During the Winter 2026 quarter, all of our meetings will be held in-person in Skye Hall 268.
Talks will run from 11a-12p PT, but feel free to show up early for socializing with the speaker.
Current organizers: Brian Collier, Filippo Mazzoli
SPRING 2026 SCHEDULE
April 3rd, 2026
Beibei Liu (Ohio State University)
Title: New insights of the critical exponent
Abstract: Hyperbolic geometry, together with group actions, is a rich field with a long history. In this talk, I will present a connection between the critical exponent of the group action and the topology and geometry of hyperbolic manifolds, focusing on using the critical exponent to characterize the finiteness property of topology and geometry. For deformations of surface group representation in PSL(2, C) and PSL(3, R), I will compare the change of the critical exponent and the deformation of the corresponding geometry on the surface. This includes joint work with Wang and ongoing projects with Martone, Mitul and Pozzetti.
April 10th, 2026
Joaquin Lema (Boston College)
Title: Epstein-Poincare surfaces for G-opers
Abstract: A complex projective structure on a surface is a (P^1, PGL_2 (C))-structure in the sense of Thurston, where P^1 is the projective plane, and PGL_2 (C) is the group of Mobius transformations. Such structures play a key role in the study of convex cocompact hyperbolic 3-manifolds.
In higher-rank complex semisimple Lie groups, there is the notion of a G-oper associated to a Riemann surface X, which is, in some sense, a generalization of complex projective structures.
With this viewpoint in mind, the goal of the talk is to explain how we can import some objects from the hyperbolic world (most notably, Epstein-Poincaré surfaces), and what they allow us to say about the geometry of opers.
April 17th, 2026
Sami Douba (University of Bonn)
Title: Zariski-closures of linear reflection groups
Abstract: We show that linear reflection groups in the sense of Vinberg are often Zariski-dense in PGL(n). Among the applications are examples of low-dimensional closed hyperbolic manifolds whose fundamental groups virtually embed as Zariski-dense subgroups of SL(n,Z), as well as some one-ended Zariski-dense subgroups of SL(n,Z) that are finitely generated but infinitely presented, for all sufficiently large n. This is joint work with Jacques Audibert, Gye-Seon Lee, and Ludovic Marquis.
April 24th , 2026 NO TALK THIS WEEK
May 1st, 2026
Mason Hart (UVA)
Title: Understanding domains of discontinuity for Anosov representations via circle actions
Abstract: The theory of Fuchsian representations is a classic subject in Teichmüller theory with noteworthy relationships with complex analysis and hyperbolic 3-manifolds. In the last two decades, there has been a driving force to better understand Anosov representations, a generalization of Fuchsian to higher rank Lie groups. One significant feature is that Anosov representations admit domains of discontinuity inside compact homogeneous spaces called flag varieties. In this talk, I will describe Fintushel's classification of circle actions on 4-manifolds and explain how this tool can be used to determine the topology of these domains of discontinuity in 6 cases. Time permitting, I may discuss my other work concerning the h-cobordism class of the fiber in the general case.
May 8th , 2026 NO TALK THIS WEEK
May 15th, 2026
Yusen Xia (UCSB)
Title: On the PPW Conjecture for Hopf-Symmetric Sets in Non-compact Rank One Symmetric Space
Abstract: Comparison results on eigenvalues of Laplace Operators are important and interesting topics in spectral geometry. In the 1990s Asbaugh and Benguria proved the Payne-Polya-Weinberger (PPW) inequality, which says balls maximize the ratio of the first two Dirichlet eigenvalues of a bounded domain in Rn. A reformulated version has been proved in the hemisphere and also hyperbolic space. Recently we generalize the PPW inequality to Hopf symmetric sets in non-compact rank one symmetric spaces.
May 22nd , 2026 NO TALK THIS WEEK