Speaker: Rostyslav Kravchenko, Northwestern University

Title: Invariant and characteristic random subgroups and their applications

Date: 01/30/2018

Time: 10:20 AM - 11:10 AM

Place: C304 Wells Hall

The invariant random subgroups (IRS) were implicitly used by Stuck and Zimmer in 1994 and defined explicitly by Abert, Glasner and Virag in 2012. We recall the definition of IRS and discuss their properties. We also define the notion of characteristic random subgroups (CRS) which are a natural analog of IRSs for the case of the group of all automorphisms. We determine CRS for free abelian groups and for free groups of finite rank. Using our results on CRS of free groups we show that for some groups of geometrical nature there are infinitely many continuous ergodic IRS.

Title: Amenability of discrete groups and their actions

Date: 01/30/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The subject of amenability essentially begins in 1900's with Lebesgue. He asked whether the properties of his integral are really fundamental and follow from more familiar integral axioms. This led to the study of positive, finitely additive and translation invariant measure on reals as well as on other spaces. In particular the study of isometry-invariant measure led to the Banach-Tarski decomposition theorem in 1924. The class of amenable groups was introduced by von Neumann in 1929, who explained why the paradox appeared only in dimensions greater or equal to three, and does not happen when we would like to decompose the two-dimensional ball. In 1940's, M. Day formally defined a class of elementary amenable groups as the largest class of groups amenability of which was known to von Naumann. He asked whether there are other groups then that. Currently there are many groups that answer von Neumann-Day's question. However, in each particular case it is algebraically difficult to show that the group is not elementary amenable, and analytically difficult to show that it is amenable. The talk is aimed to discuss recent developments and approaches in the field. In particular, it will be shown how to prove amenability of all known non-elementary amenable groups using only one single approach. We will also discuss techniques coming from random walks of groups.

Title: An Introduction to Stanley's Theory of P-Partitions, II

Date: 01/30/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

In this second lecture we will describe how Stanley associated to any labelled poset P a set of partitions having a rational generating function. Its denominator only depends on the number of elements of P and the numerator can be computed using an associated set of permutations and the major index statistic. If one bounds the size of the parts, then the major index is replaced by the number of descents.

Speaker: Arie Israel, University of Texas at Austin

Title: A new proof of the finiteness principle

Date: 01/31/2018

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

The Brudnyi-Shvartsman finiteness principle is a foundational result in the study of Whitney-type extension problems. This result provides an answer to the following question: How can we tell whether there exists a Hölder-smooth function that takes prescribed values on a given (arbitrary) subset of Euclidean space? In this talk I will describe new machinery for answering this question based on the notion of the “local complexity” of a set at a given position and scale. To complete the main induction argument we must prove that the complexity of an arbitrary set is bounded uniformly by an absolute constant. This is accomplished through an elementary lemma on the stabilization of the dynamics of a 1-parameter family of non-isotropic dilations acting on the space of positive-definite matrices. We conjecture an improvement to the constants in the stabilization lemma which would result in an improvement to the best-known constants in the finiteness principle. This is joint work with A. Frei-Pearson and B. Klartag.

Reaction-diffusion equations describe a variety of physical and biological phenomena. In this talk, I begin by presenting the classical Fisher-KPP equation and its significance to ecology. I then describe recent results on other PDEs of reaction-diffusion type, including non-local equations arising in evolutionary ecology, as well as ones that model tumor growth (joint with Inwon Kim). I will highlight the mathematical challenges and techniques that arise in the analysis of these PDEs.

Donaldson polynomials are powerful invariants associated to smooth four-manifolds. The introduction by Floer of Instanton homology groups, associated to some 3-manifolds, allowed to define analogs of such polynomials for (some) four-manifolds with boundary, that have a structure similar with a TQFT.
Wehrheim and Woodward developed a framework called "Floer field theory" which, according to the Atiyah-Floer conjecture, should permit to recover Donaldson invariants from a 2-functor from the 2-category Cob_{2+1+1} to a 2-category Symp they defined, which is an enrichment of Weinstein's symplectic category.
I will describe a framework that should permit to extend such a 2-functor to lower dimensions. This framework should permit to define new invariants in Manolescu and Woodward's symplectic instanton homology (sutured theory, equivariant version). This is work in progress.

Title: Geodesic flow in non-positive curvature: An inspiration for new techniques in ergodic theory

Date: 02/01/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We discuss some recent progress in the smooth ergodic theory of geodesic flows. This talk will be suitable for a general mathematical audience, and will start with an intuitive overview of the classic results developed by luminaries such as Anosov, Bowen and Ruelle in the well understood setting of surfaces with variable negative curvature. Efforts to understand the much more difficult case of non-positive curvature were initiated by Pesin in the 1970’s. However, despite substantial successes, the picture has remained far from complete. There has been a great deal of recent progress in this area, which has required, and motivated, the development of new machinery in the abstract theory. I will give an overview of some recent developments, including:
1) General machinery developed by Vaughn Climenhaga and myself, which gives “non-uniform" dynamical criteria for uniqueness of equilibrium measures;
2) Joint work with Keith Burns, Vaughn Climenhaga and Todd Fisher, where we apply this machinery to geodesic flow on non-positive curvature manifolds;
3) If time permits, I will also mention related joint work with Jean-Francois Lafont and Dave Constantine, where we develop the theory of equilibrium measures for geodesic flow on locally CAT(-1) spaces; these are geodesic metric spaces which generalize negative curvature Riemannian manifolds by having the “thin triangle” property.