I'll talk around the idea of using the intrinsic structure of the small category to characterize the representations (over a regular ring) of finite projective dimension.

Abstract: Consider a linearly reductive group acting nicely on a polynomial ring over a field of characteristic zero. This action endows an extra structure upon certain local cohomology modules of the ring; studying this structure has the potential to provide examples and answer open questions on local cohomology modules. We will present this theory through the example of local cohomology with support in the ideal generated by the maximal minors of a generic matrix; we will also mention how this theory can be applied in other settings (as part of work in progress).

Abstract: The representation theory of many interesting algebras, including various kinds of Cherednik algebras, can be approached via equivariant D-modules: more precisely, one can construct functors (of ``quantum Hamiltonian reduction'') from categories of equivariant D-modules to representations of the algebras. I'll describe an effective combinatorial criterion for such functors to vanish on certain equivariant D-modules---equivalently, for certain equivariant D-modules to have no nonzero group-invariant elements. I will also explain consequences of this vanishing criterion for natural t-structures on the derived categories of sheaves over quantum analogs of various interesting symplectic algebraic varieties. Most of the talk will be low-tech and will presume no prior familiarity with the terms mentioned above. This is joint work with Kevin McGerty.

Abstract: This is a report on joint work with Eric Friedlander, Julia Pevtsova and sometimes Andrei Suslin. We introduce higher rank variations on the notion of $\pi$-points as defined by Friedlander and Pevtsova for representations of finite group schemes. Using this we can define module of constant r-radical and r-socle type. Such modules determine bundles over the Grassmannian associated to the higher rank $\pi$-points in the case that the group scheme is infinitesimal of height one. When the group scheme is an elementary abelian p-group, there is universal function for computing the kernel bundles as modules over the structure sheaf of the Grassmannian of r-planes in n space. These ideas also extend to various sorts of subalgebra of restricted p-Lie algebras.

Abstract: Jerzy Weyman will continue on Vinberg Representations and Steven Sam will start on the paper by Enright, Hunziker and Pruett.

Abstract: Commutative Algebra and Algebraic Geometry Tuesdays, 3:45-6pm in Evans 939 Organizer: David Eisenbud http://hosted.msri.org/alg Date: Tuesday, Feb 19 3:45 The cone of Betti tables over a rational normal curve Speaker: Steven Sam Recent work of Eisenbud and Schreyer describes all of the linear inequalities that are satisfied by the Betti tables of graded modules over polynomial rings. There has been interest in giving analogous descriptions for other graded rings. In recent work with Kummini, we consider the homogeneous coordinate ring of a rational normal curve and relate its cone of Betti tables to the corresponding cone for a polynomial ring in 2 variables. As in the case of polynomial rings, the extremal rays are given by "pure resolutions". I will explain the idea behind this work and give conjectures for other rings of finite Cohen-Macaulay representation type. 5:00 Robust Toric Ideals Speaker: Elina Robeva An ideal of $k[x_1,..,x_n]$ is robust if it is minimally generated by a universal Gröbner basis. This rare property is shared by monomial ideals, ideals of maximal minors of generic matrices, and Lawrence ideals. In this talk we'll discuss recent attempts to classify robust toric ideals, including a complete description for ideals generated in degree two. Along the way there will be many examples, some conjectures, and plenty of counterexamples. This is joint work with Adam Boocher.

Abstract: If k is a field, the set of all representations of an associative k-algebra A on a finite-dimensional vector space V can be given the structure of an affine k-scheme, called the representation scheme Rep_V(A). According to a heuristic principle proposed by M.Kontsevich and A.Rosenberg, the family of schemes {Rep_V(A)} for a given non-commutative algebra A should be thought of as a substitute or `approximation' for `Spec(A)'. The idea is that every property or non-commutative geometric structure on A should naturally induce a corresponding geometric property or structure on Rep_V(A) for all V. This viewpoint provides a litmus test for proposed definitions of non-commutative analogues of classical geometric notions, and in recent years many interesting structures in non-commutative geometry have originated from this idea. In practice, however, if an associative algebra A possesses a property of geometric nature (for example, A is a NC complete intersection, Cohen-Macaulay, Calabi-Yau, etc.), it may happen that, for some V, the scheme Rep_V(A) fails to have a corresponding property in the usual algebro-geometric sense. The reason for this seems to be that the representation functor Rep_V is not `exact' and should be replaced by its derived functor DRep_V (in the sense of non-abelian homological algebra). The higher homology of DRep_V(A), which we call representation homology, obstructs Rep_V(A) from having the desired property and thus measures the failure of the Kontsevich-Rosenberg `approximation.' In this this talk, after reviewing the construction of DRep_V, we will present several results and a number of explicit examples confirming the above intuition. If time permits, we will also discuss other applications of representation homology, including its relation to Hochschild, cyclic homology and algebraic K-theory.

Abstract: The multiplicity of a local ring R is its most fundamental invariant. For example, it tells us how singular the ring is. The multiplicity is computed from the limit as n goes to infinity of the length of R modulo the nth power of its maximal ideal. Many other multiplicity like invariants naturally occur in commutative algebra. We discuss a number of naturally occurring limits of this type, and show that in very general rings, such limits always exist.

Abstract: 4:10 – 5:30 PM
891 Evans Hall

Counting Moduli of Quiver Representations with Relations
Speaker: Jiarui Fei

Abstract: In this talk, I will explain how to count the rational points of the GIT quotients of quiver representations with relations. I will focus on two types of algebras – one is extended from a quiver Q, and the other is Dynkin A_2 tensored with Q. For both, explicit formulas will be given. For application in the quantum algebra, we study when they are polynomial-count. We follow the similar line as quiver without relations using the Hall algebra. However, algebraic manipulations in Hall algebra will be replaced by corresponding geometric constructions. You will see many examples.

Abstract: We prove that Behrend's function is constant on Hilb^n(C^3). A calculation of motivic zeta functions shows the relevant Milnor fibers have zero Euler characteristic. As a corollary we see that Hilb^n(C^3) is generically reduced.

Abstract: I'll explain a method for computing the defining ideals of the secant and tangential varieties of Segre-Veronese varieties, and describe some connections to plethysm problems.

Abstract: Given a finite subset A of positive integers, let S(A) denote the induced semigroup ring. In other words, S(A) is the coordinate ring of the monomial curve parametrized by A. For a positive integer j, write A+(j) for the subset obtain by adding j to each element in A. In this talk, we will discuss the conjecture that the Betti numbers of the semigroup ring S(A+(j)) are eventually periodic in j. In particular, this conjecture implies that the first Betti number of S(A+(j)), which is the minimal number of equations defining the associated monomial curve, is eventually periodic in j and hence bounded for all j.

Abstract: February 05, 2013
939 Evans Hall

3:45PM
Spaces and varieties of barcodes
Speaker: Gunnar Carlsson

Persistence barcodes are very interesting and useful summaries of the shape of point cloud data sets. They can be used to understand individual data sets, or data sets whose objects themselves have shapes. The set of barcodes can be understood as a space in various ways, one as a metric space, the other as an (infinite dimensional) algebraic variety. We\'ll discuss these ideas and ask for input on more general situations.

5:00PM
Uniform annihilation of Ext modules and generation of derived categories
Speaker: Ryo Takahashi
This talk is based on ongoing joint work with Srikanth Iyengar. Let R be a commutative noetherian ring. We call a nonzerodivisor x of R a uniform annihilator if there exists an integer n>0 such that x annihilates Ext_R^n(M,N) for all finitely generated R-modules M and N. It follows from a result of Hsin-Ju Wang that R possesses a uniform annihilator if R is a complete local domain containing a field with perfect residue field. In this talk, we consider how to proceed in the general case. We also relate it with strong generation of the bounded derived category of finitely generated R-modules in the sense of Bondal and Van den Bergh.

Abstract: Let K be a regular noetherian commutative ring. I will begin by explaining the theory of rigid residue complexes over essentially finite type K-algebras, that was developed by J. Zhang and myself several years ago. Then I will talk about the geometrization of this theory: rigid residue complexes over finite type K-schemes. An important feature is that the rigid residue complex over a scheme X is a quasi-coherent sheaf in the etale topology of X. For any map f : X → Y between K-schemes there is a rigid trace homomorphism (that usually does not commute with the differentials). When the map f is proper, the rigid trace does commute with the differentials (this is the Residue Theorem), and it induces Grothendieck Duality. Then I will move to finite type Deligne-Mumford K-stacks. Any such stack \X has a rigid residue complex on it, and for any map f : \X → \Y between stacks there is a rigid trace homomorphism. These facts are rather easy consequences of the corresponding facts for schemes, together with etale descent. I will finish with the Residue Theorem, which holds when the map f : \X → \Y is proper and coarsely schematic; and the Duality Theorem, which also requires the map f to be tame. If there is time I will outline the proofs. - Lecture notes are at http://www.math.bgu.ac.il/~amyekut/lectures/resid-stacks/notes.pdf

Abstract: Intersection theory grew out of very basic questions like: "given four generic lines in \mathbb{P}^3 - how many lines intersect all four of them?" Questions like that are one of the simplest examples of an application of Schubert Calculus, a very important tool in combinatorial representation theory. It is used to describe cohomology rings, but also to construct categories which describe representation theoretic problems geometrically. It serves as the basis of more sophisticated methods of enumerative geometry, like Gromov-Witten theory and Quantum Cohomology. An amazing fact is that the same combinatorics also occurs when decomposing tensor products of representations of a semisimple complex Lie algebra.

This talk will describe some of these basic ideas and use them to explain a connection between quantum fusion products and quantum cohomology, relating Verlinde algebras and quantum cohomology rings. All this is related to questions arising in commutative and noncommutative algebraic geometry, integrable systems, representation theory, combinatorics ...

Abstract: Let $V$ be a symplectic vector space of dimension $2n$. Given a partition $\lambda$ with at most $n$ parts, there is an associated irreducible representation $\bS_{[\lambda]}(V)$ of $\Sp(V)$. This representation admits a resolution by a natural complex $L^{\lambda}_{\bullet}$, (called {\bf Littlewood complex}), whose terms are restrictions of representations of $\GL(V)$. When $\lambda$ has more than $n$ parts, the representation $\bS_{[\lambda]}(V)$ is not defined, but the Littlewood complex $L^{\lambda}_{\bullet}$ still makes sense. I will explain how to compute its homology. One finds that either $L^{\lambda}_{\bullet}$ is acyclic or it has a unique non-zero homology group, which forms an irreducible representation of $\Sp(V)$. The non-zero homology group, if it exists, can be computed by a rule reminiscent of that occurring in the Borel--Weil--Bott theorem. This result categorifies earlier results of Koike--Terada on universal character rings. The result has an interpretation in terms of commutative algebra: it calculates the Koszul homology of the ideal generated by invariants of the symplectic group on the set of vectors. One can prove analogous results for orthogonal and general linear groups. If time permits I will show how some of these ideas can be generalized to exceptional groups. The talk is based on the joint paper with Andrew Snowden and Steven Sam and another one with Steven Sam.

Abstract: This talk is based on joint work with Hailong Dao. Let R be a Cohen-Macaulay local ring. Recall that R is said to have finite CM-representation type if there are only finitely many isomorphism classes of indecomposable (maximal) Cohen-Macaulay R-modules. In this case, clearly there exists a finitely generated R-module M out of which all Cohen-Macaulay R-modules are built by taking direct sums and direct summands. The converse is also true, namely, such a module M does not exist if R has infinite CM-representation type. Now a natural question arises: what if we also allow taking syzygies and a fixed number of extensions? This is a main problem which we deal with in this talk

Abstract: Building on work of Jordan from 1870, in 1979 Harris showed that a geometric monodromy group associated to a problem in enumerative geometry is equal to the Galois group of an associated field extension. Vakil gave a geometric-combinatorial criterion that implies a Galois group contains the alternating group. With Brooks and Martin del Campo, we used Vakil's criterion to show that all Schubert problems involving lines have at least alternating Galois group. My talk will describe this background and sketch a current project to systematically determine Galois groups of all Schubert problems of moderate size on all small classical flag manifolds, investigating at least several million problems. This will use supercomputers employing several overlapping methods, including combinatorial criteria, symbolic computation, and numerical homotopy continuation, and require the development of new algorithms and software.

Abstract: In 2009 T. Ekedahl introduced some cohomological invariants for a finite group G. These relate, naturally, to invariant theory for groups and, also, to the Noether's Problem (one wonders about the rationality of the extension F(G) = F(x_g: g in G)^G over F, for a field F and a finite group G). In this talk, we introduce these invariants and we highlight some results.

Abstract: Chip-firig on graphs is a simple game with surprisingly deep connections to various parts of mathematics, and the lattice ideal of the Laplacian matrix of a graph encodes all of the combinatorics. We introduce a new family of subideals of the Laplacian lattice ideal with distinguished Gr"obner bases coming from generalized chip-firing dynamics on a graph as induced by abstract simplicial complexes on the vertex set. We will describe a conjecture generalizing a result of Postnikov and Shapiro relating the initial ideals of these binomial ideals to their power ideal deformations.

Abstract: We show that if a matroid is the truncation (=skeleton) of another matroid, then the generic initial ideal of its Stanley-Reisner ideal is level. In characteristic zero, the generic initial ideal is strongly stable and therefore admits an Artinian reduction by variables. Consequently the h-vectors of truncations of matroids satisfy Stanley's conjecture: They are Hilbert functions of Artinian monomial level algebras. This is joint work with Alexandru Constantinescu and Matteo Varbaro.

Abstract: For an acyclic quantum cluster algebra, we identify its generic quantum cluster characters with Nakajima's qt-characters defined via graded quiver varieties. As a consequence, we obtain the monoidal categorification conjecture in level 1 case. This talk is based on a joint-work with Yoshiyuki Kimura.

Abstract: Given a monomial ideal, Bayer, Peeva and Sturmfels expanded work
of Taylor by providing a criterion for when a simplicial complex
labeled by the generators of the ideal would support a free
resolution of the ideal. In this talk we focus on the situation
when the supporting complex is a simplicial tree, and discuss
some related structures.

Abstract: In this talk, I will explain how simple-minded counting can lead to some interesting results about the cluster algebra. I use Hall algebra as a key organizing tool to make our counting an enjoyable experience.

Abstract: I will present joint work with several coauthors on the description of (higher) duals of (equivariantly embedded) toric varieties and the relation with the associated polytopes/lattice configurations. The interplay between algebraic geometry and combinatorics allows us to prove algebro-geometric statements with combinatorial tools and combinatorial statements with algebro-geometric tools.

Abstract: The edge subring of a hypergraph H is the monomial subalgebra parameterized by the hyperedges of H. Its defining ideal is a toric ideal which we can understand by studying the combinatorics of H. In this talk we will survey recent results on the toric ideals of hypergraphs with a particular focus on the combinatorics of minimal generators.

Abstract: In 1990 Fröberg showed that the edge ideal of a graph has a linear resolution if and only if the complement of the graph is chordal. In this talk we will discuss the generalization of Fröberg's theorem to higher dimensions. In particular we will discuss new classes of simplicial complexes which extend the notion of a chordal graph and which give rise to a necessary condition for an ideal to have a linear resolution over any field. We will also provide a necessary and sufficient combinatorial condition for a square-free monomial ideal to have a linear resolution over fields of characteristic two.

Abstract: Let A be the union U(C_i) of a finite number of smooth plane curves C_i, such that the singular points of A are quasihomogeneous. This means that locally (at a singular point), the equation of A is homogeneous. We prove that if C is a smooth curve such that the singularities of A U C are quasihomogeneous, then there is a short exact sequence relating the bundle of logarithmic derivations on A to the bundle of logarithmic derivations on A U C. This yields an inductive tool for studying the splitting of these bundles in terms of the geometry of the divisor A|_C on C. (joint work with H. Terao and M. Yoshinaga, Hokkaido U.)

Abstract: Bring your own lunch.

Abstract: CGL (Cauchon-Goodearl-Letzter) extensions form a large, axiomatically defined class of iterated Ore extensions. Various important families, such as quantum Schubert cell algebras and quantum Weyl algebras, arise as special cases. From the point of view of noncommutative algebra, this is the "best" current definition of quantum nilpotent algebras. We will describe a general contruction of quantum clusters on all of these algebras via noncommutative unique factorization domains. The structure of their prime elements sees combinatorics that was previously specific to Weyl groups. Another aspect of this work is that quantum clusters are intrinsically constructed as unique families of prime elements of chains of subalgebras, while previous constructions relied on direct arguments with quantum minors. This is a joint work with Ken Goodearl (UC Santa Barbara).

Abstract: The sine-Gordon Y-systems and the reduced sine-Gordon Y-systems were introduced by Roberto Tateo in 90\'s in the study of the integrable deformation of the minimal models in conformal field theory by the thermodynamic Bethe ansatz method.
They are associated to continued fractions, and the periodicity property of these Y-systems was conjectured by Tateo; however, it has been only partially proved so far.

We prove the periodicity in full generality using the surface method of cluster algebras.
It turns out that there is a wonderful interplay among continued fractions, triangulations, cluster algebras, and Y-systems, which I would like to explain in this talk.

This is a joint work with Salvatore Stella.

Abstract: I'll review some of the basics of Frobenius splitting and explain how this theory has been used to study a number of varieties arising in combinatorial commutative algebra (eg. those discussed in part 3 of the Miller-Sturmfels textbook). The majority of this talk will be expository and no knowledge of Frobenius splitting will be assumed.

Abstract: I will give an overview of monoidal categorifications of Cluster algebras via Quantum groups and Nakajima quiver varieties. I will explain how these two approaches are connected.

Abstract: I will discuss recent advances in zonotopal algebra, a subject that connects analysis problems on multivariate polynomial interpolation and box splines to problems of commutative algebra concerning special polynomial ideals to problems of combinatorics involving hyperplane arrangements and lattice points in zonotopes.

Abstract: We will describe a general rigidity theorem for quantum tori. It leads to a scheme that can be used to classify the (full) automorphism groups of algebras that admit one quantum cluster (i.e. can be squeezed between a quantum affine space algebra and the corresponding quantum torus). The technique has a broad range of applications and in particular settles a couple of conjectures of Andruskiewitsch-Dumas and Launois-Lenagan. The former describes the automorphism groups of the algebras U_q^+(g) and the latter those of the algebras of square quantum matrices.

Abstract: UC Berkeley
Commutative Algebra and Algebraic Geometry Seminar
November 27, 2012
939 Evans Hall

3:45PM: Algebraic ordinary differential equations: General rational solutions and classification

Franz Winkler, Research Institute for Symbolic Computation (RISC) , J. Kepler University Linz, Austria

Consider an algebraic ODE (AODE) of the form $F(x,y,y\\')=0$, where $F$ is a tri-variate polynomial, and $y\\' = \frac{dy}{dx}$. The polynomial $F$ defines an algebraic surface, which we assume to admit a rational parametrization. Based on such a parametrization we can generically determine the existence of a rational general solution, and, in the positive case, also compute one. This method depends crucially on curve and surface parametrization and the determination of rational invariant algebraic curves. Further research is directed towards a classification of AODEs w.r.t. groups of transformations (affine, birational) preserving rational solvability.

5:00PM: Golod Algebras
Adam Boocher

Abstract: In modern number theory, it is becoming increasingly common to make computer calculations using p-adic numbers. Just as for real numbers, this involves working with finite approximations and managing the resulting errors as they propagate through the computation. In general, the most flexible framework for this seems to be the natural p-adic analogue of floating-point arithmetic. David Robbins discovered an example of a computation in which errors in p-adic arithmetic do not appear to compound as is typical: the Dodgson (Lewis Carroll) condensation recurrence for computing determinants. Based on numerical evidence, he conjectured that the loss of p-adic accuracy during a p-adic floating point computation of the condensation recurrence is controlled by the maximum valuation of any denominator appearing in the computation (i.e., by the largest precision loss at a single step rather than the sum of these). Additional numerical evidence suggests that this conjecture should follow from a purely algebraic statement applicable to arbitrary cluster algebras. Using a power series deformation of the caterpillar lemma, we prove a weaker algebraic statement which implies a direct analogue of the Robbins conjecture for some simpler recurrences (Somos-4, Somos-5, Markoff). For a recurrence derived from a general cluster algebra, we obtain a weaker analogue of the Robbins conjecture in which the bound is multiplied by a small positive integer depending on the recurrence (e.g., 3 for condensation). Joint work with Joe Buhler (CCR La Jolla).

Abstract: In this talk, we will give an introduction to Green's hyperplane restriction theorem, that gives a bound on the Hilbert function of the restriction of a symmetric algebra to a generic linear form. Moreover, we will talk about the generalization of this theorem to modules.

Abstract: In 1995, Villarreal gave a combinatorial description of the defining ideals of Rees algebras of quadratic square-free monomial ideals. In this paper we will generalize his results for hypergraphs. Our approach is based on giving a definition of closed even walks in a simplicial complex. We apply this combinatorial method to square-free monomial ideals of higher dimension.

Abstract: A quiver is an oriented graph. Quiver mutation is an elementary operation on quivers which appeared in physics in Seiberg duality in the 1990s and in mathematics in Fomin-Zelevinsky's definition of cluster algebras in 2002. In this talk, I will show how, by comparing sequences of quiver mutations, one can construct identities between products of quantum dilogarithm series. These identities generalize Faddeev-Kashaev-Volkov's classical pentagon identity and the identities obtained by Reineke. Morally, the new identities follow from Kontsevich-Soibelman's theory of Donaldson-Thomas invariants. They can be proved rigorously using the theory linking cluster algebras to quiver representations.

Abstract: The UC Berkeley Combinatorics Seminar
Fall 2012, Monday 2:10pm - 3pm, Evans Hall 939, CCN 54496
939 Evans Hall
Organizers: Florian Block, Max Glick, and Lauren Williams

Title: Polytropes and Tropical Eigenspaces
Speaker: Ngoc Tran

Abstract: The map which takes a square matrix $A$ to its polytrope is piecewise linear. We show that cones of linearity of this map form a fan partition of $\{R}^{n \times n}$, whose face lattice is anti-isomorphic to the lattice of complete set of connected relations. This fan refines the non-fan partition of $\R^{n \times n}$ corresponding to cones of linearity of the eigenvector map. Our results answer open questions in a previous work with Sturmfels and lead to a new combinatorial classification of polytropes and tropical eigenspaces.

Abstract: It is known that all ordinary powers of a Stanley-Reisner ideal I_D is Cohen-Macaylay if D is a complete intersection complex. Recently it was found that all symbolic powers of a Stanley-Reisner ideal I_D are Cohen-Macaulay if and only if D is a matroid complex. I will show how one can use tools of integer programming to prove this result. The next question is when a fixed ordinary or symbolic power of I_D is Cohen-Macaulay. For that I will present a characterization of complexes which are locally matroid or complete intersection complexes. From this it follows that if an ordinary or symbolic power of I_D is Cohen-Macaulay for some power greater than 2, then it will imply that D is a matroid or a complete intersection complex, respectively.

Abstract: Commutative Algebra and Algebraic Geometry
Evans 939
Organizer: David Eisenbud
http://hosted.msri.org/alg

Date: Tuesday, November 20

3:45: Seth Sullivant, North Carolina State U: Algebraic Statistics

Algebraic statistics advocates polynomial algebra as a tool for addressing problems in statistics and its applications. This connection is based on the fact that most statistical models are defined either parametrically or implicitly via polynomial equations. The idea is summarized by the phrase "Statistical models are semialgebraic sets". I will try to illustrate this idea with two examples, the first coming from the analysis of contingency tables, and the second arising in computational biology. I will keep the algebraic and statistical prerequisites to a minimum and keep the talk accessible to a broad audience.

5:00: Bernd Sturmfels, UC Berkeley: Commutative Algebra of Statistical Ranking

A model for statistical ranking is a family of probability distributions whose states are orderings of a fixed finite set of items. We represent the orderings as maximal chains in a graded poset. The most widely used ranking models are parameterized by rational function in the model parameters, so they define algebraic varieties. We study these varieties from the perspective of combinatorial commutative algebra. One of our models, the Plackett-Luce model, is non-toric. Five others are toric: the Birkhoff model, the ascending model, the Csiszar model, the inversion model, and the Bradley-Terry model. For these models we examine the toric algebra, its lattice polytope, and its Markov basis. This is joint work with Volkmar Welker.

Abstract: Assume that a quantum cluster algebra admits a monoidal categorification by quantum affine algebras or quantum unipotent subgroups of simply-laced type. We show that, for any chosen cluster, the dual canonical basis is a triangular basis with respect to certain linearly independent set, and the basis elements are naturally parametrized by the extended g-vectors.

Abstract: The j-multiplicity was introduced by Achiles and Manaresi in 1993 as a generalization of the Hilbert-Samuel multiplicity for arbitrary ideals in a Noetherian ring. Many of the properties and algebraic applications of the Hilbert-Samuel multiplicity of zero dimensional ideals have been extended to more general classes of ideals using the j-multiplicity. In this talk, I will review some of these properties and applications. At the end of the talk, I will briefly discuss my current research in this area.

Abstract: Rings of polynomial invariants of finite group actions are among the most classical objects in commutative algebra. There are many beautiful theorems ensuring that the invariant ring has good properties when the order of the group is invertible. However, if the order of the group is not a unit (i.e., is divisible by the characteristic of the ground field), many of these properties become more subtle. In this talk, I aim to illustrate some of the differences in invariant theory in this setting, and to describe some of my work in progress in this area.

Abstract: The join of two algebraic varieties is obtained by taking the closure of the union of all lines spanned by pairs of points, one on each variety. The secant varieties of a variety are obtained by taking the iterated join of a variety with itself. The symbolic powers of ideals arise by looking at the equations that vanish to high order on varieties. Statistical models are families of probability distributions with special structures which are used to model relationships between collections of random variables. This talk will be an elementary introduction to these topics. I will explain the interrelations between these seemingly unrelated topics, in particular, how symbolic powers can shed light on equations for secant varieties, and how theoretical results on secant varieties shed light on properties of statistical models including mixture models and the factor analysis model. Particular emphasis will be placed on combinatorial aspects including connections to graph theory.

Abstract: The UC Berkeley Combinatorics Seminar
Mondays 2:10pm - 3:00pm
939 Evans Hall
Organizers: Florian Block, Max Glick, and Lauren Williams

Brane Tilings and Cluster Algebras: the dP3 lattice
Speaker: Gregg Musiker Abstract: We discuss the link between cluster algebras and brane tilings and how this can be used to obtain combinatorial formulas for cluster variables arising from certain periodic quivers. No prior knowledge of brane tilings will be assumed. I will focus mostly on the case of the del Pezzo 3 (dP3) lattice and, in particular, work with Sicong Zhang at the 2012 REU at University of Minnesota. Connections to other examples such as Gale-Robinson sequences and Aztec Diamonds, including work with In-Jee Jeong from the 2011 REU, will also be discussed, time permitting.

Abstract: Let P be a rational polytope. The Ehrhart function counts the number of lattice points in kP for all natural numbers k. The corresponding ordinary generating function is called the Ehrhart series, and is well known to be the power series expansion of a rational function at the origin.

From an abstract viewpoint counting of lattice points can be interpreted as integration of the constant 1 with respect to the counting measure defined by the lattice. We will discuss the generalization in which the constant 1 is replaced by a polynomial, and the generalized Ehrhart function is given by the assignment $k\mapsto \sum f(x)$ where the sum is extended over the lattice points in kP.

Our approach is based on Stanley decompositions and completely algorithmic. It has recently be implemented as a computer program. We will illustrate the computations by examples from combinatorial voting theory.

Abstract: A number of generalizations of matroids have been introduced, which contain more information than the purely linear-algebraic data that matroids do. Some examples are Dress and Wenzel's _valuated matroids_ and D'Adderio and Moci's _arithmetic matroids_. This talk is propaganda for a definition: I will introduce a notion of matroid over any commutative ring, of which the foregoing objects are special cases. In the Dedekind domain case, we can prove structure theorems and define the analogue of the Tutte polynomial. This is joint work with Luca Moci.

Abstract: In the theory of cluster algebras, a prominent role is played by a family of integer vectors called the c-vectors. In this talk, I will present a study of these vectors from the point of view of representations of quivers. In particular, we will see that positive c-vectors of skew-symmetric cluster algebras are always dimension vectors of indecomposable representations without self-extensions.

Abstract: The description of cohomology tables in Boij-Soederberg theory can be thought of as saying that an arbitrary vector bundle on P^n "looks like" a certain homogeneous bundle (that is, one made from a representation of GL_n applied to the tangent bundle). I will explain this connection, and a new application of the philosophy that leads to sharp statements about the vanishing of cohomology of tensor products of bundles on P^n. This is all joint work with Frank Schreyer.

Abstract: Using complete intersections to link ideals is a classical technique, which can be traced back to Noether, Macaulay, and Severi among others. Although it started as a computational tool and a proof technique, liaison developed into a theory of independent interest. For ideals of height 3 or higher, one can consider liaison via Gorenstein ideals as a generalization of the classical liaison via complete intersection ideals of height 2. In this talk, we consider liaison via Gorenstein ideals. An ideal is called glicci if it can be linked to a complete intersection in a finite number of steps. A central question within Gorenstein liaison asks whether any Cohen-Macaulay ideal is glicci. I will introduce liaison and discuss this question, with an eye on the examples coming from determinantal ideals.

Abstract: An informal meeting in the Atrium over lunch to discuss open topics.

Abstract: Commutative Algebra and Algebraic Geometry
Evans 939
Organizer: David Eisenbud
http://hosted.msri.org/alg

Date: Tuesday, November 13

3:45: Luke Oeding (Berkeley): Eigenvectors of tensors and Waring decomposition

Waring’s problem for polynomials is to write a given polynomial as a minimal sum of powers of linear forms. The minimal number of summands required in a Waring decomposition (the Waring rank) is related to secant varieties. I will explain recent work of Landsberg and Ottaviani that unified and generalized many constructions for equations of secant varieties via vector bundle techniques. With Ottaviani we have turned this construction into effective algorithms to actually find the Waring decomposition of a polynomial (provided the Waring rank is below a certain bound). Our algorithms generalize Sylvester’s algorithm for binary forms, using an essential new ingredient – eigenvectors of tensors. Of course a naive algorithm always exists, but is rarely effective. I will explain how computations using linear algebra make our algorithms effective. Given time, I will demonstrate our Macaulay2 implementations.

5:00: Grigory Mikhalkin: Tropical curves and their phases: from face to edge models

We look at the constructions of embedded and immersed real algebraic curves in the plane. We survey and compare patchworking and tropical construction. As an example we consider topological classification of rational quintics in RP2. The first half of the talk will contain a short introduction to tropical curves and their phases.

Abstract: We introduce the notion of asymptotic triangulations of the annulus. We study their behaviour under mutation and describe their exchange graphs. Joint work with Gregoire Dupont.

Abstract: Following a recent paper by Dung (arXiv: 1209.3469v1), a (sharp) bound for the regularity of the associated graded module of a one-dimensional module is given, along with characterizations for when equality is attained. I will outline proofs of these bounds, and the extremal cases.

Abstract: Chip-firing on graphs has been studied for nearly 25 years since its independent introductions in statistical physics and graph theory. Recently, it has received some attention for its connections to the divisor theory of tropical curves and the combinatorics of lattice ideals. I will give a quick survey of these relationships and briefly describe my current research in chip-firing via gluing with emphasis on the binomial ideal case.

Abstract: Hyperdeterminants were brought into a modern light by Gelʹfand, Kapranov, and Zelevinsky in the 1990\'s. Inspired by their work, I will answer the question of what happens when you apply a hyperdeterminant to a polynomial (interpreted as a symmetric tensor). The hyperdeterminant of a polynomial factors into several irreducible factors with multiplicities. I identify these factors along with their degrees and their multiplicities, which both have a nice combinatorial interpretation. The analogous decomposition for the μ-discriminant of polynomial is also found. The methods I use to solve this algebraic problem come from geometry of dual varieties, Segre-Veronese varieties, and Chow varieties; as well as representation theory of products of general linear groups.

Abstract: Come see two talks for the price of one! (Still free) 'Codimension of deep ideals'. The Laurent embeddings of a cluster algebra A are dual to algebraic tori inside the spectrum of A. The complement of these tori is the 'deep part', where much of the algebraic complexity of A is hiding. I will talk about the codimension of this deep ideal, and an application to computing upper cluster algebras. 'Superunital domains' We will consider the subset of the positive part of a cluster algebra, on which every cluster variable is at least 1. We will talk about why this isn't a completely random thing to do, some results in finite type, and some compelling, mysterious computations."

Abstract: We describe a construction of free resolutions from higher
tensors. It provides a unifying view on a wide variety of complexes
including the Eagon-Northcott, Buchsbaum-Rim and similar complexes,
and the Eisenbud--Schreyer pure resolutions.

Abstract: Fix a polynomial ring R = K[x_1...x_n] and a homogeneous ideal I in R. Let d denote the maximal degree of generator of I. A result of Galligo, Giusti and Caviglia-Sbarra shows that reg(R/I) is at most doubly exponential in terms of d and n. Examples due to Mayer and Mayr show that any upper bound must be doubly exponential. However, all extremal examples of resolutions that have large regularity seem to have large regularity early in the resolution. So it makes sense that better upper bounds should be possible if one uses more data about the resolution than just d and n. In this talk, I\'ll show how one can prove two such bounds using the numerics of the Boij-Soederberg decomposition of the Betti table of R/I.

Abstract: We de fine higher pentagram maps on polygons in any dimension, which extend R. Schwartz's de nition of the 2D pentagram map. These maps turn out to be integrable for both closed and twisted polygons. The corresponding continuous limit of the pentagram map in dimension d is shown to be the (2,d+1)-equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D. In the 3D case we describe the corresponding spectral curve, first integrals, Liouville tori, and the motion along them. This is a joint work with Boris Khesin (University of Toronto).

Abstract: The Geometry of Generic Lagrangian Ffibres: An Illustrating Example
Abstract: The aim of the talk is to give an introduction to a strategy developed by M. Adler, P. van Moerbeke and P. Vanhaecke to study the geometry of the generic fibre of a Lagrangian fibration induced by an integrable system.

Abstract: The Arithmetical Rank of the Edge Ideals of Whisker Graphs
Abstract: A classical problem in Algebraic Geometry consists in finding the minimum number of hypersurfaces that define a certain variety. This problem can be approached from an Algebraic and Combinatorial point of view.
Given a commutative ring R with identity and an ideal I of R, the arithmetical rank of I, denoted ara(I), is the minimum number of elements of R such that the ideal generated by those elements has the same radical as I. The ideal I is called set-theoretic complete intersection (STCI) if ara(I)=ht(I).
In general if I is STCI, then I is Cohen-Macaulay, but the converse is not true. I will show that the converse holds for the edge ideals of some whisker graphs.

Abstract: In the talk I will try to explain in non-technical terms a deep and beautiful relation between the theorem of Briancon-Skoda and the monodromy of an isolated hypersurface singularity {f=0} that was discovered by John Scherk about 30 years ago. A proof of this relation involves the theory of mixed Hodge structures on the vanishing cohomology, but can be understood on an intuitive level.

Abstract: Commutative Algebra and Algebraic Geometry
Tuesdays, 3:45-6pm in Evans 939
Organizer: David Eisenbud
http://hosted.msri.org/alg

Date: Tuesday, October 30

3:45 Srikanth Iyengar: Torsion in tensor powers of modules

The starting point of this talk is the observation that if a finitely generated module M over a noetherian (commutative!) domain R is NOT free, then its d-fold tensor-product has torsion for each d > rank_RM. In fact, when R is a regular local ring, torsion shows up already when d >= dim R; this is one of the central results in Auslander\\'s 1961 paper "Modules over unramified regular local rings". Recently Celikbas, Piepmeyer, R. Wiegand and I extended this result to certain modules over isolated hypersurface singularities; the preprint will be posted on the arXiv soon. My goal is to discuss some ideas that go into its proof. The main new tool is Hochster\\'s theta-function, and is inspired by recent work of Dao.

5:00 Vu Thanh: Linear resolutions of powers of edge ideals of anti-cycles.

Let $I$ be the edge ideal of an anticycle of length at least $5$. We will show that all powers $I^k$, for $k \ge 2$ have linear resolution.

Abstract: Before studying cones of Betti tables over other other rings and with multigradings, it is good to get a picture of what the cone of Hilbert functions looks like. In some cases it is possible to give a complete picture of this cone, but in general the cone is much more complicated than in the standard graded case, suggesting that the corresponding cone of Betti table could be hard to describe.

Abstract: The Lefschetz properties are desirable properties of graded artinian algebras, which have strong consequences on the structure of their Hilbert functions. Their investigation is similar in flavor to what one does in algebraic geometry when studying the generic hyperplane section of a projective variety. The talk will provide a gentle introduction to this area of research and will illustrate the beautiful connections which occur when the above mentioned properties are translated into the language of combinatorics, algebraic geometry and representation theory. Some of these connections are quite surprising and still not completely understood.

Abstract: The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its "shortest" diagonals, and output the smaller polygon which they cut out. I will explain how the machinery of cluster algebras can be used to obtain explicit formulas for the iterates of the pentagram map. The formulas are written in terms of certain cross ratios, and involve generating functions associated with a family of posets which arose in the work of N. Elkies, G. Kuperberg, M. Larsen, and J. Propp on alternating sign matrices.

Abstract: The ring of invariants for the action of the automorphism group of the projective line on the n-fold product of the projective line is a classical object of study. The generators of this ring were determined by Kempe in the 19th century. However, the ideal of relations has been only understood recently in work of Howard, Millson, Snowden and Vakil. They prove that for n>6, the ideal of relations is generated by quadratic equations using a degeneration to a toric variety. I will report on joint work with Benjamin Howard where we compute the Hilbert functions of these rings of invariants, and further study the toric varieties arising in this degeneration. As an application we show that the second Veronese subring of the ring of invariants admits a presentation whose ideal admits a quadratic Gröbner basis.

Abstract: Over a finite field k, choose a random homogeneous polynomial f of degree d in k[x,y,z]. What is the probability that f defines a smooth planar curve? Questions like this are related to Bertini Theorems over a finite field. The first such Bertini Theorem over a finite field was proved by Bjorn Poonen, who showed that the (asymptotic) probability of smoothness may be realized as a product a of local probabilities. Recent work of myself with Melanie Matchett Wood generalizes this result. I will discuss this collection of ideas, focusing primarily on examples.

Abstract: Commutative Algebra and Algebraic Geometry
Tuesdays, 3:45-6pm in Evans 939
Organizer: David Eisenbud
http://hosted.msri.org/alg

Date: October 23

3:45 Sam Payne: Tropicalization of the moduli space of curves

Tropical geometry allows a systematic study of algebraic curves over valued fields in terms of the marked dual graphs of special fibers of models of the curve over the valuation ring. In the past several years, a number of researchers, including Caporaso, Gathmann, Kozlov, Mikhalkin, and their collaborators, have introduced and studied moduli spaces for these marked graphs, which are often called tropical curves, and estabilshed various analogies to moduli spaces of curves. I will present work that explains and extends these analogies, canonically and functorially, by applying a new generalized tropicalization map to the Deligne-Mumford compactification of the moduli space of stable curves. Berkovich spaces appear in the construction of this new tropicalization map in a natural and elementary way, but no tropical or nonarchimedean analytic background is assumed. This is joint work with D. Abramovich and L. Caporaso.}

5:00 Bernd Sturmfels: A Hilbert Scheme in Computer Vision

Multiview geometry is the study of two-dimensional images of three-dimensional scenes, a foundational subject in computer vision. We determine a universal Groebner basis for the multiview ideal of n generic cameras. As the cameras move, the multiview varieties vary in a family of dimension 11n-15. This family is the distinguished component of a multigraded Hilbert scheme with a unique Borel-fixed point. We present a combinatorial study of ideals lying on that Hilbert scheme. This is joint work with Chris Aholt and Rekha Thomas.

Abstract: I will begin with `cluster localizations', localizations of cluster algebras which are still cluster algebras. Geometrically, a cluster localization of A is dual to an open subscheme of Spec(A) with a cluster structure. Many properties of a cluster algebra can be studied locally with respect to these open patches. This local approach is very effective for `locally acyclic cluster algebras'. In this talk, I will define this class, review some of their important properties, give several examples, and present an algorithm for demonstrating that a given cluster algebra is locally acyclic.

Abstract: Sergey Fomin and Andrei Zelevinsky defined cluster algebras by recursively constructing their generators via a process called mutation. This process is closely related to various sequences of integers, arising for instance from Coxeter-Conway friezes, whose terms can be seen as specializations of the generators of specific cluster algebras. Although the fact that these sequences contain only integers is sometimes surprising from their definition, the theory of cluster algebras provides a common explanation for it: the Laurent Phenomenon. In this talk, we will first list some examples of recurrence relations of integers, then we will try to understand them from the point of view of cluster algebras.

Abstract: The UC Berkeley Combinatorics Seminar
Mondays 2:10pm - 3:00pm
939 Evans Hall
Organizers: Florian Block, Max Glick, and Lauren Williams

Combinatorics of the abelian-nonabelian correspondence
Speaker: Kaisa Taipale The Grassmannian Gr(k,n) and the product of projective spaces (P^(n-1))^k are both GIT quotients of the projective space P^(nk-1). One can relate their cohomologies and Gromov-Witten theories by the abelian-nonabelian correspondence. I will explain this geometric relationship and two combinatorial manifestations of the correspondence (through Schubert calculus and through torus localization), finishing with some unanswered questions.

Abstract: I will review the Eisenbud-Floystad-Weyman construction of equivariant pure free resolutions and explain a conjectural generalization to quadric hypersurface rings. I will discuss some current work with Andrew Snowden that tries to realize this generalization.

Abstract: Let I = (f_1,...,f_r) be a homogeneous ideal in the polynomial ring S = K[x_1,...,x_n]. The arithmetical rank of I, ara(I), is the smallest number s such that I has the same radical of the ideal J = (g_1,...,g_s) for some homogeneous polynomials g_1,...,g_s in S. One way to get lower bounds for this number is to look at the non-vanishing of local cohomology with support in I: ara(I) >= t provided that H_I^t(S) is not zero. A similar lower bound comes considering étale topology instead of Zariski's one, and basically these have been, so far, the only two successful methods to compute the arithmetical rank of certain families of ideals. Moreover, in many instances, étale topology was successful where Zariski's one failed. This fact led Lyubeznik to conjecture, in 2002, that the lower bound obtained using étale cohomology is never worse than the one obtained using Zariski's. In the talk I am going to prove that the conjecture is true if the variety defined by I is smooth and K has characteristic 0.

Abstract: In 2004, Reiner, Stanton and White noticed that to an action of a cyclic group on a finite set one can often attach a polynomial with nonnegative integer coefficients. This polynomial has the property that when it is evaluated at specific roots of unity, the result is the size of the fixed point set for an element of the group. I will introduce the notion of 'promotion' on Young tableaux and via representation theory and geometry show that the polynomial attached to this action is the Kostka-Foulkes polynomial. This is joint work with Joel Kamnitzer

Abstract: This will be a more technical version of my talk from Monday. I will begin by giving detailed descriptions of the duality pairing and some of the repercussions for studying cones of Betti tables. The rest of the talk will be more casual in nature (like a working seminar) and will depend on the questions and interests of the audience.

Abstract: We will present several methods to obtain uniform (sharp) upper bounds for the number of generators of ideals. These bounds will depend on fixing some invariants of the ideal, for instance the Hilbert function or the Hilbert polynomial, both in the graded and in the local setting. In particular we will show how classical results such as Macaulay's Theorem and some of its recent generalizations follow from a unified approach.

Abstract: Commutative Algebra and Algebraic Geometry
Tuesdays, 3:45-6pm in Evans 939
Organizer: David Eisenbud
http://hosted.msri.org/alg

Date: Oct 16

3:45: Anurag Singh: The F-pure threshold of a Calabi-Yau hypersurface

The F-pure threshold is a numerical invariant of prime characteristic singularities. It constitutes an analogue of a numerical invariant for complex singularities---the log canonical threshold---that measures local integrability. We will discuss, in detail, the calculation of F-pure thresholds of elliptic curves, and also indicate how this calculation extends to Calabi-Yau hypersurfaces. This is work in progress with Bhargav Bhatt.

5:00 Jose Rodriguez: Maximum Likelihood for Matrices with Rank Constraints

Maximum likelihood estimation is a fundamental computational task in
statistics and it also involves some beautiful mathematics. We discuss this task for determinantal varieties (matrices with rank constraints) and show how numerical algebraic geometry can be used to maximize the likelihood function. Our computational results with the software Bertini led to surprising duality conjectures. This is joint work with Bernd Sturmfels and Jon Hauenstein.

Abstract: We present work in progress on combinatorial formulas for cluster variables arising from certain periodic cluster algebras. These formulas are phrased in terms of brane tilings which arise in the string theory literature. No prior knowledge of brane tilings or string theory will be assumed, and some connections to cluster algebras and quivers will be explained. Part of this talk is based on work with In-Jee Jeong and Sicong Zhang from the 2011 and 2012 REU's at University of Minnesota.

Abstract: The cluster algebras were introduced by Fomin-Zelevinsky in a paper which appeared 10 years ago. There are connections to many different areas of mathematics, including quiver representations. One direction of research has been to try to model the ingredients in the definition of cluster algebras in “nice” categories, like module categories or related categories. In this lecture we illustrate the idea and use of “categorification” by concentrating on only one ingredient in the definition of cluster algebras. This is the operation of quiver mutation, which we define. For finite quivers (i.e. directed graphs) without oriented cycles this leads to the so-called cluster categories, which are modifications of certain module categories. For other types of quivers some stable categories of maximal Cohen-Macaulay modules over commutative Gorenstein rings are the relevant categories. We start the lecture with background material on quiver representations.

Abstract: The UC Berkeley Combinatorics Seminar
Mondays 2:10pm - 3:00pm
939 Evans Hall
Organizers: Florian Block, Max Glick, and Lauren Williams

Inverse problem in cylindrical electrical networks
Speaker: Pavlo Pylyavskyy

The inverse Dirichlet-to-Neumann problem in electrical networks asks one to recover the combinatorial structure of a network and its edge conductances from its response matrix. For planar networks embedded in a disk, the problem was studied and effectively solved by Curtis-Ingerman-Morrow, de Verdière-Gitler-Vertigan and Kenyon-Wilson. I will describe how the problem can be solved for a large class of networks embedded in a cylinder. Our approach uses an analog of the R-matrix for certain affine geometric crystals. It also makes use of Kenyon-Wilson\\'s groves. This is joint work with Thomas Lam.

Abstract: A central idea in Boij-Soederberg Theory is that there is a connection between free resolutions over the polynomial ring and sheaf cohomology on projective space. This idea emerged from Eisenbud and Schreyer's proof of the Boij-Soederberg conjectures. In joint work with David Eisenbud, we give a new perspective on this duality. I will outline the resulting duality between syzygies and sheaf cohomology, and I will explain how our duality pairing enables us to extend the scope of Boij--Soederberg theory in several directions.

Abstract: Linkage is a theory at the borders between Algebraic Geometry and Commutative Algebra aiming at classifying ideals (varieties) and their properties, based on an operation of "linkage". Since ideals in the same linkage class share several homological properties, it is important to identify the minimal elements in a linkage class (that is, the `best' and `simplest' elements in it). For example, ideals in the linkage class of a complete intersection (=licci ideals) can be very complicated, however, because of linkage they share several interesting (and subtle) properties of complete intersection ideals. Extensive work by several authors (Ballico, Bolondi, Hartshorne, Lazarsfeld, Martin-Deschamps, Migliore, Nagel, Perrin, Rao, Strano, etc.) provides a well-understood notion of minimality in the case of (homogeneous) ideals of codimension 2, However, in the general case (=non-licci ideals of codimension > 2) it is not even clear what could be a good definition of minimality and, consequently, if minimal elements exist in any linkage class. Polini and Ulrich found special ideals that are "minimal" in several regards. However, the question of finding a general notion of minimality was wide-open. In the present talk, we suggest a general notion of minimality for Cohen-Macaulay ideals, which includes the classes of ideals found by Polini-Ulrich . We prove, under reasonable assumptions, that minimal elements exist and are essentially unique. We then provide several concrete classes of ideals which are minimal. We will show applications to the question of when two ideals lie in the same even linkage class, and the Buchsbaum-Eisenbud-Horrocks' Conjecture (which suggests lower bounds for the Betti numbers of any Cohen-Macaulay ideal).

Abstract: The categorification of cluster algebras provides a powerful tool for understanding the subtle relationships between the elements of the cluster algebra. In the case of quantum cluster algebras, the categorification is only known for acyclic types. In this talk we will restrict to the special case of rank two quantum cluster algebras (with two variables in each cluster and no coefficients). I will develop the theory of quantum cluster algebras "from scratch", in particular giving a short elementary proof of the Laurent phenomenon. I will introduce the notion of valued quivers and the basic features of their representation theory, namely reflection functors. I will show how one can construct non-initial cluster variables from the representations of a valued quiver, and if there is time, I will indicate how to prove this result using only the reflection functors.

Abstract: Series Title: On the arithmetic of hyperelliptic curves

Thursday October 11th
Lecture 3: Arithmetic invariant theory
60 Evans Hall

Abstract:
I will describe a representation of the orthogonal group whose stable orbits are related to the 2-descent on hyperelliptic curves of a fixed genus with a rational Weierstrass point, and construct the principal homogeneous spaces for the Jacobian using pencils of quadrics.

Abstract: Passing from an ideal I to an initial ideal can be thought of as a flat degeneration, and as such it preserves the Hilbert function. In this talk we'll take a peek at what happens when other invariants are also preserved. Using determinantal ideals as our guide, I'll present lots of examples, and close with some new results on toric ideals.

Abstract: Series Title: On the arithmetic of hyperelliptic curves Wednesday October 10th Lecture 2: Hyperelliptic curves with a rational Weierstrass point 105 North Gate Hall Abstract: I will review the equations that define these curves, and describe the subgroup of 2-torsion in their Jacobians. After a review of the 2-descent, I will state the results which Bhargava and I have obtained on the average value of the order of the 2-Selmer group, and give some corollaries on the rational points of the curve and its Jacobian.

Abstract: The Hilbert function and the graded Betti numbers are invariants we use to study graded rings and modules. When we look at graded modules generated in degree zero over the standard graded polynomial ring, we know by Macaulay's classification exactly what the possible Hilbert functions are. However, for the graded Betti numbers, we are far from having such a complete description. What we do have, is a clear picture of what the set of Betti tables looks like up to scaling, i.e., the cone spanned by all Betti tables in a suitable vector space over the rational numbers. I will explain this picture and give an overview of the sequence of steps that led to it. When changing the grading, we don't even know much about the possible Hilbert functions and it therefore interesting to study also Hilbert functions up to scaling.

Abstract: Series Title: On the arithmetic of hyperelliptic curves Tuesday October 9th Lecture 1: The rank of elliptic curves 50 Birge Hall Abstract: I will review why elliptic curves are somewhat special, define the rank of a curve over the rational numbers, and motivate the conjecture of Birch and Swinnerton-Dyer on the L-function. After stating what is known, I will briefly introduce a new method, due to Manjul Bhargava, for studying the average rank.

Abstract: October 9 3:45: Bernd Sturmfels A Hilbert Scheme in Computer Vision Multiview geometry is the study of two-dimensional images of three-dimensional scenes, a foundational subject in computer vision. We determine a universal Groebner basis for the multiview ideal of n generic cameras. As the cameras move, the multiview varieties vary in a family of dimension 11n-15. This family is the distinguished component of a multigraded Hilbert scheme with a unique Borel-fixed point. We present a combinatorial study of ideals lying on that Hilbert scheme. This is joint work with Chris Aholt and Rekha Thomas. 5pm: Giulio Caviglia Koszul property of projections of the Veronese cubic surface Let V be the Veronese cubic surface in P^9. We classify the projections of V to P^8 whose coordinate rings are Koszul. In particular we obtain a purely theoretical proof of the Koszulness of the pinched Veronese, a result obtained originally by using filtrations, deformations and computer assisted computations. To this purpose we extend, to certain complete intersections, results of Conca, Herzog, Trung and Valla concerning homological properties of diagonal algebras. This is a joint work with Aldo Conca.

Abstract: The rings of polynomial SL(V)-invariants of configurations of vectors and linear forms in a k-dimensional complex vector space V have been explicitly described by Hermann Weyl in the 1930s. Each such ring conjecturally carries a natural cluster algebra structure (typically, many of them) whose cluster variables include Weyl's generators. In joint work with Pavlo Pylyavskyy, we describe and explore these cluster structures in the case k=3. We employ the machinery of tensor diagrams, and make a connection to the web bases introduced by G. Kuperberg.

Abstract: The UC Berkeley Combinatorics Seminar Mondays 2:10pm - 3:00pm 939 Evans Hall Organizers: Florian Block, Max Glick, and Lauren Williams On singularity confinement for the pentagram map Speaker: Max Glick The pentagram map, introduced by R. Schwartz, is a birational map on the configuration space of polygons in the projective plane. We study the singularities of the iterates of the pentagram map. We show that a typical singularity disappears after a finite number of iterations, a confinement phenomenon first discovered by Schwartz. We provide a method to bypass such a singular patch by directly constructing the first subsequent iterate that is well defined on the singular locus under consideration. The key ingredient of this construction is the notion of a decorated (twisted) polygon, and the extension of the pentagram map to the corresponding decorated configuration space.

Abstract: For a standard graded algebra R, we consider embeddings of the the poset of Hilbert functions of quotients of R into the poset of ideals of R, as a way of studying Hilbert functions. We will look at what happens when we take ring extensions. This is joint work with G. Caviglia.

Abstract: This talk will be an introductory discussion of quantum cohomology and Gromov-Witten invariants with a focus on homogeneous varieties. These ideas have enumerative interpretations and rich combinatorial structure. No new results will be discussed; the aim instead is an exposition of a beautiful area of mathematics that informs mirror symmetry and has yet-to-be-discovered connections with cluster algebras.

Abstract: We will discuss seemingly unrelated contexts in which integral closures of ideals and modules arise naturally. We will emphasize on the connection with differentials and multiplicities, and address computational issues if time permits.

Abstract: Cluster algebras are a class of commutative rings discovered by Sergey Fomin and the speaker about a decade ago. A cluster algebra of rank n has a distinguished set of generators (cluster variables) grouped into (possibly overlapping) n-subsets called clusters. These generators and relations among them are constructed recursively and can be viewed as discrete dynamical systems on a n-regular tree. The interest to cluster algebras is caused by their surprising appearance in a variety of settings, including quiver representations, Poisson geometry, Teichmuller theory, non-commutative geometry, integrable systems, quantum field theory, etc. We will discuss the foundations of the theory of cluster algebras, with the focus on their algebraic and combinatorial structural properties.

Abstract: The UC Berkeley Combinatorics Seminar
Mondays 2:10pm - 3:00pm
939 Evans Hall
Organizers: Florian Block, Max Glick, and Lauren Williams

Tropicalization method in cluster algebras
Speaker: Tomoki Nakanishi

In cluster algebras, after making several mutations of sends, you may sometimes end up with the initial seed. That is the periodicity phenomenon in cluster algebras. Periodicity is a rare event, but once you have it, you can also get the associated dilogarithm identity, plus its quantum version, for free!

There are two basic questions for periodicity: How to find it and how to prove it? The answer to the second question is given by the tropicalization method, which I explain in this talk by several examples.

The first question is more difficult, and I do not know the answer. However, we are lucky to have several (infinitely many) conjectured periodicities from the Bethe ansatz method in 90\'s, even before the birth of cluster algebras, and they are recently proved by the tropicalization method. There is always some root system behind the scene.

The talk is based on the work with R. Inoue, O. Iyama, B. Keller, and A. Kuniba.

Abstract: Linkage (or liaison) is an elegant theory standing on the border between Commutative Algebra and Algebraic Geometry. Its roots go back to the nineteenth century, although the first modern treatment appeared in 1974, in a celebrated paper written by Peskine and Szpiro. Since then, linkage has been an active area of research and has proved to be a powerful tool, successfully employed in a number of circumstances, ranging from structure theorems, to results on Hilbert scheme (smoothness, irreducible components, etc.). Moreover, in the last 20 years, the original theory developed two interesting, very different, generalizations, namely, linkage by Gorenstein ideals (or G-liaison) and residual intersection theory. Starting from the definition of linkage, we will introduce the theory, provide examples, discuss several results, and explain some of the main open questions in the theories of liaison and G-liaison.

Abstract: I will talk about ongoing work with Hacking, Keel and Kontsevich on an approach to constructing canonical bases of cluster algebras using ideas which originate in mirror symmetry. We use the technology of "scattering diagrams" and "broken lines". Scattering diagrams arose in work of Kontsevich and Soibelman to construct mirror pairs of K3 surfaces, but are now being applied much widely. In particular, seed datum for a cluster algebra generates a particularly nice scattering diagram, which incorporates but has more information than the cluster complex. Broken lines are then tropical objects in the scattering diagram whose enumeration can be used to produce canonical bases.

Abstract: Commutative algebra often abstracts geometric problems into simple questions about algebraic invariants. I will illustrate this with some open problems on the Hilbert function (a simple algebraic invariant which measures the dimensions of graded pieces of a graded ring). Geometry enters the picture when the ring is the projective coordinate ring of a variety. When the ring has a multigrading we also get some interesting combinatorics. I will emphasize the computational and combinatorial sides of this story.

Abstract: A central problem in geometric modeling is to find the implicit equations for a curve or surface defined parametrically. From the standpoint of commutative algebra, there is a strong connection between the implicit equation and the syzygies of the base locus of the parametrization. This relation is made explicit by the theory of approximation complexes, as developed by Herzog-Simis-Vasconcelos, Buse-Jouanolou, Cox, Chardin. I will give an overview of the use of syzygies in implicitization and illustrate with the case of tensor product surfaces. For tensor product surfaces of small degree, we can determine explicitly all bigraded minimal free resolutions of the base ideal. We study the singularities of these surfaces in relation to the syzygies and obtain a simplified implicit equation. The work on tensor product surfaces is joint with H. Schenck and J. Validashti.

Abstract: Given a homogeneous ideal I in a polynomial ring R over a field, one can compute a minimal graded free resolution of I (or R/I). The length of this resolution, called the projective dimension, is finite and at most the number of variables of R by Hilbert's Syzygy Theorem. When I has few generators in low degree, one expects that projdim(R/I) cannot be arbitrarily large, even when R has many variables. Stillman asked precisely for a bound on projdim(R/I) in terms of the degrees of the minimal generators of I. In this talk I will survey the history of this problem and recent developments toward a solution, including the exponential bound of Ananyan-Hochster in the case of ideals generated by quadrics, specific bounds for three-generated ideals in low degree by Eisenbud-Huneke and Engheta, and families of ideals with large projective dimension.

Abstract: Tropical Seminar schedule Location: Baker board room 10:00 Diane Maclagan (Warwick) "Commutative algebra aspects of tropical geometry" 10:45 Qingchun Ren (UC Berkeley) "The intrinsic torus of a very affine variety" 11:15 Florian Block (UC Berkeley) "The Euler characteristic of a very affine variety" 12:00 Rei Inoue (Chiba) "Tropical Geometry and Integrable systems" 12:45 - Tropical discussion over lunch (bring or order your own)

Abstract: The original motivation for the study of cluster algebras was to design an algebraic framework for understanding total positivity and canonical bases. A lot of recent activity in the field has been directed towards various constructions of ``natural" bases in cluster algebras. One of the approaches to this problem was developed several years ago in a joint work with P.Sherman where it was shown that the indecomposable positive elements form a basis over integers in any rank 2 cluster algebra of finite or affine type. It is strongly suspected (but not proved) that this property does not extend beyond affine types. In a joint work with K.Lee and L.Li we go around this difficulty by constructing a new basis in any rank 2 cluster algebra, which we call the greedy basis. It consists of a special family of indecomposable positive elements that we call greedy elements. Inspired by a recent work of K.Lee - R.Schiffler and D.Rupel, we give an explicit combinatorial expression for greedy elements using the language of Dyck paths.

Abstract: A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and Cohen-Macaulay, and its components form beautiful combinatorial patterns. For configurations in one apartment, these patterns are mixed subdivisions of simplices, and there is a tight connection to tropical convexity. In general, the components of the special fiber are rational varieties, and any blow-up of projective space along a linear subspace arrangement can arise.
We present a study of the two-dimensional building for PGL(3), joint with Dustin Cartwright, Mathias Haebich and Annette Werner, that classifies Mustafin triangles into 38 combinatorial types.

Abstract: Commutative Algebra and Algebraic Geometry
Tuesdays, 3:45-6pm in Evans 939
Organizer: David Eisenbud

Sept 18
3:45: Aldo Conca: Syzygies of Koszul algebras

Koszul algebras are certain algebras defined by quadratic relations; most quadratic algebras that arise naturally are in fact Koszul. There is no finite criterion for Koszulness, however, so it is interesting to study necessary or sufficiently conditions. I will explain what Koszul algebras are, and describe special features of their syzygies.

5:00: Elizabeth Gross: Toric Ideals of Hypergraphs

The edge subring of a graph G is the monomial subalgbera parameterized by the edges of G. It’s defining ideal, commonly referred to as the toric ideal of G, has been well-studied and several historical results tell us the same beautiful story: we can understand these ideals by understanding the combinatorics of the underlying graph. A natural extension is to consider the toric ideal of a hypergraph. In this talk, we will survey well-known results on the toric ideals of graphs and explain how these concepts generalize to hypergraphs. We will end with recent results on the toric ideals of hypergraphs.

Abstract: In a joint work with Thomas Lam we use a deformation of Serre relations to define a family of Lie groups acting on planar electrical networks. In this talk I will explain how coordinate rings of those groups can be naturally endowed with Laurent phenomenon algebra structures. A combinatorial model for certain seeds of those algebras in terms of wiring diagrams has been studied by Henriques and Speyer under the name multidimensional cube recurrence. Our machinery of Laurent phenomenon algebras allows to extend this dynamics beyond the part modeled by wiring diagrams.

Abstract: In a joint work with Thomas Lam we generalize Fomin and Zelevinsky's cluster algebras by allowing exchange polynomials to be arbitrary irreducible polynomials, rather than binomials. In the first talk I will explain the definition, and then concentrate on Laurent phenomenon algebras with linear seeds. The polytopes dual to cluster complexes of those algebras are essentially the nestohedra previously studied by Feichtner - Sturmfels, Postnikov and Zelevinsky.

Abstract: The notion of Hilbert function plays a central role in commutative algebra, in algebraic geometry and in computational algebra. The Hilbert function of a local (or graded) ring is a polynomial function and the coefficients of the corresponding polynomial, called Hilbert coefficients, may capture several numerical and homological invariants of the ring itself. Starting from classical results of S. Abhyankar, D. Northcott and P. Samuel, many papers have been written in this field which is considered an important part of the theory of blowing-up rings. In this talk we present some techniques and we will focus on some open problems which are motivating the recent research on the topic.