Activities This Week

We will explain the Geometrization Theorem proved by Perelman in 2003 and we will talk about the Virtual Fibering Theorem proved several years ago by Ian Agol and Dani Wise. I will not assume any previous knowledge of 3-manifold topology.

Abstract: In extremal graph theory, we often consider large graphs that are in the limit uniquely determined by finitely many densities of their subgraphs. The corresponding limits (so-called graphons) are called finitely forcible. Motivated by classical results in extremal combinatorics as well as by recent developments in the study of finitely forcible graphons, Lovasz and Szegedy made some conjectures about the structure of such graphons. In particular, they conjectured that the topological space of typical points of every finitely forcible graphon is compact and finitely dimensional. In joint results with D. Kral, T. Klimosova, and J. Volec, we could disprove both conjectures.

The Temperley-Lieb algebras are a family of finite dimensional algebras that are quotients of the symmetric groups algebras, or more generally the Iwahori-Hecke algebras. They appear in a number of areas of mathematics, including statistical mechanics, knot theory, quantum groups, and subfactors. We review their representation theory and give some results on an infinite dimensional generalization.

The talk discusses a convexity structure on metric spaces which we call sheltered sets. This structure arises in the study of the dynamics of the maximum cellular automaton over the binary alphabet on finitely generated groups. I will discuss relations to horoballs and dead ends in groups and present many open questions. This is work in progress with Tom Meyerovitch.