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Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes. (English) Zbl 1322.32011

Farkas, Gavril (ed.) et al., Handbook of moduli. Volume I. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-257-2/pbk; 978-1-57146-265-7/set). Advanced Lectures in Mathematics (ALM) 24, 217-258 (2013).
Summary: Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil-Petersson form. M. Mirzakhani [J. Am. Math. Soc. 20, No. 1, 1–23 (2007; Zbl 1120.32008)] proved that Weil-Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers on moduli spaces of curves. In this survey article, we discuss these results as well as some consequences and applications.
For the entire collection see [Zbl 1260.14001].

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H15 Families, moduli of curves (analytic)
53D30 Symplectic structures of moduli spaces

Citations:

Zbl 1120.32008
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