Morse theory on moduli spaces

algebraic geometryanalysis of PDEsalgebraic topologycomplex variablesdifferential geometrygeneral topologygeometric topologyK-theory and homologymetric geometrysymplectic geometry

Audience: researchers in the topic

Comments: Sarnak conjectured in the 1990s that the determinant of the Laplacian is a Morse function on the space of unit area Riemannian metrics on a given surface, and hence induces a Morse function on the moduli space of Riemann surfaces.

It is known that the systole function, defined as the length of a shortest closed geodesic with respect to the base metric, is topologically Morse on the moduli space M_{g,n}. However, it does not generate a Morse theory.

In this talk, I will introduce a family of Morse functions, defined as weighted exponential averages of all geodesic-length functions, on the Deligne-Mumford compactification (M_{g,n} bar). These functions are compatible with the Deligne-Mumford stratification and the Weil-Petersson metric, and their critical points can be characterized by a combinatorial property.

I will finally talk about homological consequences of hyperbolic geometry results via Morse theory, including a stability theorem. If time permits, I will explain how these Morse functions connect to Sarnak’s conjecture.

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CRM - Séminaire du CIRGET / Géométrie et Topologie

Series comments: Hybrid seminar of geometry and topology. Laboratory : CIRGET - www.cirget.uqam.ca The homepage of the seminar is www.cirget.uqam.ca/fr/seminaires.html

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