One such situation is that of counting copies of a given small graph H in another graph. Although this problem is very hard in general, there is a simple counting argument that counts these copies of H in the approximation produced by the Regularity Lemma.

In this talk, we present the Regularity Lemma and outline its proof. We also discuss some applications of this lemma and the counting argument mentioned above.

Let L be a finite extension of \bold Q_p. We study the space of locally L-analytic functions on the ring of integers o = o_L, and its dual, the ring of locally L-analytic distributions. We show that this ring of distributions is isomorphic to the global functions on a rigid space [^o] parameterizing L-analytic characters of o. The space [^o] turns out to be quite interesting; if L is not \bold Q_p it is isomorphic over \bold C_p to the open unit disk, but is not a disk over any discretely valued extension of L. Our methods rely on Lubin-Tate theory and some results from Tate's classic paper on p-divisible groups. We will mention some applications to representation theory and to p-adic L functions.