Mathematics

$\ell^2$ Betti numbers of countable Borel equivalence relations: Part 2

Ruiyuan (Ronnie) Chen (UIUC Math)

Abstract: We will discuss Gaboriau's 2002 paper "Invariants $\ell^2$ de relations d'équivalence et de groupes" from a descriptive set-theoretic point of view. In this second talk, we will discuss Borel fields of sets and structures (in particular, simplicial complexes and Hilbert spaces) over a countable Borel equivalence relation, and how to measure their "Borel cardinality" using Tarski's theory of cardinal algebras.

Witt groups and Real K-theory

Daniel Carmody (UIUC Math)

Abstract: Swan's theorem gives an equivalence of categories between vector bundles on a manifold and finitely-generated projective modules over the ring of continuous functions on the manifold. I'll discuss a variant of this theorem for vector bundles on a manifold with the additional data of a non-degenerate symmetric bilinear form.

Extremal problems on cycles and paths in dense graphs and hypergraphs

Alexandr Kostochka   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: A typical extremal problem in graph theory is: how many edges may have a simple n-vertex graph that does not have cycles or paths with lengths in a given set? Similar problems are important for hypergraphs for different definitions of cycles and paths (note that one can define paths and cycles in hypergraphs in several ways). We discuss some problems of this kind and recent progress on them.

Noncommutative Talagrand inequality

Marius Junge (UIUC)

Abstract: We wll show how formulate and proof mareix valued versions of Talagrand type concentration inequalities.