Seminars and Colloquia

Tuesday, September 16, 2008

01:15 pm

Quasirandomness via rotor-routers

Speaker: Jim Propp (Lowell)<br>Abstract: Quasirandom processes are deterministic processes designed to mimic some aspects of random processes. I will show how some very simple mechanisms called rotor-routers give rise to deterministic analogues of discrete random processes that share some of their asymptotic features.For information write to mkeane@wesleyan.edu

Exley Science Center - ESC 638

Tuesday, September 23, 2008

01:15 pm

Shufflings, Connections, and Couplings: Tunneling along the white light arching through the future

Speaker: Bob Burton, Corvalis, Middletown<br>Abstract: There are several models that have attracted attention in the past decade or so, and we mention a few of them. The talk will consist of showing some properties of some of these models as well as some connections between them. Credit goes to various collaborators that worked with me on parts of this, including Larry Pierce and Yevgeniy Kovchegov. Some examples that come to mind include:1.Subshifts of finite type on a graph such as Z or Z^d; Quantitative questions for d>2 seem to be difficult.2.Interation of Randomly Chosen Functions, lately often called Iterated Function Systems. These bear a similar relation with Markov Chains that processes enjoy with ergodic measure-preserving transformations. A class of examples is obtained by applying a shuffling system to a deck of cards until mixed. Perfect mixing is eventual because of symmetry because the transformation is a finite-state Bernoulli shift but the method of shuffling could affect the mixing time.3.Substitution Systems: These are well known and include the commonly [and incorrectly*] called Morse and Fibonacci systems. Note that the choice of symbols could be random.4.Uniformly distributed sequences on the unit circle or the n-torus. Often these are presented as a deterministic sequence of numbers but they usually arise from a dynamical system.5.Farey fractions and the Stern-Brocot Tree. These seem to be amazing bookkeeping systems for showing arithmetic properties of integers and rational numbers. This technique actually sheds some light of a common two dimensional model of ice and, most likely, the modular group PSL(2,Z).*recall Burtons Mother of Truth Principle: All Attributions of Originality are Untrue.

Exley Science Center - ESC 638

Tuesday, November 11, 2008

01:15 pm

Recurrent orbits on $L^\infty$ spaces

Speaker: Katsuhiko Matsuzaki, Okayama University and Wesleyan University<br>Abstract:For a shift operator on the space of all bilateral sequences of real numbers with $L^\infty$ norm, we give an example of a recurrent orbit that is not accumulated by periodic orbits. We also consider this problem for the $L^\infty$ space on a discrete group equipped with the group action.

Exley Science Center - ESC 638

Tuesday, November 25, 2008

01:15 pm

outer billiards, tilings, and the modular group

Speaker: Richard Schwartz, Providence<br>Abstract: outer billiards is a dynamical system based on a convex planar shape in the plane. I'll explain how the system gives rise to interesting and not-well-understood tilings of the plane. I will also discuss connections to Diophantine approximation and (in special cases) the modular group. In particular, I'll talk about my recent resolution of the Moser-Neumann question, which was probably the central question in the subject. The Moser-Neumann question, around since 1960, asked if one could have unbounded orbits in an outer billiards system.

Exley Science Center - ESC 638

Tuesday, December 02, 2008

01:00 pm

Using global invariant manifolds to understand metastability in Burgers equation with small viscosity

Speaker: Margaret Beck, Brown Applied Mathematics<br>Abstract: Metastability is a phenomenon that refers to large periods of time in which solutions remain near unstable states, or families of unstable states, before finally converging to their stable asymptotic limit. This has been observed is a variety of PDEs, including the Navier-Stokes equations, and it is particularly important in applications because, if solutions exhibit transient but long-lived behavior, it is often this behavior that one actually observes in practice. To better understand metastability, we investigate it in the context of a relatively simple PDE, Burgers equation. When the viscosity is small, it has been observed numerically that solutions look for long times like "N-waves," the stable solution in the zero viscosity limit, before finally converging to a self-similar "diffusion wave," the stable solution for nonzero viscosity. We show that there exist global invariant manifolds in the phase space of Burgers equation that can be used to provide a geometric understanding of this phenomenon. This is joint work with Gene Wayne of Boston University.

Exley Science Center - ESC 638

Monday, April 20, 2009

03:00 pm

The window of a surface in a hyperbolic 3-manifold

Speaker: Genevieve Walsh, Medford<br>Abstract: We characterize the maximal product region, or window of a hyperbolic 3-manifold cut along a surface. This is done by looking at the combinatorics of the limit set.

Exley Science Center - ESC 638

Monday, April 18, 2011

04:10 pm - 06:00 pm

Exceptional Points For Cocompact Fuchsian Groups

Speaker: Joseph Fera, Wesleyan UniversityAbstract: Let G be a cocompact Fuchsian group acting on the hyperbolic plane H. If G covers a compact hyperbolic surface of genus g (at least 2), then almost every Dirichlet region for G has 12g-6 sides. In this talk, we discuss the exceptional points for G, i.e., the points in H associated to Dirichlet regions for G with strictly less than 12g-6 sides. More specifically, we show that uncountably many exceptional points exist for any cocompact group. We also define and prove the existence of higher order exceptional points for any such group.

ESC 638