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Speaker: Laura DeMarco, University of Chicago, Host: Baker
Title: Dynamics at the boundary of moduli space

Abstract: The moduli space of rational maps, M(d), is the collection of all holomorphic self-maps of the Riemann sphere of degree d > 1, modulo the action by conjugation of the group of Möbius transformations. I will discuss the limiting dynamics of rational maps at the boundary of M(d), from algebraic, geometric, and ergodic theoretic points of view. The ideas were motivated by relations to Teichmüller theory, the study of moduli spaces in algebraic geometry, and the theory of entropy of a dynamical system.

Speaker: Fabio Martinelli, University of Roma (Tre), Host: Tetali
Title: Glauber dynamics and reconstruction problems on trees

Abstract: We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics for various spin models on a tree, including those with hard--core constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard-core lattice gas (independent sets). Specifically for the Ising model, we show that the mixing time on an n-vertex regular tree with plus boundary remains O(n\log n) at all temperatures (in contrast to the free boundary case, where the mixing time is not bounded by any fixed polynomial at low temperatures). Our results apply also to reconstruction problems on a tree in a noisy channel. Finally we will describe some recent results on the basin of attraction under the Glauber dynamics of the pure phases of the system. Based on joint works with i) A. Sinclair, D. Weitz and ii) P. Caputo.

Speaker: Idris Assani, University of North Carolina at Chapel Hill
Title: Return Times and Birkhoff theorems in L¹ of product spaces

Abstract

Speaker: Ovidiu Costin, Rutgers University, Host: Garoufalidis
Title: Borel summability and the study of the Schrödinger equation in time-periodic fields of arbitrary strength

Abstract: Generalized Borel summation is a powerful regularization tool in ordinary and partial differential equations in singular limits. I will discuss the key ideas of generalized Borel summability and transseries and apply the methods on the Schrödinger equation with time-periodic external forcing when this forcing is not necessarily small. This study is central to the problem of ionization of atoms in laser and microwave fields. The results reflect qualitative departures from the behavior expected from the small-field approximation (perturbation theory) and agree with experimental results. The talk will focus on the results, as well as on the new methods developed in the process.

Speaker: Assaf Naor, Microsoft Research, Redmond, WA, Host: Tetali
Title: The Lipschitz extension problem

Abstract: The Lipschitz extension problem asks for conditions on a pair of metric spaces X,Y such that every Y-valued Lipschitz map on a subset of X can be extended to all of X with only a bounded multiplicative loss in the Lipschitz constant. This problem dates back to the work of Kirszbraun and Whitney in the 1930s, and has been extensively investigated in the past two decades. In this talk we will present the main known results on the Lipschitz extension problem, as well as several recent breakthroughs. The techniques used in the proofs are based on geometric, probabilistic and combinatorial arguments.

Speaker: Prof. Jim Geelen, University of Waterloo, Host: Thomas
Title: Inequivalent representations of matroids

Abstract: We are given a finite field F, a finite set S, and a nonnegative integer rank, r(X), assigned to each subset of E. An F-representation of (E,r) is an embedding of the elements of E into a projective space over F respecting the prescribed ranks. Give one such representation, of the pair (E,r), it is easy to construct others using a projective transformation. However, projective transformations are not always sufficient to generate all possible representations. We show that the number of inequivalent representations of any "sufficiently connected" pair (E,r) is bounded by a constant depending only on the size of F. This leads to a succinct certificate for proving non-F-representability. This is joint work with Bert Gerards and Geoff Whittle.

The talk is intended to be accessible to a general mathematics audience.

Spring 2004
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Spring 2003
Fall 2002

| ACM Seminar | Algebra/Topology | CDSNS Colloquium | CDSNS Seminar | Colloquium | Combinatorics | Geometry/Topology | Graph Theory | Math Seminars | Nonlinear Science | PDEs Seminar | Research Horizons | Stochastics |