Department of Mathematics

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Abstracts

Week of February 20, 2022

February 21, 2022
Group Actions and Dynamics	The maximal spectral gap of a hyperbolic surface	4:00pm - Zoom	A hyperbolic surface is a surface with metric of constant curvature -1. The spectral gap between the first two eigenvalues of the Laplacian on a closed hyperbolic surface contains a good deal of information about the surface, including its connectivity, dynamical properties of its geodesic flow, and error terms in geodesic counting problems. For arithmetic hyperbolic surfaces the spectral gap is also the subject of one of the biggest open problems in automorphic forms: Selberg’s eigenvalue conjecture. It was an open problem from the 1970s whether there exist a sequence of closed hyperbolic sur- faces with genera tending to infinity and spectral gap tending to 1/4. (The value 1/4 here is the asymptotically optimal one.) Recently we proved that this is indeed possible. I’ll discuss the very interesting background of this problem in detail as well as some ideas of the proof. This is joint work with Will Hide.
Geometry, Symmetry and Physics	Modular forms of half-integral weight on G_2	4:30pm - Zoom	Abstract: Classical holomorphic modular forms are number-theoretic objects that have been intensely studied. The split exceptional group G_2 does not support a theory of holomorphic modular forms, but it does possess so-called quaternionic modular forms. These are a special class of automorphic forms that appear to behave similarly to holomorphic modular forms. In the talk, I will describe a theory of modular forms of half-integral weight on G_2 and other exceptional groups. In particular, we prove the existence of a modular form of weight 1/2 on G_2 whose Fourier coefficients are related to the 2-torsion in the narrow class groups of totally real cubic fields. This is joint work with Spencer Leslie.

February 22, 2022
Geometry & Topology	Peripheral birationality for 3-dimensional convex co-compact $PSL_2\mathbb{C}$ varieties	4:15pm - LOM 214	It is a consequence of a well-known result of Ahlfors and Bers that the $PSL_2\mathbb{C}$ character associated to a convex co-compact hyperbolic 3-manifold is determined by its peripheral data. In this talk we will show how this map extends to a birational isomorphism of the corresponding $PSL_2\mathbb{C}$ character varieties, so in particular it is generically a 1-to-1 map. Analogous results were proven by Dunfield in the single cusp case, and by Klaff and Tillmann for finite volume hyperbolic 3-manifolds. This is joint work with Ian Agol.
Algebra and Number Theory Seminar	$L$-function for $\mathrm{Sp}(4) \times \mathrm{GL}(2)$ via a non-unique model	4:30pm -	We prove a conjecture of Ginzburg and Soudry (2020 IMRN) on an integral representation for the tensor product partial L-function for $\mathrm{Sp}(4) \times \mathrm{GL}(2)$ which is derived from the twisted doubling method of Cai, Friedberg, Ginzburg, and Kaplan. We show that the integral unfolds to a non-unique model and analyze it using the New Way method of Piatetski-Shapiro and Rallis.

February 23, 2022
Applied Mathematics	Partial Permutation Synchronization via Cycle-Edge Message Passing	12:00pm - https://yale.zoom.us/j/97458245891	Abstract: The problem of partial permutation synchronization (PPS) provides a global mathematical formulation for the multiple image matching problem. In this matching problem, one is provided with possibly corrupted matches (i.e., partial permutations) between keypoints in pairs of images and the underlying task is to match keypoints in each image to universal 3D scene points (resulting in other partial permutations). For structure from motion (SfM) common datasets, previous PPS algorithms for image matching often become computationally intractable and demand an exceedingly large amount of memory. We address this issue by extending the recent framework of Cycle-Edge Message Passing (CEMP) to the setting of PPS despite the fact that partial permutations do not have a full group structure. We emphasize mathematical difficulties that arise when extending CEMP to PPS and also explain the mathematical guarantees for the performance of the modified CEMP algorithm in the setting of adversarial corruption and sufficiently small noise. If time allows we will demonstrate the state-of-the-art accuracy of our overall product for solving PPS within SfM. This is a joint work with Shaohan Li and Yunpeng Shi.
Colloquium	Positroids, knots, and Catalan numbers	4:15pm -	Abstract: A classical result states that the Poincare polynomial of a Grassmannian manifold is given by a Gaussian polynomial, the q-analogue of the binomial coefficient. I will discuss joint work with Pasha Galashin where we prove a variant of this relating the topology of a positroid variety and (q,t)-Catalan numbers. Positroid varieties are remarkable subvarieties of the Grassmannian that have made appearances in total positivity, scattering amplitudes, and Schubert calculus. Our work relies on a connection between the geometry of these spaces and link homology groups of certain “positroid links”.

February 24, 2022
Algebra and Geometry lecture series	An informal introduction to categorical actions of groups, Lecture 4	4:00pm - https://yale.zoom.us/j/92613729337

February 25, 2022
Friday Mornings	Friday Morning Seminar	9:00am -	We discuss topics of common interest in the areas of geometry, probability, and combinatorics.
Graduate Student Seminar	Schur-Weyl duality and its generalizations	12:00pm -	Schur-Weyl duality was originally defined by Schur in 1901, which builds a correspondence between the representations of the symmetric group and the polynomial representations of the general linear group. In this talk, I will first walk through the classical Schur-Weyl duality, which states that the symmetric group and the general linear group generate each other’s centralizers. Then I will discuss some possible generalizations of the classical Schur-Weyl duality, for example, the duality between the orthogonal group and the Brauer algebra.
Geometric Analysis and Application	Metric SYZ conjecture	2:00pm -	Abstract: One possible interpretation of the SYZ conjecture is that for a polarized family of CY manifolds near the large complex structure limit, there is a special Lagrangian fibration on the generic region of the CY manifold. Generic here means a set with a large percentage of the CY measure, and the percentage tends to 100% in the limit. I will discuss some recent progress on this version of the SYZ conjecture, with some emphasis on the special case of the Fermat family.