The moduli space of Riemann surfaces and the Weil-Petersson metric
Connections for Women: Holomorphic Differentials in Mathematics and Physics August 15, 2019 - August 16, 2019
Location: MSRI: Simons Auditorium
53Cxx - Global differential geometry [See also 51H25, 58-XX; for related bundle theory, see 55Rxx, 57Rxx]
The subject of this talk is the moduli space of Riemann surfaces, which is the set of isometry classes of constant curvature metrics on a surface. The cotangent space of the moduli space is given by holomorphic quadratic differentials, and there is a natural Weil-Petersson metric defined by an $L^2$-type pairing. I will discuss the behavior of the moduli space when approaching the boundary of the Deligne-Mumford compactification, and show how to use tools from microlocal analysis to understand the degeneration of the Weil-Petersson metric.
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