Activities This Week

In the talk, I will discuss the Turner-Turaev formalism for unoriented Topological Quantum Field Theory (TQFT). Building upon this formalism, I will introduce an analogous version for (d+1)-dimensional TQFT for pro-p Poincare duality groups. In the case of d = 1, this enables us to study cobordisms and TQFTs for both the maximal pro-p quotient of absolute Galois groups of p-adic fields and pi_1(X)^p, the pro-p completions of fundamental groups of surfaces. This generalisation gives a framework for arithmetic TQFTs and strengthens the analogies within arithmetic topology, which relates p-adic fields to surfaces (oriented mod p^r). I will explain the classification of TQFTS for the (1+1)-dimensional case, in terms of Frobenius algebras with some extra structure.

If time permits, I will explain how we define a Dijkgraaf Witten like theory, to get formulas for counting G-covers of X, where X is either a surface, or a p adic field, and G is a p-group (these formulas are similar to the ones given by Mednykh for surfaces using TQFTs, and by Masakazu Yamagishi using a more algebraic approach). I will also try to outline how we plan to also get similar formulas for Hom(\pi_1(X)^p,G), where G=GL_n(k) for k=F_{p^r} or Z/p^rZ.

The talk is based on joint work with Oren Ben-Bassat.

Cornulier conjectured that two completely solvable Lie groups are quasiisometric if and only if they are isomorphic. This is a very difficult and very open problem. In this talk I will present some of the structure theory of solvable Lie groups and focus on the importance of sublinear distortions to this theory. I will review some of the important work of Cornulier and Tessera on Dehn functions of these groups. Finally, I will present recent results that contribute to their QI classification, which are interesting (also) because they are based on a (sublinear) weak form of QI.

Based on joint work with Gabriel Pallier.