Seminars and Colloquia by Series

Thrifty approximations of convex bodies by polytopes

Series
School of Mathematics Colloquium
Time
Thursday, March 7, 2013 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alexander BarvinokUniversity of Michigan
Given a d-dimensional convex body C containing the origin in its interior and a real t>1, we seek to construct a polytope P with as few vertices as possible such that P is contained in C and C is contained in tP. I plan to present a construction which breaks some long-held records and is nearly optimal for a wide range of parameters d and t. The construction uses the maximum volume ellipsoid, the John decomposition of the identity and its recent sparsification by Batson, Spielman and Srivastava, Chebyshev polynomials, and some tensor algebra.

Discrepancy of multidimensional Kronecker sequences.

Series
School of Mathematics Colloquium
Time
Thursday, February 28, 2013 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dmitry DolgopyatUniv. of Maryland
The classical Weyl equidistribution theorem says that if v is a non-resonant vector then the sequence v, 2v, 3v... is uniformly distributed on a torus. In this talk we discuss the rate of convergence to the uniform distribution. This is a joint work with Bassam Fayad.

Applications of Algebraic Geometry in Statistics

Series
School of Mathematics Colloquium
Time
Thursday, February 21, 2013 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mathias DrtonUniversity of Washington
Statistical modeling amounts to specifying a set of candidates for what the probability distribution of an observed random quantity might be. Many models used in practice are of an algebraic nature in thatthey are defined in terms of a polynomial parametrization. The goal of this talk is to exemplify how techniques from computational algebraic geometry may be used to solve statistical problems thatconcern algebraic models. The focus will be on applications in hypothesis testing and parameter identification, for which we will survey some of the known results and open problems.

Random Matrices: Law of the Determinant

Series
School of Mathematics Colloquium
Time
Friday, February 8, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Van VuYale University
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.

The curve complex of a surface

Series
School of Mathematics Colloquium
Time
Friday, December 7, 2012 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joan BirmanColumbia University

Please Note: 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.

Population persistence in the face of demographic and environmental uncertainty

Series
School of Mathematics Colloquium
Time
Thursday, November 8, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sebastain SchreiberUC Davis
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.

Isoperimetric inequalities in Gaussian Space

Series
School of Mathematics Colloquium
Time
Thursday, October 25, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Elchanan MosselUC Berkeley, Statistics
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.

Operator Monotone Functions of Several Variables

Series
School of Mathematics Colloquium
Time
Thursday, October 18, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
John McCarthyWashington University - St. Louis
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.

Lifts of Convex Sets and Cone Factorizations

Series
School of Mathematics Colloquium
Time
Thursday, October 4, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Rekha ThomasUniversity of Washington
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.

Genericity of chaotic behavior

Series
School of Mathematics Colloquium
Time
Thursday, September 27, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yakov PesinPenn State
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.

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