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We show that the analogue of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.

This volume contains the written account of the Bonn seminar on arith-metic geometry 2003/2004. It gives a coherent exposition of the theory of intersections of modular correspondences. The focus of the seminar is the formula for the intersection number of arithmetic modular correspondences due to Gross and Keating. Other topics treated are Hurwitz's theorem on the intersection of modular correspondences over the field of complex numbers, and the relation of the arithmetic intersection numbers to Fourier coefficients of Siegel-Eisenstein series. Also included is background material on one-dimensional formal groups and their endomorphisms, and on quadratic forms over the ring of p-adic integers.

We define Stasheff polytopes in the spaces of tropical points of cluster A -varieties. We study the supports of products of elements of canonical bases for cluster A -varieties. We prove that, for the cluster A -variety of type A , such supports are Stasheff polytopes.

We prove that the bounded derived category of coherent sheaves of the Brill-Noether variety G^r_d (C) that parametrizing linear series of degree d and dimension r on a general smooth projective curve C is indecomposable when d ≤ g(C)−1.