Mirzakhani's volume recursion and approach for the Witten-Kontsevich theorem on moduli tautological intersection numbers (Q2848317)

This survey article summarizes the contents of five lectures presented by the author at the 2011 Park City Mathematics Institute program on moduli spaces of Riemann surfaces. These lectures were directed at graduate students and researchers interested in this subject, and their main goal was to explain some of the very recent, highly spectacular developments in the study of the Weil-Petersson geometry of moduli spaces of curves and their applications. More precisely, the focus is on the pioneering work of \textit{M. Mirzakhani} that appeared in several papers published between 2007 and 2010 [Invent. Math. 167, No. 1, 179--222 (2007; Zbl 1125.30039); and J. Am. Math. Soc. 20, No. 1, 1--23 (2007; Zbl 1120.32008)]. In these (and in some subsequent papers, M. Mirzakhani established an explicit recursion formula for the rational coefficients of the polynomials describing the Weil-Petersson volumes of the moduli spaces of Riemann surfaces with geodesic boundary components of prescribed lengths, applied symplectic reduction to prove that these polynomial coefficients are related to the intersection numbers of certain tautological classes on the Deligne-Mumford compactification \(\overline M_{g,n}\) of the moduli space of genus \(g\) curves with \(n\) marked points, and finally used these results to give a new proof of the famous Witten-Kontsevich formula in this context [\textit{M. Kontsevich}, Commun. Math. Phys. 147, No. 1, 1--23 (1992; Zbl 0756.35081)].NEWLINENEWLINE All this is lucidly depicted in the paper under review, the structure of which is as follows:NEWLINENEWLINE After a very enlightening introduction to the topic and its history, Lecture 1 provides the necessary background material on Teichmüller spaces, mapping class groups, hyperbolic geometry, Weil-Petersson metrics, moduli spaces of Riemann surfaces, and tautological cohomology classes. Moreover, a first overview of Mirzakhani's main results is given at the end of this section. While Lecture 2 reviews G. McShane's universal identity for a sum of lengths of simple geodesics on a punctured torus and its generalization by M. Mirzakhani to hyperbolic surfaces with geodesic boundaries of given lengths, Lecture 3 describes Mirzakhani's above-mentioned Weil-Petersson volume formula for the corresponding moduli spaces. In this context, \textit{G. McShane's} identity [Invent. Math. 132, No. 3, 607--632 (1998; Zbl 0916.30039)] as well as Mirzakhani's generalization play a crucial role. Lecture 4 explains Mirzakhani's symplectic reduction procedure for the underlying Teichmüller spaces of hyperbolic surfaces, and Lecture 5 finally discusses the applications of these results to the pattern of intersection numbers of tautological classes on \(\overline M_{g,n}\) that is, to the former Witten-Kontsevich conjecture.NEWLINENEWLINE In this context, some further recent approaches by M. Mulase, B. Safnuk, N. Do, A. Okounkov, R. Pandharipande, and others are also briefly touched upon. The present survey article ends with a section containing the questions for the problem sessions accompanying the single lectures. Apart from M. Mirzakhani's original papers cited above, the author's recent book [Families of Riemann surfaces and Weil-Petersson geometry. CBMS Regional Conference Series in Mathematics 113. Providence, RI: American Mathematical Society (AMS), vii, 118 p. (2010; Zbl 1198.30049)] is a general reference for more details and further reading.NEWLINENEWLINEFor the entire collection see [Zbl 1272.30002].