Dynamics n' Analysis Seminar Archive

April 20, 2009

The window of a surface in a hyperbolic 3-manifold

Speaker: Genevieve Walsh, Medford
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.

December 02, 2008

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

Speaker: Margaret Beck, Brown Applied Mathematics
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.

November 25, 2008

outer billiards, tilings, and the modular group

Speaker: Richard Schwartz, Providence
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.

November 11, 2008

Recurrent orbits on $L^\infty$ spaces

Speaker: Katsuhiko Matsuzaki, Okayama University and Wesleyan University
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.

September 23, 2008

Tunneling along the white light arching through the future

Speaker: Bob Burton, Corvalis, Middletown
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.

September 16, 2008

Quasirandomness via rotor-routers

Speaker: Jim Propp (Lowell)
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

March 04, 2008

Theorems of Stone-Weierstrass Type

Speaker: Stu Sidney, (Storrs)
Abstract: The standard (real) Stone-Weierstrass theorem says that if E is a vector space of continuous functions on a compact Hausdorff space X that contains the constant functions and separates the points of X, and if E is either a lattice or an algebra under pointwise operations, then E is dense in C(X). The lattice and algebra conditions can be reformulated to say that E is closed under composition with the absolute value function and the squaring function respectively. In 1963 de Leeuw and Katznelson generalized both versions by showing that if E is closed under composition with some non-affine function, then E is dense in C(X). Of course, if it is also hypothesized that E is uniformly closed -- equivalently, that E is complete in the supremum norm -- then the conclusion becomes E = C(X). But what if E is complete in some other norm that dominates the supremum norm? This makes E into a (real) Banach function space, and from the early 1960s on there has been slow and steady progress on the following problem: If a non-affine function f operates by composition on a Banach function space E, under what additional conditions on f and/or E can we deduce that E = C(X)? We shall recount some of this history, with the emphasis being on conditions on E, and perhaps discuss some work in progress along these lines.

February 12, 2008

Finitary isomorphisms of renewal-like processes

Speaker: Stephen Shea, Wesleyan University
Abstract: We will examine a particular type of stochastic process in which the occurence of a state (a renewal state) yields complete independence of past and future events. If all states in a process have this renewal property, we say the process is Markov. In 1979, Keane and Smorodinsky showed that finite state Markov chains are finitarily isomorphic to Bernoulli schemes. Informally, a finitary isomorphism is a coding from one process to another such that almost all coordinates in the image can be determined by finitely many coordinates in the pre-image. We would like to know when we can generalize Keane and Smorodinsky's marker and filler methods of '79 to produce a finitary result for the above described renewal-like processes. I will show that a restriction on the return times of the renewal state in our process is sufficient for such a result.

November 28, 2007

Square summable variations and uniqueness of g-measures

Speaker: Anders Oberg, Uppsala and Amherst
Abstract: Oberg: In this talk I will describe some recent developments on the uniqueness of g-measures and convergence of the iterates of the corresponding transfer operator. The key condition is square summability of the variations of the probability potentials. This is a survey of results from three papers co-authored with Anders Johansson, and one of them also with Mark Pollicott.

October 30, 2007

Exhaustive Weakly Wandering Sequences

Speaker: Stanley Eigen, Northeastern University

Abstract: A sequence of integers $\{n_i:i=0,1,\cdots\}$ is an {\it exhaustive weakly wandering} sequence for a transformation $T$ if for some measurable set $W$, $\DS X=\cup _{i=0}^{\infty}T^{n_i}W (disj)$. We introduce a hereditary {\it Property H} for a sequence of integers associated with an infinite measure preserving ergodic transformation $T$, and show that it is a sufficient condition for the sequence to be an exhaustive weakly wandering sequence for $T$. We then show that every infinite measure preserving ergodic transformation admits sequences that possess Property (H), and observe that Property (H) is inherited by all subsequences of a sequence that possess it. As a Corollary, we obtain an application to tiling the set of integers $\mathbb Z$ with infinite subsets.

Please note that this seminar starts at 1:10pm.

October 23, 2007

Recurrence and Strong Recurrence in Ergodic Theory

Speaker: Arshag Hajian, Northeastern University
Title: Recurrence and Strong Recurrence in Ergodic Theory

April 03, 2007

ETDS Seminar - Jesse Ratzkin, University of Connecticut

Rigidity and deformations of constant mean curvature surface