Abstracts

Week of April 27, 2025

April 28, 2025
Group Actions and Dynamics Singularity of stationary measures and Patterson-Sullivan measures for mapping class groups 1:00pm -
KT 801

Kaimanovich-Masur showed that for a non-elementary finitely generated random walk on the mapping class group of a surface, almost every sample path in the Teichmüller space converges to the boundary. This gives the hitting measure on the boundary of the Teichmüller space, which is shown to be the unique stationary measure. They conjectured that the stationary measure is singular to all Patterson-Sullivan (or, conformal) measures for the group generated by the random walk. In this talk, I will present an affirmative answer to this conjecture for a certain class of random walks, showing the singularity with all Patterson-Sullivan measures. The proof is based on our generalization of Tukia's rigidity theorem for Kleinian groups to a wide class of group actions, which also generalizes Mostow's rigidity. If time permits, we will also discuss analogous singularity results for any finitely generated Kleinian groups and some discrete subgroups of higher rank Lie groups.

This is joint work with Andrew Zimmer.
Group Actions and Dynamics Chaos in polygonal billiards 4:00pm -
KT 207

We discuss how chaos, i.e., sensitivity to initial conditions, arises in the setting of polygonal billiards. In particular, we give a complete classification of the rational polygons whose billiard flow is weak mixing in almost every direction, proving a longstanding conjecture of Gutkin. This is joint work with Jon Chaika and Giovanni Forni. No previous knowledge on the subject will be assumed.
Analysis Counterexamples to Strichartz estimates and gallery waves for the irrotational compressible Euler equation in a vacuum setting 4:00pm -
KT 205

We consider the free boundary problem for the irrotational compressible Euler equation in a vacuum setting. By using the irrotationality condition in the Eulerian formulation of Ifrim and Tataru, we derive a formulation of the problem in terms of the velocity potential function, which turns out to be an acoustic wave equation that is widely used in solar seismology. This paper is a first step towards understanding what Strichartz estimates are achievable for the aforementioned equation. Our object of study is the corresponding linearized problem in a model case, in which our domain is represented by the upper half-space. For this, we investigate the geodesics corresponding to the resulting acoustic metric, which have multiple periodic reflections next to the boundary. Inspired by their dynamics, we define a class of whispering gallery type modes associated to our problem, and prove Strichartz estimates for them. By using a construction akin to a wave packet, we also prove that one necessarily has a loss of derivatives in the Strichartz estimates for the acoustic wave equation satisfied by the potential function. In particular, this suggests that the low regularity well-posedness result obtained by Ifrim and Tataru might be optimal, at least in a certain frequency regime. To the best of our knowledge, these are the first results of this kind for the irrotational vacuum compressible Euler equations.
Geometry, Symmetry and Physics Lie-Theoretic Generalization of Some Hilbert Schemes 4:30pm -
KT 801

Haiman’s construction of the Hilbert scheme of points on the plane and its isospectral variant has several different generalizations to other reductive Lie algebras. We explore these constructions and single out a particularly interesting candidate among these. This yields a class of varieties with conical symplectic singularities. In types ABC, and conjecturally in general, the varieties we propose are hyper-Kähler rotations of (possibly singular) Calogero–Moser spaces and their fixed points correspond to two-sided cells in the Weyl group. Time permitting, I will explain how the geometry of these varieties encodes Hochschild homology of Soergel bimodules as well as topological properties of affine Springer fibers.
April 29, 2025
Geometry & Topology Totally geodesic submanifolds of Teichmüller space 4:00pm -
KT 207

The classification of totally geodesic (i.e., rigid) submanifolds is a fundamental question in Riemannian geometry. Following work of Eskin, McMullen, Mukamel, and Wright, these submanifolds have played a fundamental role in the discovery of new phenomena in Teichmüller dynamics. I will discuss recent progress towards a complete classification of these submanifolds. This is joint work with Alex Wright. No previous familiarity with the subject will be assumed.
April 30, 2025
Piatetski-Shapiro Memorial Lecture Origins of geometric Langlands 4:00pm -
KT 101

I will tell (part of) the origin story of geometric Langlands. Motivated by zeta functions, we will discuss algebraic geometry over finite fields and the “sheaf-function dictionary” following Weil, Grothendieck, and Deligne. We will introduce geometric Langlands as part of this philosophy. Finally, we will survey some recent developments extending these older ideas. The goal of the talk is to give a big picture perspective without getting bogged down in technicalities.
May 1, 2025
Quantum Topology and Field Theory Inverse Hamiltonian reduction for affine W-algebras in type A 4:30pm -
KT 801

Inverse Hamiltonian reduction refers to a series of conjectural relations between W-algebras corresponding to distinct nilpotent orbits in a Lie algebra. I will outline a proof of this conjecture in type A that relies on novel geometric methods. Along the way, we shall encounter a technique for localising vertex algebras and, time permitting, speak briefly on the deformation theory thereof. To build intuition, I shall focus on the finite type analogue of this story, where such techniques are more commonplace. This talk is based on joint work with Dylan Butson. 
May 2, 2025
Social events THESIS DEFENSE: Framed Bundles, Abelianization, and the Ooguri-Vafa Space 1:30pm -
KT 801

Moduli spaces of Higgs bundles were introduced by Hitchin in 1987 and are now ubiquitous in geometry. They are closely related to other fundamental spaces such as character varieties, and they carry an incredibly rich structure, including a hyperkähler metric (a Riemannian metric compatible with three complex structures). In this talk I will give a gentle introduction to all of these objects, and describe some recent progress towards describing the metric near the singular locus of the Hitchin fibration.
Algebra and Geometry lecture series Vertex algebra and moduli of Higgs bundles IV 3:00pm -
KT801