UIUC Dept. of Mathematics Seminar Calendar

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Tuesday, November 8, 2011

Ergodic Theory
11:00 am in 347 Altgeld Hall, Tuesday, November 8, 2011

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Submitted by jathreya.

Ilie Ugarcovici (DePaul)
The structure and spectrum of Heisenberg odometers
Abstract: Odometer systems are a well studied class of examples in the classical theory of measurable and topological dynamical systems. They can be viewed measure theoretically as cutting and stacking transformations of the unit interval. Alternatively, they can be viewed algebraically as a $Z$-action on an inverse limit space of increasing quotient groups of $Z$. Recently, Cortez and Petite defined such odometer actions for any discrete, finitely generated and residually finite group. It follows from the work of Mackey that all such actions have discrete spectrum. This talk is mainly about odometer actions defined over the discrete Heisenberg group. We provide a complete classification of Heisenberg odometers based on the structure of their defining subgroups and we construct examples of each class. We also determine explicitly those representations of the Heisenberg group which appear in the spectral decomposition of a Heisenberg odometer, as a function of the defining subgroups. This is joint work with S. Lightwood and A. Sahin.

Harmonic Analysis and Differential Equations
1:00 pm in 347 Altgeld Hall, Tuesday, November 8, 2011

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Submitted by berdogan.

Yi Hu (Department of Mathematics, University of Illinois)
Discrete Fourier restriction associated with Schrodinger and KdV type equations
Abstract: The talk I will give is based on the joint work of Prof. Xiaochun Li and I. The standard way of solving nonlinear Schrodinger or KdV equations is to rewrite them into the equivalent form of integral equations and use Picard iteration. One key tool in controlling the nonlinear term is the Strichartz estimate. However, when we consider the periodic equations, the exact periodic type analogue of continuous Strichartz estimate fails, so it forces us to find some new inequality of the same type. The periodic Strichartz estimate is in the form of exponential sums, which is also equivalent to the discrete Fourier restriction estimate. In this talk I’ll present the results we got for this type of restriction. The main step is a decomposition of the kernel which helps us get the level set estimate. Then we could get the sharp upper bound when $p$ is large. With Strichartz type estimates, as well as some classical technique, we also proved the sharp results of local well-posedness of general KdV equations and fifth order KdV type equations with polynomial nonlinear terms. In the last I will mention some topics that worth pursuing.

Number Theory
1:00 pm in 241 Altgeld Hall, Tuesday, November 8, 2011

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Submitted by berndt.

Donald Mu-Tsun Tsai (Department of Mathematics, University of Illinois)
Representations as Sums of Consecutive Integers in Sequences
Abstract: For any set $A\subseteq\mathbb N$, regarding as a strictly increasing sequence, define for each natural number $n$ the function $r_A(n)$ to be the number of ways to represent $n$ as a sum of consecutive elements in $A$. We prove that the average order of $r_{\mathbb N}(x)$ is asymptotically the same as that of $\sum r_{A_i}(x)$, where $\{A_i\}$ is a partition of $\mathbb N$, provided that the partition is not too ``fine''. We also give several applications of this result. Based on joint work with Prof. Zaharescu.

Algebra, Geometry and Combinatoric
2:00 pm in 345 Altgeld Hall, Tuesday, November 8, 2011

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Submitted by darayon2.

Iwan Duursma (Department of Mathematics, University of Illinois)
Pivot distributions and Weierstrass nongaps
Abstract: We show how various properties of a linear code are captured by the collection of its dimension-length profiles. The dimension-length profile of a matrix describes the column positions of the pivots after the matrix is brought into row echelon form. The collection of dimension-length profiles describes the distribution of the pivots considered over all possible column permutations of the matrix. A code is MDS if and only if for every permutation of the columns in the generator matrix the pivot columns are the leading columns. The zeta function of a code that we introduced in previous work will be used to describe the deviation from this distribution when the code is not MDS. We will point out how the zeta function is the exact analogue of the zeta function for curves over a finite field and how for curves it describes the distribution of Weierstrass nongaps.

Geometry Seminar
2:00 pm in 243 Altgeld Hall, Tuesday, November 8, 2011

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Peter Loeb (Department of Mathematics, University of Illinois)
End Compactifications
Abstract: The title is not a political slogan. Simple examples of end compactifications are the two point compactification of the real line and the one point compactification of the complex plane. This joint work with Matt Insall and Malgorzata Marciniak applies to quite general connected and locally connected spaces. We are using the insights of Robinson's nonstandard analysis as a powerful tool to extend and simplify previous work of Hans Freudenthal and work on groups of Isaac Goldbring. For the nonstandard extension of a metric space, the monad of a standard point x is the set of all points infinitely close to x. Monads of standard points can also be defined for non-metric spaces. Topological ends are equivalence classes of points that are not in such monads. This is the starting point that greatly simplifies the theory and illuminates various examples. The talk will begin with the needed introduction to nonstandard analysis.

Algebraic Geometry
3:00 pm in 243 Altgeld Hall, Tuesday, November 8, 2011

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Submitted by schenck.

Jesse Kass (University of Michigan)
Local Structure of the Compactified Jacobian
Abstract: The Jacobian variety of a non-singular curve is a basic tool in algebraic geometry, and a fundamental question is: How to extend this construction to singular curves? Starting with work of Igusa in the 1950's, a great deal of effort has gone into answering this question. Today we have a detailed understanding of how to assign a degenerate Jacobian to a singular curve, but our understanding of the geometry of the resulting object is less extensive. In my talk, I describe the local geometry of the Caporaso-Pandharipande degenerate Jacobian associated to a stable curve. This work is joint with Sebastian Casalaina-Martin and Filippo Viviani.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, November 8, 2011

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Submitted by west.

A.M. Raigorodskii (Lomonosov Moscow State University and Moscow Institute of Physics and Technology)
From combinatorial geometry to Ramsey theory
Abstract: This talk consider problems on the boundary of combinatorial geometry and Ramsey theory. We start with two classical and closely connected problems: Borsuk's problem and the Nelson--Hadwiger problem. The first problem is to find the the least $t$ such that every bounded non-singleton set in ${\Bbb R}^n $ can be cut into $t$ parts of smaller diameter. The second problem deals with the chromatic number of Euclidean space, which is the smallest number of colors needed to paint all the points in ${\mathbb R}^n$ so that any two points separated by distance 1 receive different colors.

The two problems admit graph theoretic interpretations that bring them close together. In the case of the Nelson--Hadwiger problem, one studies the chromatic number of a distance graph $G$, where $V(G)\subseteq {\mathbb R}^n$ and $E(G)$ consists of some pairs of vertices separated by distance $1$. In the case of Borsuk's problem, distance graphs are substituted by graphs of diameters. Here $V(G)$ is a (finite) set in ${\mathbb R}^n$ and vertices are joined by an edge if and only if the distance between them is the diameter of the set $V(G)$.

The above interpretations not only provide a common language for both problems, but also they give rise to some important questions. In particular, classical questions of Ramsey theory may be naturally asked about distance graphs and graphs of diameters.

We will first give a brief overview of the Borsuk and Nelson--Hadwiger problems.Then we will discuss Ramsey-theoretic aspects of both problems. For example, we will discuss graphs with small cliques and large chromatic numbers (small independence numbers) and graphs having both high girth and high chromatic number.