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).

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).

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.

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.

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.

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

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).

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).

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.

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.

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

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).

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.

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).

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.

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.

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

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).

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.

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).

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.

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

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).

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).

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.

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

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).

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).

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.

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

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).

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).

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.

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

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).

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).

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.

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).

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).

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.

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

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).

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).

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.

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

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).

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).

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.

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).