James Parkinson, School of Mathematics and Statistics, The University of Sydney Wednesday 12th May 14:05-14:55pm, New Law School Seminar 030 (Building F10). "Trees" are ubiquitous in pure and applied mathematics, arising in such diverse areas as the study of branching processes and the representation theory of the special linear group SL_2 over a p-adic field. A homogeneous tree is a tree in which each vertex has the same valency. A random walk on the vertices of a homogeneous tree is called isotropic if the transition probabilities p(x,y) of the walk only depend on the graph distance between x and y. In this talk we outline how some basic harmonic analysis can be used to derive a precise asymptotic formula for the n-step transition probabilities p_n(x,y) of an isotropic random walk as n approaches infinity, with x and y fixed vertices (ie, a local limit theorem). This work is classical, dating back (at least) to S. Sawyer in 1978. But the technique is rather robust: Recently it has been vastly generalised to study random walks on "affine buildings" by Cartwright, Woess, Parkinson, Shapira and others, thereby giving local limit theorems on semi-simple Lie groups over p-adic fields.