L6

We will look at a number of interesting examples — some proven, others merely conjectured — of Hamburger moment sequences in combinatorics. We will consider ways in which this positivity may be expected: for instance, in different types of combinatorial statistics on perfect matchings that encode moments of noncommutative analogues of the classical Central Limit Theorem. We will also consider situations in which this positivity may be surprising, and where proving it would open up new approaches to a class of very hard open problems in combinatorics.

Abstract: It’s a classical result by Huber that the number of closed geodesics of length bounded by L on a closed hyperbolic surface S is asymptotic to exp(L)/L as L grows. This result has been generalized in many directions, for example by counting certain subsets of closed geodesics. One such result is the asymptotic growth of those that are homologically trivial, proved independently by both by Phillips-Sarnak and Katsura-Sunada. A homologically trivial curve can be written as a product of commutators, and in this talk we will look at those that can be written as a product of g commutators (in a sense, those that bound a genus g subsurface) and obtain their asymptotic growth. As a special case, our methods give a geometric proof of Huber’s classical theorem. This is joint work with Juan Souto. 

We introduce a new class of groups with Thompson-like group properties. In the surface case, the asymptotic mapping class group contains mapping class groups of finite type surfaces with boundary. In dimension three, it contains automorphism groups of all finite rank free groups. I will explain how asymptotic mapping class groups act on a CAT(0) cube complex which allows us to show that they are of type F_infinity. 

This is joint work with Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig and Xaolei Wu.

The classification of measure preserving actions up to orbit equivalence has attracted a lot of interest in the last 25 years. The goal of this talk is to survey the major discoveries in the field, including Popa's cocycle and orbit equivalence superrigidity theorem and discuss some recent superrigidity results for dense subgroups of Lie groups acting by translation.

A central question in infinite group theory is to determine how much global information about a group is encoded in its set of finite quotients. In this talk, we will discuss this problem in the case of algebraically fibered groups, which naturally generalise fundamental groups of compact manifolds that fiber over the circle. The study of such groups exploits the relationships between the geometry of the classifying space, the dynamics of the monodromy map, and the algebra of the group, and as such draws from all of these areas.

It was shown by Balasubramanya that any acylindrically hyperbolic group (a natural generalisation of a hyperbolic group) must act acylindrically and non-elementarily on some quasi-tree. It is therefore sensible to ask to what extent this is true for trees, i.e. given an acylindrically hyperbolic group, does it admit a non-elementary acylindrical action on some simplicial tree? In this talk I will introduce the concepts of acylindrically hyperbolic and acylindrically arboreal groups and discuss some particularly interesting examples of acylindrically hyperbolic groups which do and do not act acylindrically on trees.

Oka theory is about the validity of the h-principle in complex analysis and geometry. In this expository lecture, I will trace its main developments, from the classical results of Kiyoshi Oka (1939) and Hans Grauert (1958), through the seminal work of Mikhail Gromov (1989), to the introduction of Oka manifolds (2009) and the present state of knowledge. The lecture does not assume any prior exposure to this theory.

We will survey work of Birman-Craggs, Johnson, and Sato on the abelianization of the level 2 congruence group of the mapping class group of a surface, and of the corresponding Torelli group. We will then describe recent work of Lewis providing a common framework for both abelianizations, with applications including a partial answer to a question of Johnson.