## Monopoles and the Sen Conjecture

#### Michael SingerUniversity College London

The Sen conjecture, made in 1994, makes precise predictions about the existence of L^2 harmonic forms on the monopole moduli spaces. For each positive integer k, the moduli space M_k of monopoles of charge k is a non-compact smooth manifold of dimension 4k, carrying a natural hyperkaehler metric. Thus studying Sen’s conjecture requires a good understanding of the asymptotic structure of M_k and its metric. This is a challenging analytical problem because of the non-compactness of M_k and because its asymptotic structure is at least as complicated as the partitions of k. For k=2, the metric was written down explicitly by Atiyah and Hitchin, and partial results are known in other cases.

The talk will introduce the main characters in this story and describe recent work aimed at proving Sen’s conjecture.

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