“模空间的几何研讨会”学术报告
报告一
报告题目：Counting simple geodesics on surfaces
报告时间：2018年4月21日（星期六）13:30—14:30
报告地点：苏州大学本部维格堂319
报告人：Greg McShane (Universite Grenoble Alpes)
报告提要：I will discuss the work of Mirzakhani, Rivin, myself and others on counting
simple geodesics and applications to integration over moduli space.
报告二
报告题目：McShane-type identities for quasifuchsian representations of nonorientable
surfaces
报告时间：2018年4月21日（星期六）14:50—15:50
报告地点：苏州大学本部维格堂319
报告人：Yi Huang (清华大学丘成桐数学科学中心)
报告提要： We adapt Bers' double uniformization for nonorientable surfaces and show
that the space QF(N) of quasifuchsian representations for a nonorientable surface N is the
Teichmueller space T (dN) of an orientable double of N. We then utilize the inherited complex
structure of QF(N) = T(dN) to show that Norbury's McShane identities for nonorientable
cusped hyperbolic surfaces N generalizes to quasifuchsian representations and punctured torus bundles for N.
报告三
报告题目：Growth of the Weil--Petersson inradius of moduli space
报告时间：2018年4月21日（星期六）16:20—17:20
报告地点：苏州大学本部维格堂319
报告人：吴云辉 (清华大学丘成桐数学科学中心)
报告提要：In this paper we study the systole function along Weil--Petersson geodesics.
We show that the square root of the systole function is uniformly Lipschitz on Teichmuller
space endowed with the Weil--Petersson metric. As an application, we study the growth of the Weil--Petersson inradius of moduli space of Riemann surfaces of genus g with n punctures as a function of g and n. We show that the Weil--Petersson inradius is comparable to p ln g with respect to g, and is comparable to 1 with respect to n. Moreover, we also study the asymptotic behavior, as g goes to infinity, of the Weil--Petersson volumes of geodesic balls of finite radii in Teichmuller space. We show that they behave like o(1/g^{(3-ε)g}) as g goes to infinity, where ε > 0 is arbitrary.
报告四
报告题目：The monodromy of meromorphic projective structures
报告时间：2018年4月22日（星期日）08:30—09:30
报告地点：苏州大学本部维格堂319
报告人：Dylan G.L. Allegretti (University of Sheffield)
报告提要：A projective structure on an oriented surface S is an atlas of charts mapping
open subsets of S into the Riemann sphere. There is a natural map from the space of projective structures to the PGL(2; C) character variety of S which sends a projective structure
to its monodromy representation. In this talk, I will describe a meromorphic analog of this
construction. I will introduce a moduli space parametrizing projective structures with poles at
a discrete set of points. I will explain how, in this setting, the object parametrizing monodromy
data is a type of cluster variety. This is joint work with Tom Bridgeland.
报告题目：Quantum traces for Fock--Goncharov coordinates
报告时间：2018年4月22日（星期日）09:45—10:45
报告地点：苏州大学本部维格堂319
报告人：Daniel Charles Douglas (University of Southern California)
报告提要：We describe work-in-progress generalizing the SL_2 quantum trace map of Bonahon and Wong (2010) to the case of SL_n. The SL_2 quantum trace is a homomorphism from the Kauffman bracket skein algebra of a punctured surface to a certain noncommutative algebra which can be thought of as a quantum Teichmuller space. The construction is modeled on the classical trace of monodromies of hyperbolic structures on surfaces. Our current work focuses on SL_3, where convex projective structures play the central role, as developed by Fock and Goncharov. Another distinction is the appearance of the HOMFLY-PT skein algebra in place of the Kauffman bracket skein algebra.
报告六
报告题目：Generalized McShane's identities on strictly convex projective structure
and beyond
报告时间：2018年4月22日（星期日）11:00—12:00
报告地点：苏州大学本部维格堂319
报告人：孙哲 (清华大学丘成桐数学科学中心)
报告提要： Goncharov and Shen introduced a Landau--Ginzberg potential on the Fock--Goncharov A_{G;S} moduli space, where G is a semisimple Lie group and S is a ciliated surface.
They used the potential to formulate a mirror symmetry via Geometric Satake Correspondence. This potential is the Markoff equation for A_{PSL(2;R); S_{1;1}}. When S = S_{g;m}, such a potential can be written as a sum of rank(G)*m partial potentials. We obtain a family of generalized McShane's identities by splitting these partial potentials for A_{PSL(n;R); S_{g;m}}. We also find some interesting new phenomena in higher rank case, like triple ratio is bounded in mapping class group orbit. As applications, we find a generalized collar lemma which involves λ_1 / λ_2 length spectral, discreteness of that spectral etc. In further research, we would like to ask: how can we integrate to obtain the generalized Mirzakhani's topological recursion with W_n constraint? (Joint work with Yi Huang).