Activities This Week

The philosophy of arithmetic topology, first established by Mazur, gives an analogy relating arithmetic to lower dimensional topology. Under this philosophy, one gets a dictionary, relating between Number fields and 3-manifolds, primes and knots, and p-adic fields and surfaces. In the talk I will try and explain why these surprising analogies were drawn. I will also outline some recent work of the speaker, which further establishes this connection for the local case, using tools such as graphs of groups and Bass–Serre theory.

The dimer model, also referred to as domino tilings or perfect matching, are tilings of the $Z^d$ lattice by boxes exactly one of whose sides has length 2 and the rest have length 1. This is a very well-studied statistical physics model in two dimensions with many tools like height functions and Kasteleyn determinant representation coming to its aid. The higher dimensional picture is a little daunting because most of these tools are limited to two dimensions. In this talk I will describe what techniques can be extended to higher dimensions and give a brief account of a large deviations principle for dimer tilings in three dimensions that we prove analogous to the results by Cohn, Kenyon and Propp (2000).

This is joint work with Scott Sheffield and Catherine Wolfram.