I will say several words about Maryam Mirzakhani and then will present her results on asymptotic count of simple closed geodesics on a Riemann surface. At the end of the talk I will mention a development of the results of Mirzakhani in the case when the genus of the surface is large. This development is obtained jointly with V. Delecroix, E. Goujard and P. Zograf, using recent progress due to A. Aggarwal. The talk would be informal, not always rigorous, and the proofs would be omitted. As a compensation I will do my best to make it accessible to students of the third year. The text of the presentation is in English, but (if the audience does not object) the talk will be given in Russian.
A meander is a topological configuration of a line (the road) and a transverse curve (the river) in the plane, or equivalently a pair of simple closed curves on the sphere. They appear in combinatorics, theoretical physics, and computational biology. Counting meanders with a given number of intersections (bridges) is still an open problem. Similarly one can define meanders on higher genus surfaces: they correspond to topological configurations of pairs of simple closed curves on surfaces of higher genus. In this talk I will present some results, joint with V. Delecroix, P. Zograf and A. Zorich, on the counting of meanders and their variants with additional combinatorial constraints, as well as their large genus asymptotics. We will see in particular that meanders can be encoded by some other combinatorial objects, namely square-tiled surfaces, that are easier to count.
Moduli spaces of Riemann surfaces (complex algebraic curves) of genus g with n marked points carry a natural Kaehler metric called Weil-Petersson metric. This metric and the corresponding volumes play an important role in geometry and dynamics of moduli spaces, as well as in topological quantum gravity. In this talk I will explain how the Weil-Petersson volumes behave for growing g and n.
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