Title: Morse theory on moduli of curves and stability phenomena

Speaker: Changejie Chen, CRM

Abstract: In 1997, Sarnak conjectured that the determinant of the Laplacian is a Morse function on the space of unit area Riemannian metrics on a given real surface, and hence induces a Morse function on its moduli space. Meanwhile, the systole function, defined as the length of a shortest essential closed geodesic with respect to the base Riemannian metric, is topologically Morse on the Teichmüller space of n-dimensional flat tori (due to Ash) and of Riemann surfaces of genus g with n marked points (due to Akrout), though it does not yield a classical 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 named eutaxy. I will talk about the index gap theorem for these functions and its homological consequences, in the form of a stability theorem for the homology of moduli spaces of stable curves. If time permits, I will explain how these Morse functions connect to Sarnak’s conjecture.