Tony Guttmann Department of Mathematics and Statistics, University of Melbourne The analytic structure of lattice models Wednesday 21th, April 14:05-15:55pm, Carslaw Building Room 359. Very few interesting lattice models are exactly solvable even in two dimensions, let alone three. The free-energy and spontaneous magnetization of the two-dimensional Ising model, and the specific heat of the 8-vertex model, as well as certain properties of the hard hexagon model constitute the bulk of solved models in two dimensions. Models such as the self-avoiding walk, percolation (both ordinary and directed), site (bond) trees, all remain unsolved. In three dimensions the list of solvable models is even shorter. We recently observed that most of the solved models were differentiably finite. That is, they satisify a linear differential equation with polynomial coefficients. We then proposed a numerical method for conjecturing which models are likely to satisfy such a d.e. All the above-mentioned unsolved models appear not to be differentiably finite. More recently, Andrew Rechnitzer has shown how, in some cases, these conjectures can be made rigorous, and has proved that the generating function for self-avoiding polygons, directed bond animals, bond trees and general bond animals on the square lattice are all not differentiably finite. In all cases the generating function is shown to have a natural boundary in the complex plane. The proofs can be extended to the hypercubic lattices. This feature is also shared by the susceptibility of the Ising model, though a polynomial time algorithm for the generation of the coefficients has been obtained by Orrick et. al. We will describe both the numerical and rigorous methods in this approach to lattice models.