This pre-talk will introduce some basic ideas for studying complete noncompact three-manifolds with nonnegative scalar curvature using harmonic functions. We will focus on the geometry of harmonic level sets and explain how the Bochner formula, the Gauss equation, Gauss–Bonnet, and the coarea formula work together to produce useful geometric estimates. 

One can construct an A-model analogue of the Gauss-Manin connection using genus 0 Gromov-Witten invariants, which also make perfect sense over fields of positive characteristics and p-adic fields. I will explain how arithmetic gadgets including p-curvature, Fontaine-Laffaile modules, and over-convergent Frobenii (conjecturally) arise in this context, and discuss certain extensions in the q-difference setting based on quasimap K-theory. This is based on joint works with Lee, Pomerleano, and Seidel.

Abstract: The metaplectic geometric Langlands program can be understood in two ways: as a twisted analogue of the geometric Langlands program, and as a geometric analogue of the Langlands program for covering groups, namely the metaplectic Langlands program. One of the key structures in the geometric Langlands program is the spectral action, which roughly says that the spectral side acts on the automorphic side, allowing one to spectrally decompose the automorphic category over the moduli of \check{G}-local systems. This can be viewed as a geometric incarnation of the automorphic-to-Galois direction.

One may hope that the spectral action carries over to the metaplectic setting, and this expectation was formulated by Gaitsgory and Lysenko under the name “metaplectic vanishing conjecture.” In this talk, I will discuss my recent work on constructing the spectral action in the Betti metaplectic context. I will focus on what changes in the metaplectic setting and on the categorical framework for studying factorizable actions of local systems of symmetric monoidal categories.

Geometric Representation Theory Seminar:  [16]https://sites.google.com/view/yale-grt-seminar/home [16]

In this talk, I will discuss a joint work with Samuel J. Taylor, where we consider embedded non-fiber surfaces in a closed hyperbolic 3-manifold transverse to a pseudo-Anosov flow. Utilizing foundational work on Kleinian surface groups, we connect the hyperbolic geometry of the surface to the dynamics of the flow via the curve graph. More specifically, the flow gives a pair of special multicurves on the surface, and we show that they are also geometrically distinguished, enabling us to extract geometric information, such as the position of short curves and the volume, from the relation of the pair of multicurves in the curve graph. Our result also connects to recent study of the geometry of endperiodic mapping tori.

A recent manuscript written by Claude under the direction of Levent Alpoge constructs a 1-parameter family of compact complex threefolds diffeomorphic to S^6. I will describe the main geometric ideas behind the construction and outline the proof. If time permits, I will remark on possible future research directions.

Abstract: Classical Harish-Chandra bimodules are related to many objects in representation theory such as infinite dimensional representations of complex groups viewed as real groups, category O and character sheaves. The notion of Harish-Chandra bimodules for quantum groups was introduced in the works about topological quantum field theory of surfaces. I will mention some results about quantum Harish-Chandra bimodules at odd order roots of unity, connect them to affine Soergel bimodules and coherent sheaves on Steinberg variety.

Seminar website: https://sites.google.com/view/yale-grt-seminar/home [16]

I will talk about the existence of bounded harmonic functions on hyperbolic manifolds and more generally locally symmetric spaces, explains how it relates to the fundamental group acting ergodically on the Furstenberg boundary, and discuss some new and old results about this topic . Based on joint work with Subhadip Dey and Mikolaj Fraczyk.

McMullen constructed automorphisms of complex K3 surfaces with positive entropy—and thus chaotic dynamics on a large set—that nonetheless have a Siegel disk: an open region on which the automorphism is holomorphically conjugate to an irrational rotation. Strikingly, such an automorphism forces the K3 surface to be non-projective, so McMullen’s examples cannot be written down using explicit equations. Instead they are obtained indirectly via the global Torelli theorem.

In this talk we construct a p-adic analogue: a positive-entropy automorphism with a Siegel disk on a non-projective rigid-analytic K3 surface over a p-adic field. Our construction produces the surface directly, without the Torelli theorem, which makes it considerably more flexible.

Abstract: In joint work with Gurbir Dhillon, we prove a universal monodromic extension of Bezrukavnikov’s two realizations of the affine Hecke category over the integers. I will discuss some ingredients that go into to the proof of the Arkhipov-Bezrukavnikov equivalence in this setting. These include analysis of multi-graded rings, deformation between characteristic zero and positive characteristic, and the tilting property of Whittaker averaged central sheaves. The proof of the latter is based on ideas of Faergeman-Raskin and Bezrukavnikov-Morton-Ferguson.