Department of Mathematics

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Abstracts

Week of October 24, 2021

October 25, 2021
Group Actions and Dynamics	Lattice actions, escape of mass, and arithmetic structures	4:00pm - Zoom	I will outline some ideas in recent work with D. Fisher and S. Hurtado establishing Zimmer’s conjecture for actions of general lattices in SL(n,R). This extends our earlier results for actions of cocompact lattices and of SL(n,Z). Problems involving escape of mass and “escape of Lyapunov exponents” arise when the lattice is not cocompact and I will outline some ideas used to avoid such behavior.
Geometry, Symmetry and Physics	Complex K-theory of dual Hitchin systems	4:30pm - Zoom	Abstract: Let G and G’ be Langlands dual reductive groups (e.g. SL(n) and PGL(n)). According to a theorem by Donagi-Pantev, the generic fibres of the moduli spaces of G-Higgs bundles and G’-Higgs bundles are dual abelian varieties and are therefore derived equivalent. It is an interesting open problem to prove existence of a derived equivalence over the full Hitchin base. I will report on joint work in progress with Shiyu Shen, in which we construct a K-theoretic shadow thereof: natural equivalences between complex K-theory spectra for certain moduli spaces of Higgs bundles (in type A). Zoom link: https://yale.zoom.us/j/99305994163, contact the organizers (Gurbir Dhillon and Junliang Shen) for the passcode.

October 26, 2021
Geometry & Topology	Uniform models for hyperbolic 3-manifolds	4:15pm - LOM 214	I'll discuss the problem of predicting the internal structure of a 3-manifold from its end invariants. While this has a pretty satisfying answer for manifolds homeomorphic to (surface)x(interval), we do not have a complete answer in general which is --uniform over an entire deformation space--. In joint work with Ken Bromberg and Dick Canary, we construct such a model for manifolds with incompressible boundary, prove it gives estimates in one direction, and use this to establish an old theorem of Thurston whose original proof is not known.
Algebra and Number Theory Seminar	Unitary Friedberg-Jacquet periods and central values of L functions	4:30pm - Zoom	Let G be a reductive group over a number field F and H a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on G over H(F)\H(A_F). They are often related to special values of certain L functions. One of the most notable case is when (G,H)=(U(n+1)☓U(n), U(n)), and these periods are related to central values of Rankin-Selberg L functions on GL(n+1)☓GL(n). In this talk, I will explain my work in progress with Wei Zhang that studies central values of standard L functions on GL(2n) using (G,H)=(U(2n), U(n)☓U(n)) and some variants. I shall explain the conjecture and a relative trace formula approach to study it. We prove the required fundamental lemma using a limit of the Jacquet-Rallis fundamental lemma and Hironaka’s characterization of spherical functions on the space of Hermitian matrices. Also, the question admits an arithmetic analogy.

October 27, 2021
Applied Mathematics/Analysis Seminar	Smooth Extension and Interpolation of Data	2:30pm - https://yale.zoom.us/j/2188028533	Abstract: Whitney’s Extension Problem asks how we can tell if a function defined on a compact subset can be extended to a C^m function on all of R^n. Following Fefferman’s solution in 2006, much attention has been brought to the finite set analogue: Given N data points, how can we fit a smooth function through them? In this talk we will provide an introduction to this area of research, discussing some of the speaker’s recent work (joint with Fushuai Jiang and Garving K. Luli) on constrained problems where the function is required to be nonnegative or pass through certain convex sets.
Undergraduate Seminar	Putnam Seminar	4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214	The Putnam seminar meets every Wednesday from 4 to 5:30 in LOM 214. As always, everyone is warmly welcomed to come to hang out, learn more cool math, and meet folks. The seminar is casual, and folks can come and go as they like. See Pat Devlin’s webpage (and/or contact him) for more information. Folks can sign up for the mailing list here: https://forms.gle/nYPx72KVJxJcgLha8

October 28, 2021
Algebra and Geometry lecture series	Quantizations in charateristic p. Lecture 7	4:00pm - https://yale.zoom.us/j/99019019033 (password was emailed by Ivan)	See Algebra and Geometry lecture series (yale.edu)
Analysis	Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds	4:15pm -	Abstract: This presentation, which is based on the work Sun [2], is dedicated to describing the complete integrability of the Benjamin-Ono (BO) equation on the line when restricted to every N-soliton mani- fold, denoted by UN. We construct (generalized) action{angle coordinates which establish a real analytic symplectomorphism from UN onto some open convex subset of R2N and allow to solve the equation by quadrature for any such initial datum. As a consequence, UN is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by Gerard-Kappeler [1]. The global well-posedness of the BO equation on UN is given by a polynomial characterization and a spectral char- acterization of the manifold UN. Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the innitesimal generator of the very shift semigroup. The construction of action-angle coordinates for each UN constitutes a rst step towards the soliton resolution conjecture of the BO equation on the line.

October 29, 2021
Geometric Analysis and Application	Volume preserving mean curvature flows in hyperbolic space	2:00pm -	Abstract: The VPMCF is a normalized mean curvature flow with a global forcing term. For a compact hypersurface, it flows while keeping the enclosed volume unchanged. Unlike the MCF, the singularity for such a “global” flow is not well-understood, yet it has potentially many applications in many geometry problems. Based on joint work with Lin (Santa Cruz) and Zhang (Sydney), we consider this VPMCF in hyperbolic space, and prove a dynamical stability theorem near spheres.