Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, November 15, 2005.

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Tuesday, November 15, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, November 15, 2005

The Stern sequence

Bruce Reznick (UIUC Math)

Abstract: This talk is an advertisement for a Spring 2006 Math 595 course devoted to the Stern sequence. Some of the basic properties of the Stern sequence (s(0) =0, s(1) = 1, s(2n) = s(n), s(2n+1) = s(n) + s(n+1)) will be described, as well as other mathematical objects which can be understood through the Stern sequence. These objects reside in number theory, combinatorics, analysis, geometry and elsewhere.

1:00 pm in Illini Hall rm 1,Tuesday, November 15, 2005

Building higher dimensional relatively hyperbolic groups

Igor Belegradek (Georgia Tech)

Abstract: Gromov suggested in mid 80s that relative hyperbolization of polyhedra should be a source of relatively hyperbolic groups. In the talk I shall hint on how to prove this claim, and then exhibit many closed aspherical manifolds with hyperbolic or relatively hyperbolic fundamental groups. For example, I prove that if G is hyperbolic relative to H_1, ..., H_k, and if G is the fundamental group of a closed triangulated aspherical n-manifold, then G embeds as a retract into some group G' such that G' is hyperbolic relative to H_1, ..., H_k, and G' is the fundamental group of a closed triangulated aspherical (n+1)-manifold.

2:00 pm in 241 Altgeld Hall,Tuesday, November 15, 2005

A description of sofic groups in terms of nonstandard analysis

Evgeny Gordon (Eastern Illinois University)

Abstract: Sofic groups were first defined by M. Gromov as a common generalization of amenable groups and residually finite groups. B. Weiss proved that the old problem of Gottschalk on surjunctive groups has a positive solution for sofic groups. The problem of existence of non-sofic groups remains open. In this talk we discuss this topic in terms of nonstandard analysis, which allows us to simplify some definitions and proofs.

2:00 pm in 347 Altgeld Hall,Tuesday, November 15, 2005

Evans functions, Jost functions, and Fredholm determinants

Yuri Latiushkin   [email] (University of Missouri Math)

Abstract: I am going to report on recent results (joint with Fritz Gesztesy and Konstantin Makarov) relating the three objects listed in the title. The Evans function is a widely known Wronskian-type object, which is used to detect isolated eigenvalues of differential operators obtained as linearizations of nonlinear partial differential equations along such special solutions as traveling waves. The Evans function has become a main tool in detecting instability for the special solutions. The Jost function is a classical object well-familiar from scattering theory. The Fredholm determinants considered in the talk are the modified Fredholm determinants of certain Birman-Schwinger type integral operators with semi-separable kernels. Using the theory of Lyapunov exponents, we give a definition of the Evans function for quite general first order nonatonomous matrix differential equations on the line in the context of perturbation theory. Unlike the commonly used definition, our definition of the Evans function is coordinate-free, and thus the Evans function is uniquely defined by the differential equation and its perturbation. We show that the Evans function is a generalization of the classical Jost function associated with the Schrodinger equation on the line, and, in particular, that the Evans function for the first order system corresponding to the Schrodinger equation coincides with the Jost function. However, the central result of the talk is a formula relating the Evans function and the modified Fredholm determinant of the so-called ``sandwiched resolvent". This result is obtained under the assumption that the unperturbed equation has an exponential dichotomy on the line, and that the perturbation has certain exponential decay at infinities controlled by the general Lyapunov exponents of the unperturbed system.

2:00 pm in 243 Altgeld Hall,Tuesday, November 15, 2005

Generalized billiards inside an infinite strip.

Prof. Gregory Galperin (EIU Department of Mathematics and Computer Science)

Abstract: Consider billiards inside an infinite strip with a prescribed periodic law of reflection off the bottom and top strip's boundaries. Each of the boundaries is equipped with the same periodic lattice, where the number of lattice's nodes between any two consecutive reflection points may be prescribed arbitrarily. Such billiards are called "generalized" billiards. Some of them satisfy the so-called "universality law of reflection" and can be called for simplicity "universal billiards." The universal law of reflection allows giving the full structure of the set of infinite trajectories in the strip and an algorithm for finding such trajectories. It turns out that a spatial chaos in such "universal" billiards can be found as well as the exact value of the spatial entropy. The speaker will present interesting examples of universal billiards together with general theorems about them.

2:00 pm in 152 Henry,Tuesday, November 15, 2005

1. K-Theory
2. Introduction to Multiple Cover Formulas

Josh Mullet (UIUC Math)

Abstract: Some of you may recall question "Bergvelt One" from last time. The answer is K-Theory. If you don't remember, show up to find out what the question was! Also, time permitting, we will try again to begin discussing multiple cover formulas in Gromov-Witten theory.

3:00 pm in 3:00 pm 243 Altgeld Hall,Tuesday, November 15, 2005

Reduced Genus-One Gromov-Witten Invariants and Applications

Aleksey Zinger   [email] (SUNY Stony Brook)

Abstract: I will describe a "part" of the standard GW-invariant, which under ideal conditions counts genus-one curves without any genus-zero contribution. In contrast to the standard GW-invariant, the resulting reduced GW-invariant has the expected behavior with respect to certain embeddings. These invariants have applications to computing the standard genus-one GW-invariants of complete intersections as well as some enumerative genus-one invariants of sufficiently positive complete intersections.

3:00 pm in 241 Altgeld Hall,Tuesday, November 15, 2005

Edge-choosability of planar graphs with no adjacent triangles

Daniel Cranston (UIUC Computer Science)

Abstract: Vizing conjectured that every graph G is (\Delta(G)+1)-edge-choosable, where \Delta(G) denotes the maximum vertex degree of G; this would strengthen Vizing's Theorem that G is (\Delta(G)+1)-edge-colorable. For planar graphs, this has been proved for \Delta(G)\ge 9, and for \Delta\ge 6 when triangles sharing a vertex are forbidden and when 4-cycles are forbidden.

We have improved these results in two ways. If G is a planar graph with no 3-faces sharing an edge and \Delta(G) >= 7, then G is (\Delta(G)+1)-edge-choosable. We also show that if G is a planar graph with \Delta(G) = 5 and G has no 4-cycles then G is 6-edge-choosable. Both of our results use discharging.

3:00 pm in 341 Altgeld Hall ,Tuesday, November 15, 2005

The von Neumann algebra generated by t-gaussians

Eric Ricard (Universite de Franche-Comte)

Abstract: The t-gaussians come from a deformation of the the classical free semi-circular random variables. We give a complete description of the von Neumann algebras that they generate.