Friday, February 8, 2013 - 15:00 , Location: Skiles 005 , Van Vu , Yale University , Organizer: Greg Blekherman
Random matrix theory is a fast developing topic with connections to so many areas of mathematics: probability, number theory, combinatorics, data analysis, mathematical physics, to mention a few. The determinant is one of the most studied matrix functionals. In our talk, we are going to give a brief survey on the studies of this functional, dated back to Turan in the 1940s. The main focus will be on recent developments that establish the limiting law in various models.
Friday, December 7, 2012 - 16:00 , Location: Skiles 006 , Joan Birman , Columbia University , Organizer: John Etnyre
Kickoff of the Tech Topology Conference from December 7-9, 2012.
This will be a Colloquium talk, aimed at a general audience. The topic is the curve complex, introduced by Harvey in 1974. It's a simplicial complex, and was introduced as a tool to study mapping class groups of surfaces. I will discuss recent joint work with Bill Menasco about new local pathology in the curve complex, namely that its geodesics can have dead ends and even double dead ends.
Thursday, November 8, 2012 - 11:00 , Location: Skiles 006 , Sebastain Schreiber , UC Davis , email@example.com , Organizer: Greg Blekherman
Populations, whether they be viral particles, bio-chemicals, plants or animals, are subject to intrinsic and extrinsic sources of stochasticity. This stochasticity in conjunction with nonlinear interactions between individuals determines to what extinct populations are able to persist in the long-term. Understanding the precise nature of these interactive effects is a central issue in population biology from theoretical, empirical, and applied perspectives. For the first part of this talk, I will discuss, briefly, the relationship between attractors of deterministic models and quasi-stationary distributions of their stochastic, finite population counterpoints i.e. models accounting for demographic stochasticity. These results shed some insight into when persistence should be observed over long time frames despite extinction being inevitable. For the second part of the talk, I will discuss results on stochastic persistence and boundedness for stochastic models accounting for environmental (but not demographic) noise. Stochastic boundedness asserts that asymptotically the population process tends to remain in compact sets. In contrast, stochastic persistence requires that the population process tends to be "repelled" by some "extinction set." Using these results, I will illustrate how environmental noise can facilitate coexistence of competing species and how dispersal in stochastic environments can rescue locally extinction prone populations. Empirical work on Kansas prairies, acorn woodpecker populations, and microcosm experiments demonstrating these phenomena will be discussed.
Thursday, October 25, 2012 - 11:00 , Location: Skiles 006 , Elchanan Mossel , UC Berkeley, Statistics , firstname.lastname@example.org , Organizer: Greg Blekherman
Isoperimetric problems in Gaussian spaces have been studied since the 1970s. The study of these problems involve geometric measure theory, symmetrization techniques, spherical geometry and the study of diffusions associated with the heat equation. I will discuss some of the main ideas and results in this area along with some new results jointly with Joe Neeman.
Thursday, October 18, 2012 - 11:00 , Location: Skiles 006 , John McCarthy , Washington University - St. Louis , Organizer:
Self-adjoint $n$-by-$n$ matrices have a natural partial ordering, namely $ A \leq B $ if the matrix $ B - A$ is positive semi-definite. In 1934 K. Loewner characterized functions that preserve this ordering; these functions are called $n$-matrix monotone. The condition depends on the dimension $n$, but if a function is $n$-matrix monotone for all $n$, then it must extend analytically to a function that maps the upper half-plane to itself. I will describe Loewner's results, and then discuss what happens if one wants to characterize functions $f$ of two (or more) variables that are matrix monotone in the following sense: If $ A = (A_1, A_2)$ and $B = (B_1,B_2)$ are pairs of commuting self-adjoint $n$-by-$n$ matrices, with $A_1 \leq B_1 $ and $A_2 \leq B_2$, then $f(A) \leq f (B)$. This talk is based on joint work with Jim Agler and Nicholas Young.
Thursday, October 4, 2012 - 11:00 , Location: Skiles 006 , Rekha Thomas , University of Washington , Organizer: Greg Blekherman
A basic strategy for linear optimization over a complicated convex set is to try to express the set as the projection of a simpler convex set that admits efficient algorithms. This philosophy underlies all "lift-and-project" methods in optimization which attempt to find polyhedral or spectrahedral lifts of complicated sets. In this talk I will explain how the existence of a lift is equivalent to the ability to factorize a certain operator associated to the convex set through a cone. This theorem extends a result of Yannakakis who showed that polyhedral lifts of polytopes are controlled by the nonnegative factorizations of the slack matrix of the polytope. The connection between cone lifts and cone factorizations of convex sets yields a uniform framework within which to view all lift-and-project methods, as well as offers new tools for understanding convex sets. I will survey this evolving area and the main results that have emerged thus far.
Thursday, September 27, 2012 - 11:00 , Location: Skiles 006 , Yakov Pesin , Penn State , Organizer: Greg Blekherman
It is well-known that a deterministic dynamical system can exhibit stochastic behavior that is due to the fact that instability along typical trajectories of the system drives orbits apart, while compactness of the phase space forces them back together. The consequent unending dispersal and return of nearby trajectories is one of the hallmarks of chaos. The hyperbolic theory of dynamical systems provides a mathematical foundation for the paradigm that is widely known as "deterministic chaos" -- the appearance of irregular chaotic motions in purely deterministic dynamical systems. This phenomenon is considered as one of the most fundamental discoveries in the theory of dynamical systems in the second part of the last century. The hyperbolic behavior can be interpreted in various ways and the weakest one is associated with dynamical systems with non-zero Lyapunov exponents. I will discuss the still-open problem of whether dynamical systems with non-zero Lyapunov exponents are typical. I will outline some recent results in this direction. The genericity problem is closely related to two other important problems in dynamics on whether systems with nonzero Lyapunov exponents exist on any phase space and whether nonzero exponents can coexist with zero exponents in a robust way.
Thursday, September 6, 2012 - 11:00 , Location: Klaus 1116 , Noga Alon , Tel Aviv Uniersity , Organizer: Greg Blekherman
The study of random Cayley graphs of finite groups is related to the investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing the motivation and focusing on the question of estimating the chromatic number of a random Cayley graph of a given group with a prescribed number of generators. The investigation of this problem combines combinatorial, algebraic and probabilistic tools. Several intriguing questions that remain open will be mentioned as well.
Tuesday, April 24, 2012 - 16:00 , Location: Skiles 005 , Prof. Andrea Bertozzi , UCLA Math , Organizer: Sung Ha Kang
There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, methods in linear and nonlinear partial differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes. Examples will consider both "bottom up" and "top down" approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory.