Seminars and Colloquia by Series

What is Weak KAM Theory?

Series
Job Candidate Talk
Time
Tuesday, March 3, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Albert Fathi – ENS Lyon –
The goal of this lecture is to explain and motivate the connection between Aubry-Mather theory (Dynamical Systems), and viscosity solutions of the Hamilton-Jacobi equations (PDE). The connection is the content of weak KAM Theory. The talk should be accessible to the ''generic" mathematician. No a priori knowledge of any of the two subjects is assumed.

q-Ehrhart polynomials and knots

Series
Geometry Topology Seminar
Time
Monday, March 2, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Roland van der Veen – University of Amsterdam –
We will start by counting lattice points in a polytope and showhow this produces many familiar objects in mathematics.For example if one scales the polytope, the number of lattice points givesrise to the Ehrhart polynomials, including binomals and other well knownfunctions.Things get more interesting once we take a weighted sum over the latticepoints instead of just counting them. I will explain how toextend Ehrhart's theory in this case and discuss an application to knottheory. We will derive a new state sum for the colored HOMFLYpolynomial using q-Ehrhart polynomials, following my recent preprint Arxiv1501.00123.

Existence and Stability of Radially Symmetric Solutions to the Swift--Hohenberg Equation

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 2, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Scott McCalla – Montana State University
The existence, stability, and bifurcation structure of localized radially symmetric solutions to the Swift--Hohenberg equation is explored both numerically through continuation and analytically through the use of geometric blow-up techniques. The bifurcation structure for these solutions is elucidated by formally treating the dimension as a continuous parameter in the equations. This reveals a family of solutions with an anomalous amplitude scaling that is far larger than expected from a formal scaling in the far field. One key advantage of the geometric blow-up techniques is that a priori knowledge of this scaling is unnecessary as it naturally emerges from the construction. The stability of these patterned states will also be discussed.

Torsion in Homology for Hyperbolic 3-Manifolds

Series
Geometry Topology Seminar
Time
Monday, March 2, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Pere Menal-Ferrer – Georgia Tech
How is the homological torsion of a hyperbolic 3-manifold related to its geometry? In this talk, I will explain some techniques to address this general question. In particular, I will discuss in detail the case of arithmetic manifolds, where the situation is presumably easier to understand.

Introduction to regularity theory of second order Hamilton-Jacobi-Bellman equations

Series
PDE Working Seminar
Time
Friday, February 27, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Andrzej Sweich – GeorgiaTech –
I will give a series of elementary lectures presenting basic regularity theory of second order HJB equations. I will introduce the notion of viscosity solution and I will discuss basic techniques, including probabilistic techniques and representation formulas. Regularity results will be discussed in three cases: degenerate elliptic/parabolic, weakly nondegenerate, and uniformly elliptic/parabolic.

Burgers equation with random forcing

Series
Stochastics Seminar
Time
Thursday, February 26, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yuri Bakhtin – Courant Institute of Mathematical Sciences, New York University
Ergodic theory of randomly forced space-time homogeneous Burgers equation in noncompact setting has been developed in a recent paper by Eric Cator , Kostya Khanin, and myself. The analysis is based on first passage percolation methods that allow to study coalescing one-sided action minimizers and construct the global solution via Busemann functions. i will talk about this theory and its extension to the case of space-continuous kick forcing. In this setting, the minimizers do not coalesce, so for the ergodic program to go through, one must use new soft results on their behavior to define generalized Busemann functions along appropriate subsequences.

RESCHEDULED: Describing geometry and symmetry in cryo-EM datasets using algebra

Series
Mathematical Biology Seminar
Time
Thursday, February 26, 2015 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
David Dynerman – University of Wisconsin-Madison
Cryo-electron microscopy (cryo-EM) is a microscopy technique used to discover the 3D structure of molecules from very noisy images. We discuss how algebra can describe two aspects of cryo-EM datasets. First, we'll describe common lines datasets. Common lines are lines of intersection between cryo-EM images in 3D. They are a crucial ingredient in some 2D to 3D reconstruction algorithms, and they can be characterized by polynomial equalities and inequalities. Second, we'll show how 3D symmetries of a molecule can be detected from only 2D cryo-EM images, without performing full 3D reconstruction.

Intersection theory and the Horn inequalities for invariant subspaces

Series
Analysis Seminar
Time
Wednesday, February 25, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wing Li – Georgia Institute of Technology
It is well known that the Horn inequalities characterize the relationship of eigenvalues of Hermitian matrices A, B, and A+B. At the same time, similar inequalities characterize the relationship of the sizes of the Jordan models of a nilpotent matrix, of its restriction to an invariant subspace, and of its compression to the orthogonal complement. In this talk, we provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C_0 (such operator can be thought of as the infinite dimensional generalization of matrices, that is an operator will be annihilated by an H-infinity function), of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where ‘inequality’ is replaced by ‘divisibility’. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. Our approach also explains why the same combinatorics solves the eigenvalue and the Jordan form problems. This talk is based on the joint work with H. Bercovici.

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