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
WINTER AND SPRING 2026 SCHEDULE
January 9th, 2026 NO TALK THIS WEEK
January 16th, 2026
Brian Collier (UCR)
Title: Counting connected components of character varieties.
Abstract: In this talk I will present the classification of the connected component count of character varieties of surface groups into real semisimple Lie groups. In particular, I will discuss how for compact and complex groups, the components are classified by topological invariants. while for real groups, something more interesting can occur.
January 23rd, 2026
Fernando Al Assal (University of Wisconsin Madison)
Title: Asymptotic properties of essential surfaces
Abstract: Let M be a real hyperbolic 3-manifold. A sequence of distinct (non-commensurable) essential closed surfaces in M is asymptotically geodesic if their principal curvatures go uniformly to zero. When M is closed, these sequences exist abundantly by the Kahn-Markovic surface subgroup theorem, and we will discuss the fact that such surfaces are always asymptotically dense, even though they do not always equidistribute. We will also talk about the fact that such sequences do not exist when M is geometrically finite of infinite volume. This is joint work with Ben Lowe.
Finally, time permitting, we will describe an ongoing project with Mitul Islam and Filippo Mazzoli aiming to build essential surfaces in a 2-complex dimensional complex hyperbolic manifold that have a prescribed ratio of their Toledo number by their Euler characteristic.
January 30th, 2026
Sarah Yeakel (UCR)
Title: Isovariant Homotopy Theory
Abstract: Given a finite group G and two G-spaces, an isovariant map between them is a continuous function that preserves both the G-action and isotropy subgroups. The category of G-spaces with isovariant maps has a nice notion of homotopy theory where homotopy equivalences can capture interesting structure on manifolds. In this talk, we'll discuss some homotopical results in this category and hopefully an interesting application in progress with Inbar Klang.
February 6th, 2026
Tengren Zhang (National University of Singapore)
Title: Title: A rigidity theorem for complex Kleinian groups
Abstract: The notion of hyperconvexity for representations into PGL(d,R) plays a central role in the study of positive representations in Higher Teichmuller theory; for instance, this property is responsible for ensuring good regularity properties of the limit sets of positive representations. On the other hand, hyperconvex representations into PGL(d,C) are much more rigid. In this talk, I will explain two rigidity results about hyperconvex representations into PGL(d,C). This is joint work with Richard Canary and Andrew Zimmer.
February 13th, 2026
Fred Wilhelm (UCR)
Title: PL-Stability, finiteness and dimension 4
Abstract: See PDF attached.
February 20th, 2026
Title: A geometric correspondence for reparameterizations of geodesic flows of hyperbolic groups
Abstract: Given a closed hyperbolic surface, its Teichmüller space is a classical construction that parameterizes the geometric actions of its fundamental group on the hyperbolic plane. This space can be enlarged to a metric space that encodes all its geometric actions on Gromov hyperbolic spaces. This enlarged space is infinite dimensional, as it contains all its Hitchin components, as well as other geometric/probabilistic spaces defined in terms of the fundamental group. In this talk I will present a joint work with Stephen Cantrell and Dídac Martínez-Granado, in which we describe an explicit correspondence between this space and the space of reparameterizations of the geodesic flow of the surface. Our argument relies on Green metrics, which encode the behavior of random walks on the fundamental group.
February 27th, 2026
Kostantinos Tsouvalas (Max Planck Institute for Mathematics in the Sciences. Leipzig)
Title: Anosov amalgamations and hyperbolic groups indiscrete in rank 1
Abstract: Anosov representations form a stable and rich class of discrete subgroups of semisimple Lie groups that today is recognized as the correct higher rank generalization of rank 1 convex cocompact subgroups of Lie groups. In this talk, I will discuss joint work with Subhadip Dey on constructing new classes of Anosov groups arising from amalgamations and HNN extensions of Anosov representations; many of these new examples of groups fail to embed as discrete subgroups of rank 1 Lie groups. No previous knowledge on Anosov representations will be required for this talk.
March 6th, 2026
Parker Evans (Washington University St. Louis)
Title: Geometrizing Surface Group Representations
Abstract: The Teichmuller space T(S) of a closed surface S is a moduli space where each point represents a hyperbolic metric on the surface S. Interpreted appropriately, each of these hyperbolic metrics is encoded by a representation of the fundamental group of S to PSL(2,R), the group of isometries of the hyperbolic plane. This talk concerns a similar story with the Lie group PSL(2,R) replaced by the exceptional split real Lie group G2’ of type G2. That is, we shall “geometrize” surface group representations to G2’ as holonomies of some (explicitly constructed) locally homogenous (G,X)-manifolds. Along the way, we encounter pseudoholomorphic curves in a non-compact pseudosphere that carry a (T,N,B)-framing analogous to that of space curves in Euclidean 3-space. These curves play a key role in the construction. Time permitting, we discuss how this specific G_2’ recipe relates to a broader construction that unifies other approaches to geometrize representations in rank two. This talk concerns joint work with Colin Davalo.
March 13th, 2026
Rodrigo Pereira (University of Porto)
Title: Zero velocity Lagrangians in the moduli space of Higgs bundles
Abstract: Lagrangians of the moduli space of Higgs bundles are a fundamental piece of its study
under the lens of mirror symmetry. We propose a new class of Lagrangians defined via a "zero
velocity condition" imposed on the natural action of the non-zero complex numbers on the
moduli space, motivated by a certain "infinitesimal conormal principle". After describing their
construction, we prove a formula for the number of intersection points of this Lagrangian with
the generic fiber of the Hitchin map, with the goal of determining its mirror hyperholomorphic
bundle. This is joint work with Peter Gothen and André Oliveira.
March 20th, 2026 NO TALK THIS WEEK
March 27th, 2026 NO TALK THIS WEEK (Spring break)
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
TBA
Title: TBA
Abstract: TBA
May 1st, 2026
Mason Hart (UVA)
Title:
Abstract:
May 8th , 2026
Sara Maloni (UVA)
Title: TBA
Abstract: TBA