Date October 26, 2025
Venue B627, ShuangQing Complex Building (双清综合楼)
Organizers
Xiang He Tsinghua University
Fan Kang Tsinghua University
Yunhui Wu Tsinghua University
8：00-8：50 Joe Thomas (Durham University)
9：00-9：50 Yuhao Xue (IHES)
9:50-10:10 Tea Break
10：10-11：00 Yang Shen (Fudan University)
11：10-12：00 Xiang He (Tsinghua University)
Title: Optimal spectral gap for Weil Petersson random surfaces via the polynomial method
Joe Thomas (Durham University)
Abstract: The first non-zero eigenvalue, or spectral gap, of the Laplacian on a closed hyperbolic surface encodes important geometric and dynamical information about a surface. In this talk, I will discuss the typical size of the spectral gap for a random surface with large genus sampled with respect to the Weil-Petersson probability measure. In particular, I will explain joint work with Will Hide and Davide Macera where we obtain a spectral gap with a polynomial error rate. Our result uses a fusion of the polynomial method used in recent breakthroughs on the strong convergence of group representations with the trace formula for hyperbolic surfaces.
Title: Comparison between extremal length and hyperbolic length on hyperbolic Riemann surfaces
Yuhao Xue (IHES)
Abstract: On a hyperbolic Riemann surface, we discuss the relation between the extremal length and the hyperbolic length of a free homotopy class of simple closed curve. As an application, we show that the separating extremal length systole of a Weil-Petersson random hyperbolic surface is approximately (4log(g)-8loglog(g))/π, where g is the genus of surface. This is a joint work with Yunhui Wu.
Title: Constructions of expander graphs and surfaces
Yang Shen (Fudan University)
Abstract: In this talk, we will review some relative results of spectrums on graphs and hyperbolic surfaces, and introduce a new model of random graphs. Based on the study of such model, we constructed a sequence of non-compact hyperbolic surfaces with uniform positive spectral gaps. This is a joint work with Qi Guo and Bobo Hua.
Title: Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces
Xiang He (Tsinghua University)
Abstract: In this talk, we give two new upper bounds on the multiplicity of Laplacian eigenvalues for closed hyperbolic surfaces in terms of the number of short closed geodesics and the genus g. For example, we show that if the number of short closed geodesics is sublinear in g, then the multiplicity of the first eigenvalue is also sublinear in g. This makes new progress on a conjecture by Colin de Verdière in the mid 1980s. This is a joint work with Yunhui Wu and Haohao Zhang.
