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Series: Research Horizons Seminar

Hosts: Yao Li and Ricardo Restrepo

Suppose you want to stir a pot of soup with several spoons. What is the most efficient way to do this? Thurston's theory of surface homeomorphisms gives us a concrete way to analyze this question. That is, to each mixing pattern we can associate a real number called the entropy. We'll start from scratch with a simple example, state the Nielsen-Thurston classification of surface homeomorphisms, and give some open questions about entropies of surface homeomorphisms.

Series: Research Horizons Seminar

Hosts: Yao Li and Ricardo Restrepo.

Dr. Millman is the Director of the Center for Education Integrating Science, Mathematics & Computing (CEISMC) and professor of mathematics at the Georgia Institute of Technology. He is a first hand expert in mathematics education and K-12 mathematics teacher education. Complementing the previous panel discussion on jobs in academia and industry, Dr. Milman will lead the discussion on teaching jobs.

Series: Research Horizons Seminar

Hosts: Yao Li and Ricardo Restrepo

Modern Economic Theory is largely based on the concept of Nash Equilibrium. In its simplest form this is an essentially statics notion. I'll introduce a simple model for the origin of money (Kiotaki and Wright, JPE 1989) and use it to introduce a more general (dynamic) concept of Nash Equilibrium and my understanding of its relation to Dynamical Systems Theory and Statistical Mechanics.

Series: Research Horizons Seminar

Hosts: Yao Li and Ricardo Restrepo

We consider compressible fluid flows in Lagrangian coordinates in one space dimension. We assume that the fluid self-interacts through a force field generated by the fluid. We explain how this flow can be described by a differential inclusion on the space of transport maps, when the sticky particle dynamics is assumed. We prove a stability result for solutions of this system. Global existence then follows from a discrete particle approximation.

Series: Research Horizons Seminar

Hosts: Yao Li and Ricardo Restrepo

This will be an expository talk on the study of orthogonal polynomials on the real line and on the unit circle. Topics include recurrence relations, recurrence coefficients and simple examples. The talk will conclude with applications of orthogonal polynomials to other areas of research.

Series: Research Horizons Seminar

Hosts: Yao and Ricardo

Consider self-adjoint operators $A, B, C : \mathcal{H} \to \mathcal{H}$ on a finite-dimensional Hilbert space such that $A + B + C = 0$. Let $\{\lambda_j (A)\}$, $\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ be sequences of eigenvalues of $A, B$, and $C$ counting multiplicity, arranged in decreasing order. In 1962, A. Horn conjectured that the relations of $\{\lambda_j (A)\}$,$\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ can be characterized by a set of inequalities defined inductively. This problem was eventually solved by A. Klyachko and Knutson-Tao in the late 1990s. Recently together with H. Bercovici, Collins, Dykema, and Timotin, we are able to find a proof to show that the inequalities are valid for self-adjoint elements that satisfies the relation $A+B+C=0$, and the proof can be applied to finite von Neumann algebra. The major difficulty in our argument is to show that certain generalized Schubert cells have nonempty intersection. In the finite dimensional case, it follows from the classical intersection theory. However, there is no readily available intersection theory for von Neumann algebras. Our argument requiresa good understanding of the combinatorial structure of honeycombs, and produces an actual element in the intersection algorithmically, and it seems to be new even in finite dimensions.

Series: Research Horizons Seminar

Hosts: Yao Li and Ricardo Restrepo

The Research Horizons seminar this week will be a panel discussion on the job market for mathematicians. Professor Doug Ulmer and Luca Dieci will give a presentation with general information on the academic job market and the experience of our recent students, in and out of academia. The panel will then take questions from the audience.

Series: Research Horizons Seminar

Hosts: Yao Li and Ricardo Restrepo

I will consider mathematical models of decision making based on dynamics in the neighborhood of unstable equilibria and involving random perturbations due to small noise. I will report results on the vanishing noise limit for these systems, providing precise predictions about the statistics of decision making times and sequences of unstable equilibria visited by the process. Mathematically, the results are based on the analysis of random Poincare maps in the neighborhood of each equilibrium point. I will discuss applications to neuroscience and psychology along with some experimental data.

Series: Research Horizons Seminar

Hosts: Yao Li and Ricardo Restrepo

When an object is small enough that quantum mechanics matters, many of its physical properties, such as energy levels, are determined by the eigenvalues of some linear operators. For quantum wires, waveguides, and graphs, geometry and topology show up in the operators and affect the set of eigenvalues, known as the spectrum. It turns out that the spectrum can't be just any sequence of numbers, both because of some general theorems about the eigenvalues and because of inequalities involving the shape. I'll discuss some of the extreme cases that test the theorems and inequalities and connect them to the shapes of the structures and to algebraic properties of the operators.To understand this lecture it would be helpful to know a little about PDEs and eigenvalues, but no knowledge of quantum mechanics will be needed.

Series: Research Horizons Seminar

Hosted by Yao Li and Ricardo Restrepo.

Deciding how to unknot a knotted piece of string (with its ends glued together) is not only a difficult problem in the real world, it is also a difficult and long studied problem in mathematics. (There are several notions of what one might mean by "unknotting" and I will leave the exact meaning a bit vague in this abstract.) In the past mathematicians have used a vast array of techniques --- from geometry to algebra, and even PDEs --- to study this question. I will discuss this question and (partially) recast it in terms of 4 dimensional topology. This new perspective will allow us to use a powerful new knot invariant called Khovanov Homology to study the problem. I will give an overview of Khovanov Homology and indicate how to study our unknotting question using it.