Spring 2012

- Date:
**Wednesday, 1/18/12 in Stevenson 1431****Stuart White, University of Glasgow**- Title:
**Kadison-Kastler stability for operator algebras** - Abstract: Kadison and Kastler equipped the set of all C*-subalgebras of B(H) with a natural metric and conjectured that sufficiently close algebras should be isomorphic. This is a uniform metric: two algebras A and B are close in the Kadison-Kastler metric if every operator in the unit ball of A can be well approximated in the unit ball of B and vice versa. Kadison and Kastler's conjecture was established in the 70's when one algebra is an injective von Neumann algebra. In this talk I'll give discuss some recent progress for non-injective von Neumann algebras (mainly arising as crossed products) and discuss the connections between this problem and the similarity problem.

- Date:
**1/20/12****Colloquium Lecture in Stevenson 5211: Noah Snyder, Columbia University**

- Date:
**2/10/12****Adebisi Agboola, UC Santa Barbara**- Title:
**L-functions and Galois structure** - Abstract: This will be a colloquium-style talk giving a survey of certain classical results concerning the Galois structure of rings of integers. Starting with an elementary result in Galois theory (the normal basis theorem), one is led to a remarkable connection between the Galois module structure of rings on integers on the one hand, and the behaviour of certain analytical objects called Artin L-functions. I shall discuss this connection and how it fits into a much broader picture than one might have at first suspected.

- Date:
**2/17/12****Darren Creutz, Vanderbilt University**- Title:
**Property (T) for Certain Totally Disconnected Groups Related to a Conjecture of Margulis and Zimmer** - Abstract: I will present some of my work on reduced cohomology and property (T) for totally disconnected groups and dense countable subgroups. The primary application of this work is to show property (T) for a class of totally disconnected groups arising from a conjecture of Margulis and Zimmer regarding the classification of all commensurated subgroups of lattices in higher-rank Lie groups. The key idea in our work is to expand on Kleiner's work on the energy of a cocycle (the idea of which goes back to Mok) and derive a very general result about energy and reduced cohomology. This is joint work with Yehuda Shalom.

- Date:
**2/24/12****Peter Sarnak, Princeton University**- Title:
**Thin integer Matrix Groups** - Abstract: Infinite index subgroups of integer matrix groups like SL(n,Z) which are Zariski dense in SL(n), arise in geometric diophantine problems (eg Integral Apollonian packings), as monodromy groups associated with families of algebraic varieties, as reflection groups... One of the key features needed when applying such groups in number theoretic problems is that the congruence graphs associated with these groups are "expanders". We will introduce and explain these ideas and review some recent developments and applications.

- Date:
**Monday, 2/27/12****Ralph Kaufmann, Purdue University**- Title:
**The arc operad and its possible relation to planar algebras.** - Abstract: Both the arc operad and planar algebras are built from graphs on surfaces. Although similar in nature they were constructed with different aspects of geometry and applications in mind. With hindsight there is a striking similarity in one particular application. To elucidate this possible connection, we will proceed by introducing the arc operad and its discretization. The latter gives rise to actions on the Hochschild complex of a (Frobenius) algebra. These operations include Deligne's conjecture, its cyclic version and string topology. This discretization is also what is closely related to planar diagrams as we shall discuss.

- Date:
**Monday, 3/26/12****Thomas Sinclair, UCLA**- Title:
**II_1 factors of negatively curved groups.** - Abstract: I will present some structural results for II_1 factors of products of hyperbolic groups and their ergodic actions. Applications will be given to the measure equivalence theory of such groups. This is joint work with Ionut Chifan and Bogdan Udrea.

- Date:
**4/6/12****Darren Creutz, Vanderbilt University**- Title:
**Stabilizers for Ergodic Actions of Commensurators, I.** - Abstract: A strong generalization of the Margulis Normal Subgroup Theorem, due to Stuck and Zimmer, states that any properly ergodic finite measure-preserving action of an irreducible lattice in a center-free semisimple Lie group with all simple factors of higher-rank is essentially free. We present a similar result generalizing the Creutz-Shalom Normal Subgroup Theorem for Commensurators of Lattices to actions of commensurators. As a consequence, we show that S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) and that lattices in certain product groups do as well. In particular, any nontrivial ergodic measure-preserving action of $\mathrm{PSL}_{n}(\mathbb{Q})$, for $n \geq 3$, is essentially free. This is joint work with Jesse Peterson.

- Date:
**Monday, 4/9/12****Jesse Peterson, Vanderbilt University**- Title:
**Stabilizers for Ergodic Actions of Commensurators, II.** - Abstract: A strong generalization of the Margulis Normal Subgroup Theorem, due to Stuck and Zimmer, states that any properly ergodic finite measure-preserving action of an irreducible lattice in a center-free semisimple Lie group with all simple factors of higher-rank is essentially free. We present a similar result generalizing the Creutz-Shalom Normal Subgroup Theorem for Commensurators of Lattices to actions of commensurators. As a consequence, we show that S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) and that lattices in certain product groups do as well. In particular, any nontrivial ergodic measure-preserving action of $\mathrm{PSL}_{n}(\mathbb{Q})$, for $n \geq 3$, is essentially free. This is joint work with Darren Creutz.

- Date:
**4/13/12****Marius Dadarlat, Purdue University**- Title:
**Topological invariants of spaces and groups obtained by matricial deformations.** - Abstract: Starting with elementary examples, we shall discuss how deformations into matrix algebras of the algebra of continuous functions on a space X relate to the K-theory of X. We will then explore similar phenomena in the context of discrete groups and their convolution algebras and will elaborate on connections with index theory. If time allows, we will draw some parallels with the theory of topological insulators.

- Date:
**4/20/12****Michael Hull, Vanderbilt University**- Title:
**Quasi-coycles on groups with hyperbolically embedded subgroups.** - Abstract: (Joint work with Denis Osin). Bounded cohomology has been used in proving many different types of rigidity results, including measure equivalence and orbit equivalence of groups as well as rigidity of group von Neumann algebras. Proving the non-triviality of the second bounded cohomology of a group is often done by constructing quasi-cocycles. We give a general construction for building quasi-cocycles on the class of groups which contain hyperbolically embedded subgroups, a generalized version of relative hyperbolicity introduced recently by Dahmani, Guirardel, and Osin. As a corollary we can shows that any group G containing a (proper and infinite) hyperbolically embedded subgroup has infinite dimensional second bounded cohomology with coefficents in l^2(G).

- End of Spring Semester.