Conférencier: Chris Kottke, New College of Florida

Résumé: The known complete non-compact hyperkahler manifolds include several families of moduli spaces, including the moduli spaces of SU(2) monopoles on R^3 and the Hilbert schemes of points on C^2, among others. Beyond dimension 4, the asymptotic geometries of these spaces are not uniform, but exhibit singularities `at infinity’, presenting a challenge for geometric analysis. I will report on a framework for geometric analysis for a broad class of `quasi-fibered boundary’ (QFB) metrics. The point of view is to consider compactifications of these spaces as manifolds with corners, which can also be thought of as resolutions of certain stratified spaces. Through a pseudodifferential parametrix construction for the Hodge de Rham operator and an analysis relating weighted L2 cohomology with intersection cohomology, we prove a new case of Sen’s conjecture for the L2 cohomology of the charge 3 monopole moduli space, and of the Vafa-Witten conjecture for the L2 cohomology of Hilbert schemes in all cases. This is joint work with F. Rochon.