Department of Mathematics

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Abstracts

Week of February 23, 2025

February 24, 2025
Group Actions and Dynamics	Angles between Oseledets spaces	4:00pm - KT207	This talk is based on joint work with Pablo Lessa. We provide an example of a probability distribution on the group GL(2,R) with finite first moment such that the corresponding random product of i.i.d. matrices has two distinct Lyapunov exponents, but the angle between the Oseledets directions is not log-integrable. We prove that, on the other hand, if the second moment is finite, then this angle, if defined, is log-integrable. Next, we turn our attention to general GL(2,R)-cocycles over ergodic automorphisms, and ask ourselves if there is any criterion for log-integrability of the angle between the Oseledets directions in terms of a suitable integrability condition. The answer is negative. In fact, we show the following flexibility result: given any ergodic automorphism T of a non-atomic Lebesgue probability space, we can find a GL(2,R)-cocycle over T whose Lyapunov exponents and joint distribution of Oseledets spaces are prescribed a priori, and meeting any prescribed integrability condition.
Geometry, Symmetry and Physics	Hall Structures, Intrinsic Donaldson–Thomas Theory, and Cohomology	4:30pm - KT 801	For a general algebraic stack X, we will present combinatorial structures underlying the connected components of the stack of filtrations of X. This allows us to define analogues of the Hall algebra in different flavors, except that we do not get algebras but a more general kind of structure. Classically these Hall algebras were only defined when X parametrizes objects in an abelian category, while our construction is general. We will then discuss applications of the theory. In the motivic setting, we define a notion of Euler characteristic for a stack and, in the (-1)-shifted symplectic case, we give an intrinsic definition of Donaldson–Thomas invariants. The construction relies on a no-pole theorem. The invariants depend on the choice of a so-called stability measure. The space of such measures is a unipotent algebraic group that governs how invariants change under wall-crossing. In the cohomological setting, we get an explicit form of the decomposition theorem for the map from the stack to its good moduli space, in the smooth, 0-symplectic, and (-1)-symplectic case, assuming tangent space representations at closed points are orthogonally symmetric. This is joint work over different projects with Chenjing Bu, Ben Davison, Daniel Halpern-Leistner, Tasuki Kinjo and Tudor Pădurariu.

February 25, 2025
Applied Mathematics	Geometric Manifold Learning	4:00pm - LOM 215	The primary theme of this talk is geodesics. First I’ll introduce the Calculus of Variations and explain the variational approach to geodesics from Riemannian geometry. A novel result on the length of the shortest closed geodesic on 2-spheres will be presented. Next, I’ll consider these ideas in a data setting, where data diffusion has been a powerful tool for studying manifold geometry. I’ll discuss NeuralFIM, a diffusion-based method that learns a differentiable representation of the data, allowing for computation of the Fisher Information Metric (FIM). One can then use this Riemannian metric to compute volumes and geodesics on the data manifold. NeuralFIM is joint work with the Krishnaswamy Lab.
	TBA	4:00pm -
Geometry & Topology	Morse theory on the moduli space of Riemann surfaces	4:00pm - KT 207	It is known that the systole function, defined to be the length of a shortest closed geodesic, is topologically Morse on the moduli space of Riemann/hyperbolic surfaces, proved by Hugo Akrout. However, Morse theory cannot be applied as the function is not differentiable and the base space is noncompact. We construct a family of weighted exponential averages of all geodesic-length functions, and show that they are Morse on the Deligne-Mumford compactification of the moduli space. We will also characterize the critical points and Morse indices, and from certain properties of them we may find conclusions on the homology of the moduli space by Morse theory.

February 27, 2025
Analysis	Lieb-Thirring Inequalities: What we know and what we want to know	4:00pm - Zoom	https://yale.zoom.us/j/95303636613 Lieb--Thirring inequalities are functional inequalities that generalize Sobolev inequalities and that have proved to be powerful tools in several questions from mathematical physics, PDEs and functional analysis. Recently, in the spirit of Lieb--Thirring inequalities, certain inequalities in harmonic analysis were extended to the setting of orthonormal functions. In this talk we give a gentle introduction to classical aspects of the subject, some recent progress and some open problems.
Quantum Topology and Field Theory	Gluing cluster structures	4:30pm - KT 801	Many interesting algebraic varieties appearing in low-dimensional topology and representation theory (for example various kinds of surface character varieties, or subvarieties of simple Lie groups or their flag manifolds) are known to admit cluster Poisson structures. Given some geometrically defined morphism between two such varieties, it is natural to ask whether it respects the corresponding cluster structures in a suitable sense. I will explain a kind of ‘gluing procedure’ for certain special kinds of cluster structures, which leads to a positive answer to the question above for morphisms of character varieties associated to cutting a surface along a simple closed curve, as well for morphisms between BFN Coulomb branches of quiver gauge theories obtained by restricting a factor of the gauge group to its maximal torus. Based on joint work with Alexander Shapiro.

February 28, 2025
Friday Morning Seminar	Symplectic forms on the space of circle patterns	10:00am - KT 801	We consider circle patterns on surfaces with complex projective structures. We investigate two symplectic forms pulled back to the deformation space of circle patterns. The first one is Goldman’s symplectic form on the space of complex projective structures on closed surfaces. The other is the Weil-Petersson symplectic form on the Teichmüller space of punctured surfaces. We show that their pullbacks to the space of circle patterns coincide. It is applied to prove the smoothness of the deformation space, which is an essential step to the conjecture that the space of circle patterns is homeomorphic to the Teichmüller space of the closed surface.
Algebra and Geometry lecture series	Vertex algebras and moduli of Higgs bundles II	3:00pm -	This a continuation of the lecture of last week. In the second lecture, we explain how characters of these vertex algebras often satisfy (quasi-)modular linear differential equations, with logarithmic terms in characters related to extensions of vertex algebra modules.