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Postdoc Seminars

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Past Postdoc

  1. MT Postdoc Seminar: Jet Spaces and Diophantine Geometry

    Location: MSRI: Simons Auditorium
    Speakers: Taylor Dupuy (University of New Mexico)

    We will explain how to obtain effective Mordell-Lang and Manin-Mumford using jet space techniques in the characteristic zero function field setting.

    Updated on May 16, 2014 12:58 PM PDT
  2. AT Postdoc Seminar

    Location: MSRI: Simons Auditorium
    Speakers: Joseph Hirsh (Massachusetts Institute of Technology)
    Created on Feb 07, 2014 09:37 AM PST
  3. AT Postdoc Seminar: Homotopy and arithmetic: a duality playground

    Location: MSRI: Simons Auditorium
    Speakers: Vesna Stojanoska (Massachusetts Institute of Technology)

    Homotopy theory can be thought of as the study of geometric objects and continuous deformations between them, and then iterating the idea as the deformations themselves form geometric objects. One result of this iteration is that it replaces morphism sets with topological spaces, thus remembering a lot more information. There are many examples to show that the approach of replacing sets with spaces in a meaningful way can lead to remarkable developments. In this talk, I will explain some of my recent work in the case of implementing homotopy theory in arithmetic in a way which produces new results and relationships between some classical notions of duality in both fields.

    Updated on Apr 25, 2014 11:19 AM PDT
  4. AT Postdoc Seminar: The Mirror Symmetry Conjecture and Cobordisms

    Location: MSRI: Baker Board Room
    Speakers: Hiro Tanaka (Harvard University)

    This talk--aimed for a general audience of neither topologists nor model theorists--will discuss applications of cobordisms to Kontsevich's mirror symmetry conjecture. We'll begin by stating a rough version of the
    conjecture, which builds a bridge between symplectic geometry on one hand, and on the other hand, algebraic geometry over the complex numbers. We then discuss how the theory of cobordisms, which studies when two manifolds can be the boundary of another manifold, sheds light on how to generalize the mirror symmetry conjecture, while giving us information about objects in symplectic geometry. (For example, two Lagrangians related by a compact cobordism are equivalent in the Fukaya category.)

    Updated on Apr 17, 2014 05:00 PM PDT
  5. AT Postdoc Seminar: Galois equivariance and stable motivic homotopy theory

    Location: MSRI: Simons Auditorium
    Speakers: Kyle Ormsby (MIT / Reed College)

    We will explore the relationships between Galois theory, groups acting on spaces, and motivic homotopy theory. Ultimately, for R a real closed field, we will discover that that there is a full and faithful embedding of the stable Gal(R[i]/R)-equivariant homotopy category into the stable motivic homotopy category over R.

    Updated on Mar 28, 2014 01:16 PM PDT
  6. AT Postdoc Seminar: Uses of commutative rings in homotopy theory

    Location: MSRI: Simons Auditorium
    Speakers: Sean Tilson (Universität Osnabrück)

    Homotopy theorists try to gain geometric information and insight through the use of algebraic invariants. Specifically, these invariants are useful in determining whether or not two spaces can be equivalent. We will begin with an example to demonstrate the usefulness of cohomology and some of the extra structure it possesses, such as cup products and power operations. This extra structure provides a very strong invariant of the space. As these invariants are representable functors, this extra structure is coming from the representing object. Indeed, cohomology theories possess products and power operations when they are represented by objects called commutative ring spectra. We then shift focus to studying commutative ring spectra on their own and try to detect what maps of commutative ring spectra might look like.

    Updated on Mar 28, 2014 10:11 AM PDT
  7. MT Postdoc Seminar: Strong minimality of the $j$-function

    Location: MSRI: Simons Auditorium
    Speakers: James Freitag (University of California, Berkeley)

    In this talk, we will be working in with the theory of differentially closed fields of characteristic zero; essentially this theory says that every differential equation which might have a solution in some field extension already has a solution in the differentially closed field. After introducing this theory in a bit of detail, we will sketch a proof of the strong minimality of the differential equation satisfied by the classical $j$-function starting from Pila's modular Ax-Lindemann-Weierstrass theorem. This resolves an open question about the existence of a geometrically trivial strongly minimal set which is not $\aleph _0$-categorical. If time allows, we will discuss some finiteness applications for intersections of certain sets in modular curves. This is joint work with Tom Scanlon.

    Updated on Mar 21, 2014 11:13 AM PDT
  8. MT Postdoc Seminar: Connections between Ramsey Theory and Model Theory.

    Location: MSRI: Simons Auditorium
    Speakers: Cameron Hill (Wesleyan University)

    One of the great insights of model theory is the observation that very mundane-looking "combinatorial configurations" carry a huge amount of geometric information about a structure. In this talk, I will explain what we mean by "combinatorial configuration," and then I will sketch out how configurations can be "smoothed out" to yield Ramsey classes, which can themselves be analyzed using model-theoretic tools. I will also discuss the kinds of model-theoretic dividing lines that can be defined just through the interaction of structures with Ramsey classes.

    Updated on Mar 21, 2014 11:24 AM PDT
  9. AT Postdoc Seminar: Why do algebraic topologists care about categories?

    Location: MSRI: Simons Auditorium
    Speakers: Angelica Osorno (Reed College)

    The study of category theory was started by Eilenberg and MacLane, in their effort to codify the axioms for homology. Category theory provides a language to express the different structures that we see in topology, and in most of mathematics. Categories also play another role in algebraic topology. Via the classifying space construction, topologists use categories to build spaces whose geometry encodes the algebraic structure of the category. This construction is a fruitful way of producing important examples of spaces used in algebraic topology. In this talk we will describe how this process works, starting from classic examples and ending with some recent work.

    Updated on Mar 14, 2014 10:27 AM PDT
  10. AT Postdoc Seminar: Mumford Conjecture, Characteristic Classes, Manifold Bundles, and the Tautological Ring

    Location: Space Science Lab Conference Room
    Speakers: Ilya Grigoriev (University of Chicago)

    I will describe a topologists' perspective on the history of the study of an object that Mumford called "the tautological ring" and its generalizations.

    The tautological ring was originally defined as a subring of the cohomology of the moduli space of Riemann surfaces, but can also be studied as a ring of characteristic classes of topological bundles. This point of view led to a proof of Mumford's conjecture, stating that the tautological ring coincides with the entire cohomology of the moduli space in a "stable range", as well as to some generalizations of this result. If time permits, I will explain what we know about the tautological ring outside the stable range.

    Updated on Feb 21, 2014 09:07 AM PST
  11. MT Postdoc Seminar

    Location: Space Science Lab Conference Room
    Speakers: Artem Chernikov (L'Institut de Mathématiques de Jussieu)
    Updated on Feb 07, 2014 03:51 PM PST
  12. AT Postdoc Seminar: Groups, Fixed Points, and Algebraic Topology

    Location: MSRI: Simons Auditorium
    Speakers: Anna Marie Bohmann (Northwestern University)

    In algebraic topology, one key way of understanding group actions on spaces is by considering families of fixed points under subgroups.  In this talk, we will discuss this basic structure and its fundamental role in understanding equivariant algebraic topology.  I will then describe some recent joint work with A. Osorno that builds on fixed point information to create equivariant cohomology theories.

    Updated on Feb 21, 2014 09:02 AM PST
  13. MT Postdoc Seminar: Finite VC-dimension in model theory and elsewhere

    Location: MSRI: Simons Auditorium
    Speakers: Pierre Simon (Centre National de la Recherche Scientifique (CNRS))

    I will present a combinatorial property---finite VC-dimension---which appeared independently in various parts of mathematics.

    In model theory it is called "NIP" and is used notably in the study of ordered and valued fields. In probability theory, it is related to "learnable classes". In combinatorics, classes of finite VC-dimension behave a lot like families of convex subsets of euclidean space. I will also talk about Banach spaces and topological dynamics.

    The talk will be accessible to postdocs of both programs.

    Updated on Feb 20, 2014 02:34 PM PST
  14. Growth of groups using Euler characteristics

    Location: MSRI: Simons Auditorium
    Speakers: Alexander Young (University of Washington)

    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.

    Updated on May 10, 2013 10:59 AM PDT
  15. New computations of the Riemann zeta function

    Location: MSRI: Simons Auditorium
    Speakers: Jonathan Bober

    I'll describe the implementation of Hiary's O(t1/3) algorithm and the computations that we have been running using it. Some highlights include the 10^32nd zero (and a few hundred of its neighbors, all of which lie on the critical line), values of S(T) which are larger than 3, and values of zeta larger than 14000.
    Updated on Feb 19, 2014 08:53 AM PST
  16. Averages of central L-values

    Location: MSRI: Simons Auditorium
    Speakers: TBA

    Updated on Apr 01, 2011 08:17 AM PDT
  17. Non-Degeneracy of an Elliptic-Free Boundary Problem

    Location: MSRI: Simons Auditorium
    Speakers: Betul Orcan (University of Texas)

    In this talk, we will consider a free boundary problem with a
    very general free boundary condition and analyze the non-degeneracy of the
    largest subsolution near the free boundary.
    Updated on Jul 07, 2014 08:16 AM PDT
  18. Postdoctoral Seminars FBP

    Location: MSRI: Baker Board Room

    Pizza Lunch

    Updated on Jan 24, 2011 08:17 AM PST
  19. Imaging Edges in Random Media

    Location: MSRI: Simons Auditorium
    Speakers: Fernando Guevara Vasquez

    Pizza Lunch

    Consider the problem of imaging a reflector (target) from recordings of the echoes resulting from probing the medium with waves emanating from an array of transducers (the array response matrix). We present an algorithm that selectively illuminates the edges or the interior of an extended target by choosing particular subspaces of the array response matrix. For a homogeneous background medium, we characterize these subspaces in terms of the singular functions of a space and wave number restricting operator, which are also called generalized prolate spheroidal wave functions. We discuss results indicating what can be expected from using this algorithm when the medium fluctuates around a constant background medium and the fluctuations can be modeled as a random field.
    Updated on May 13, 2013 11:01 PM PDT
  20. Integrable Equations for Random Matrix Spectral Gap Probabilities

    Location: MSRI: Simons Auditorium
    Speakers: Igor Rumanov

    Pizza Lunch

    Connections are exposed between integrable equations for spectral gap probabilities of unitary invariant ensembles of random matrices (UE) derived by different --- Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) --- methods. Simple universal relations are obtained between these probabilities and their ratios on one side, and variables of the approach using resolvent kernels of Fredholm operators on the other side. A unified description of UE is developed in terms of universal, i.e. independent of the specific probability measure, PDEs for gap probabilities, using the correspondence of TW and ASvM variables. These considerations are based on the three-term recurrence for orthogonal polynomials (OP) and one-dimensional Toda lattice (or Toda-AKNS) integrable hierarchy whose flows are the continuous transformations between different OP bases. Similar connections exist for coupled UE. The gap probabilities for one-matrix Gaussian UE (GUE) or joint gap probabilities for coupled GUE satisfy various PDEs whose number grows with the number of spectral endpoints. With the above connections serving as a guide, minimal complete sets of independent lowest order PDEs for the GUE and for the largest eigenvalues of two-matrix coupled GUE are found.
    Updated on May 13, 2013 11:01 PM PDT
  21. The Inverse Calderon Problem for Schrödinger Operator on Riemann Surfaces

    Location: MSRI: Simons Auditorium
    Speakers: Leo tzou

    Pizza Lunch

    We show that on a smooth compact Riemann surface with boundary (M0, g) the Dirichletto- Neumann map of the Schrödinger operator â g + V determines uniquely the potential V . This seemingly analytical problem turns out to have connections with ideas in symplectic geometry and differential topology. We will discuss how these geometrical features arise and the techniques we use to treat them. This is joint work with Colin Guillarmou of CNRS Nice. The speaker is partially supported by NSF Grant No. DMS-0807502 during this work.
    Updated on May 13, 2013 11:01 PM PDT
  22. E. Nordenstam's Talk

    Location: MSRI: Simons Auditorium
    Speakers: Eric Nordenstam

    Pizza Lunch

    Updated on May 13, 2013 11:01 PM PDT
  23. Resistor Networks and Optimal Grids for Electrical Impedance Tomography with Partial Boundary Measurements

    Location: MSRI: Baker Board Room
    Speakers: Alexander Mamonov

    Pizza Lunch

    The problem of Electrical Impedance Tomography (EIT) with partial boundary measurements is to determine the electric conductivity inside a body from the simultaneous measurements of direct currents and voltages on a subset of its boundary. Even in the case of full boundary measurements the non-linear inverse problem is known to be exponentially ill-conditioned. Thus, any numerical method of solving the EIT problem must employ some form of regularization. We propose to regularize the problem by using sparse representations of the unknown conductivity on adaptive finite volume grids known as the optimal grids. Then the discretized partial data EIT problem can be reduced to solving the discrete inverse problems for resistor networks. Two distinct approaches implementing this strategy are presented. The first approach uses the results for the EIT problem with full boundary measurements, which rely on the use of resistor networks with circular graph topology. The optimal grids for such networks are essentially one dimensional objects, which can be computed explicitly. We solve the partial data problem by reducing it to the full data case using the theory of extremal quasiconformal (Teichmuller) mappings. The second approach is based on resistor networks with the pyramidal graph topology. Such network topology is better suited for the partial data problem, since it allows for explicit treatment of the inaccessible part of the boundary. We present a method of computing the optimal grids for the networks with general topology (including pyramidal), which is based on the sensitivity analysis of both the continuum and the discrete EIT problems. We present extensive numerical results for the two approaches. We demonstrate both the optimal grids and the reconstructions of smooth and discontinuous conductivities in a variety of domains. The numerical results show two main advantages of our approaches compared to the traditional optimization-based methods. First, the inversion based on resistor networks is orders of magnitude faster than any iterative algorithm. Second, our approaches are able to correctly reconstruct the conductivities of very high contrast, which usually present a challenge to the iterative or linearization-based inversion methods.
    Updated on May 13, 2013 11:01 PM PDT