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Week of February 9, 2025

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February 10, 2025
Group Actions and Dynamics [3] Exponential mixing and counting conjugacy classes for Anosov subgroups Pratyush Sarkar - UCSD 4:00pm -
KT 207
Analysis [4] Multiscale decompositions of Hardy spaces Jacques Peyriere - University of Paris Orsay 4:00pm -
KT 809B
Geometry, Symmetry and Physics [5] Geometric Eisenstein Series on Fargues–Fontaine Curves Linus Hamann - Harvard University 4:30pm -
KT 801
February 11, 2025
Geometry & Topology [6] Geometry of the regular representation of hyperbolic groups Antoine Song - California Institute of Technology 4:00pm -
KT 207
February 12, 2025
Colloquium [7] From harmonic maps to hyperbolic surfaces via random matrices Antoine Song - California Institute of Technology 4:00pm -
KT 101
February 13, 2025
Analysis [4] Construction of multi-soliton solutions for semilinear equations in dimension 3 Istvan Kadar - Princeton University 4:00pm -
KT 207
Quantum Topology and Field Theory [8] Multiple polylogarithms and the Steinberg module Daniil Rudenko - University of Chicago 4:30pm -
KT 801
February 14, 2025
Friday Mornings [9] TBA 10:00am -
KT 801

Abstracts

Week of February 9, 2025

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February 10, 2025
Group Actions and Dynamics [3] Exponential mixing and counting conjugacy classes for Anosov subgroups 4:00pm -
KT 207

The celebrated prime geodesic theorem gives the asymptotic formula for the number of primitive closed geodesics according to their length in a closed hyperbolic manifold, i.e., for uniform lattices in SO(n, 1). What’s more, the error term is power-saving. This was generalized with the error term for convex cocompact subgroups by Naud and Stoyanov. It is natural to wonder whether one can generalize this further to the higher rank setting such as SL(n, R). This was done without the error term for Anosov subgroups by Sambarino. In keeping with a principle from Margulis's thesis, we go further and establish exponential mixing of an appropriate dynamical system and use that to produce a power-saving error term.

Analysis [4] Multiscale decompositions of Hardy spaces 4:00pm -
KT 809B

We consider orthogonal expansions of holomorphic functions in
the unit disk or in the upper half-plane whose terms are Blaschke
products (the so-called phase unwinding expansions). We also construct
holomorphic wavelets in the upper half-plane and use them to get some
explicit phase unwinding.

Geometry, Symmetry and Physics [5] Geometric Eisenstein Series on Fargues–Fontaine Curves 4:30pm -
KT 801

Let G be a reductive group over the p-adic numbers with P = MU a parabolic subgroup. A basic fact in smooth representation theory is that parabolic induction preserves the property of being admissible.  In this talk, we will discuss the analogue of this in the geometrization of the local Langlands program. In particular, smooth representations will be replaced by sheaves on Bun_G, the moduli stack of G-bundles on the Fargues–Fontaine curve, and parabolic induction will be replaced by a geometric Eisenstein functor carrying sheaves on Bun_M to sheaves on Bun_G. The property of being admissible translates into the rather bizarre property of being ULA over a point, which is a new phenomenon native to analytic variants of the geometric Langlands program. The main result we will discuss is that the geometric Eisenstein functor sends sheaves which are ULA over a point on Bun_M to sheaves which are ULA over a point on Bun_G. This generalizes the basic fact on admissibility mentioned at the beginning, and much more interestingly shows that various gluing functors on Bun_G send admissible representations to admissible representations.  Along the way, we hope to explain some of the similarities and differences between the usual geometric Langlands programs and the Fargues–Scholze geometric Langlands program, mostly stemming from the differences between l-adic sheaves on algebraic and p-adic analytic spaces, respectively. 

February 11, 2025
Geometry & Topology [6] Geometry of the regular representation of hyperbolic groups 4:00pm -
KT 207

I will discuss geometric properties of the regular representation of hyperbolic groups. Given a torsion free hyperbolic group G, one can build a natural quotient Q of a Hilbert sphere from the regular representation of G. I am interested in the geometry of this infinite dimensional Riemannian space Q, and also of its ultralimit Q_omega. This ultralimit encodes the asymptotic geometry of Q, for instance possible limits of the regular representation. I will mention an application: the spherical volume (a topological invariant defined by Besson-Courttois-Gallot) of a negatively curved manifold is realized by a minimal surface inside the corresponding Q_omega. This result is motivated by the problem of constructing a "minimal surface geometry" on closed manifolds.

February 12, 2025
Colloquium [7] From harmonic maps to hyperbolic surfaces via random matrices 4:00pm -
KT 101

Abstract: Harmonic maps and hyperbolic surfaces are among the most studied special objects in Differential Geometry. Harmonic maps into Riemannian manifolds are a nonlinear generalization of harmonic functions. Hyperbolic surfaces are surfaces with constant Gaussian curvature equal to -1. In this talk, I will describe a phenomenon connecting the two notions: often, “random” harmonic maps from surfaces to Euclidean spheres have images which are almost hyperbolic surfaces with high probability. Among other ingredients, this connection relies on a new invariant for unitary representations of surface groups, and on the concept of strong convergence appearing in random matrix theory. 

February 13, 2025
Analysis [4] Construction of multi-soliton solutions for semilinear equations in dimension 3 4:00pm -
KT 207

The existence of multi black hole solutions in General Relativity is one of the expectations from the final state conjecture, the analogue of soliton resolution. In this talk, I will present preliminary works in this direction via a semilinear toy model in dimension 3. In particular, I show 1) an algorithm to construct approximate solutions to the energy critical wave equation that converge to a sum of solitons at an arbitrary polynomial rate in (t-r); 2) a robust method to solve the remaining error terms for the nonlinear equation. The methods apply to energy supercritical problems.

Quantum Topology and Field Theory [8] Multiple polylogarithms and the Steinberg module 4:30pm -
KT 801

 Multiple polylogarithms appear to be central for many seemingly unrelated areas of mathematics: volumes of hyperbolic polytopes, scissors congruence, algebraic K-theory, special values of zeta functions, etc. Despite the existence of this wide network of connections, the most fundamental properties of these functions, predicted by the Goncharov program, remain conjectural. I will talk about the recent progress in the Goncharov program, which is based on the connection between multiple polylogarithms and the Steinberg module of Q. The talk is based on the joint work with Steven Charlton and Danylo Radchenko. 

February 14, 2025
Friday Mornings [9] TBA 10:00am -
KT 801
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[1] https://calendar.math.yale.edu/list/calendar/grid/week/2025-W06 [2] https://calendar.math.yale.edu/list/calendar/grid/week/2025-W08 [3] https://calendar.math.yale.edu/seminars/group-actions-and-dynamics [4] https://calendar.math.yale.edu/seminars/analysis [5] https://calendar.math.yale.edu/seminars/geometry-symmetry-and-physics [6] https://calendar.math.yale.edu/seminars/geometry-topology [7] https://calendar.math.yale.edu/seminars/colloquium [8] https://calendar.math.yale.edu/seminars/quantum-topology-and-field-theory [9] https://calendar.math.yale.edu/seminars/friday-mornings [10] https://calendar.math.yale.edu/list/calendar/grid/week/abstract/2025-W06 [11] https://calendar.math.yale.edu/list/calendar/grid/week/abstract/2025-W08