Nicolas.Smoot

Affiliation: University of Vienna

Title Of Talk: Partition Congruences and Topology

Abstract: The integer partition function p(n) counts the number of ways to add positive integers to n (p(4)=5, since we have 4, 3+1, 2+2, 2+1+1, 1+1+1+1). Ramanujan discovered that p(n) is divisible by any given power of 5 whenever 24n-1 is so divisible (He also discovered similar properties for p(n) with respect to powers of 7 and 11). This remarkable set of congruence properties derives from the fact that p(n) is counted by the coefficients of a certain modular form. Indeed, analogous divisibility properties have been found for the coefficients of a great variety of different modular forms---many of which count interesting arithmetical objects. However, the difficulty of proving these congruence families can vary substantially; some are proved with routine techniques, while others are standing conjectures. Part of the difficulty is due to the topology of the modular curve associated with a given congruence family. In this talk we discuss some new results in the theory of partition congruences, in which topological considerations play a central role.

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