The analytic structure of lattice models --- looking inside unsolved problems.

Abstract: What it means for a problem to be "solved" really depends on who you talk to; you can square the circle if you are happy to use more than a ruler and compass. In this talk, I will describe work that has lead to "unsolvability" results for a wide class of combinatorial problems that arise in statistical physics.

The starting point for these results are techniques which allow us to peek inside the solutions of these problems without actually having to solve them. The structure of the solutions proves that they do not belong to the most pervasive class of functions in mathematical physics, so called differentiably-finite functions. This, in turn, shows that the models cannot be solved using many traditional combinatorial methods.