Mathematics

The work of Goldston-Pintz-Yildirim on gaps between successive primes

Kevin Ford (UIUC Math )

Abstract: Dan Goldston, Janos Pintz and Cem Yildirim recently made a huge breakthrough in the study of small gaps between prime numbers. We give a survery of their results and techniques, and describe how the new methods relate to classical sieve theory.

The coset space of a Borel subgroup of a Polish group

Slawomir Solecki (UIUC Math)

Abstract: The following type of question was first considered by Kechris and Louveau. Let H be a Borel group. Consider all orbit equivalence relations induced by continuous actions of H on Polish spaces. Does the complexity of these equivalence relations determine whether or not H admits a (necessarily unique) Polish group topology whose Borel structure coincides with the original Borel structure on H? Complexity of equivalence relations is to be calibrated here by comparing them, in terms of Borel reducibility, with some canonical equivalence relations. I will show how to answer this question for Abelian H. In fact, I will present general theorems leading to this result, which apply also to many non-Abelian groups.

Homology and bounded dynamics for surface homeomorphisms

Christopher Leininger (UIUC Department of Mathematics)

Abstract: I'll discuss why a trivial action on homology for a pseudo-Anosov homeomorphism of a closed surface implies a positive lower bound for its topological entropy, independent of genus. I'll define all the terms, explain the significance, and sketch the proof which draws on complex analysis, hyperbolic geometry, and combinatorial topology--a potpourri of geometry! This is joint work with Benson Farb and Dan Margalit.

A description of sofic groups in terms of nonstandard analysis, cont.

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.

The editing distance in graphs

Ryan Martin (Iowa State University)

Abstract: An edge-operation on a graph G is defined to be either the deletion of an existing edge or the addition of a nonexisting edge. Given a family of graphs F, the editing distance from G to F is the smallest number of edge-operations needed to modify G into a graph from F. In this talk, we fix a graph H and consider F_n,H, the set of all graphs on n vertices that have no induced copy of H. We provide bounds for the maximum over all n-vertex graphs G of the editing distance from G to F_n,H, using an invariant we call the binary chromatic number of the graph H. We give asymptotically tight bounds for that distance when H is self-complementary and exact results for several small graphs H. This is joint work with Maria Axenovich and André Kézdy.

Understanding multiple Dirichlet series

Benjamin B. Brubaker (Stanford University )

Abstract: Multiple Dirichlet series (MDS) are Dirichlet series in several complex variables. The MDS are said to be "perfect" if they possess meromorphic continuation to the entire complex space in which they are defined. Ad hoc examples of perfect MDS were given in the 90's by Bump, Friedberg, and Hoffstein, with important applications to the Birch-Swinnerton-Dyer conjecture, Langlands functoriality and moment conjectures. More recent examples with similar applications have appeared in the last couple of years due to Chinta, Diaconu, and myself. In this talk, I'll briefly mention these examples and their applications. Then I'll explain how in very recent work we have developed a theoretical framework for these MDS (both results and additional conjectures), producing large classes of new perfect multiple Dirichlet series coming from the harmonic analysis of Eisenstein series associated to certain Lie groups (which we'll review along the way). The description of these Dirichlet series involves combinatorics of Lie algebra representations, such as Young tableaux and Gelfand-Tsetlin patterns, leading to surprising new applications of MDS. The work described is joint with Dan Bump and Sol Friedberg. Host: S. Ahlgren

Is the whole really greater than the sum of its parts? Exploring partitions of numbers.

Stephanie Treneer (UIUC Department of Mathematics)

Abstract: A partition of a positive integer n is a sequence of positive integers that sum to n. The partition function p(n) counts the partitions of n without regard to order. This deceptively simple function has led to a rich theory. We'll look at two elementary methods for analyzing partitions: Ferrers graphs and generating functions, and then briefly discuss how the theory of modular forms has led to some recent surprising results about p(n).