Abstracts

Week of February 14, 2021

February 15, 2021
Group Actions and Dynamics Infinite volume and infinite injectivity radius 10:15am -
Zoom

Abstract. We answer a question of Margulis by proving the following: Let G be a higher rank simple Lie group and let Λ ≤ G be a discrete subgroup of infinite covolume, then the locally symmetric space Λ\G/K admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We establish similar results for higher rank semisimple groups with Kazhdan’s property (T). We prove a stiffness result for discrete stationary random subgroups in higher rank groups and a stationary variant of the Stuck-Zimmer theorem for higher rank semisimple groups with property (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete.

Joint work with Mikolaj Fraczyk
Geometry, Symmetry and Physics Smooth locus of twisted affine Schubert varieties and twisted affine Demazure modules 4:30pm -

Zhu proved a duality theorem between level one affine Demazure modules and function rings of torus fixed point subschemes of affine Schubert varieties in affine Grassmannian. Using his methods and results, we prove a similar duality theorem between level one twisted affine Demazure modules and twisted affine Schubert varieties for absolutely special parahoric group schemes. As a consequence, we determine the smooth locus of all twisted affine Schubert varieties for many types of parahoric group scheme. This confirms a conjecture of Haines and Richarz for these types of group schemes. If time permits, I will also talk about how this duality theorem is related to the Frenkel-Kac isomorphism for twisted affine Lie algebras, and also the fusion product for twisted affine Demazure modules. This is a joint work with Marc Besson.
February 16, 2021
Algebra and Number Theory Seminar Monoidal Structures on GL(2)-Modules and Abstractly Automorphic Representations 9:00am -
Zoom

Consider the function field $F$ of a smooth curve over $\mathbb{F}_q$, with $q \neq 2$.

L-functions of automorphic representations of $\mathrm{GL}(2)$ over $F$ are important objects for studying the arithmetic properties of the field $F$. Unfortunately, they can be defined in two different ways: one by Godement--Jacquet, and one by Jacquet--Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation.

Each of these L-functions is the GCD of a family of zeta integrals associated to test data. I will categorify the question, by showing that there is a correspondence between the two families of zeta integrals, instead of just their L-functions. The resulting comparison of test data will induce an exotic symmetric monoidal structure on the category of representations of $\mathrm{GL}(2)$.

It turns out that an appropriate space of automorphic functions is a commutative algebra with respect to this symmetric monoidal structure. I will outline this construction, and show how it can be used to construct a category of automorphic representations.
Geometry & Topology Exotic real projective Dehn surgery space 4:00pm -
https://yale.zoom.us/j/96501374645

We study properly convex real projective structures on closed 3-manifolds. A hyperbolic structure is one special example, and in some cases the hyperbolic structure may be deformed non-trivially as a convex projective structure. However, such deformations seem to be exceedingly rare. By contrast, we show that many closed hyperbolic manifolds admit a second convex projective structure not obtained through deformation. We find these examples through a theory of properly convex projective Dehn filling, generalizing Thurston’s picture of hyperbolic Dehn surgery space. Joint work with Sam Ballas, Gye-Seon Lee, and Ludovic Marquis.
February 17, 2021
Applied Mathematics Confidence budget matching for sequential budgeted learning 1:00pm -
Zoom Meeting ID: 97670014308

Abstract: A core element in sequential decision making problems, such as contextual bandits and reinforcement learning, is the feedback on the quality of the performed actions. However, in many real-world applications, such feedback is restricted. In this work, we study decision making problems with querying budget, that is, when the total amount of feedback is restricted by a hard budget and the agent can choose when to query for feedback. We propose a simple algorithmic principle which we refer to as Confidence Budget Matching (CBM), analyze its performance on a variety of sequential budgeted learning problems, and establish its robustness relatively to more naive approaches.

email tatianna.curtis@yale.edu for info.