Published on Department of Mathematics (https://math.yale.edu)

Week of December 4, 2022

December 5, 2022
Applied Mathematics	Real Space Quantum Optics in Periodic Media and 1 Dimensional Dirac Points	Joseph Kraisler - Columbia University	3:00pm - AKW 200
Group Actions, Geometry and Dynamics	Intersection number and intersection points of closed geodesics on hyperbolic surfaces	Tina Torkaman - Harvard University	4:00pm - LOM 206
Geometry, Symmetry and Physics	Cohomological Non Abelian Hodge Theorem for curves in positive characteristic	Mark de Cataldo - Stony Brook University	4:30pm - LOM214

December 6, 2022
Geometry & Topology	Geometry of geodesic currents	Jenya Sapir - Binghamton University	3:00pm - LOM 202
Walter Feit Memorial Lecture	Equivariant birational geometry	Yuri Tschinkel - Simons Foundation & NYU	4:30pm -

December 7, 2022
Colloquium	Braid varieties	Evgeny Gorskiy - UC Davis	4:15pm -

December 9, 2022
Friday Morning Seminar	Friday Morning Seminar		10:30am - LOM 205
Graduate Student Seminar	TBD	Linh Tran - Yale	12:00pm - LOM
Geometric Analysis and Application	TBA	Zhihan Wang - Princeton University	2:00pm - TBA

Abstracts

Week of December 4, 2022

December 5, 2022
Applied Mathematics	Real Space Quantum Optics in Periodic Media and 1 Dimensional Dirac Points	3:00pm - AKW 200	Abstract: We will start by introducing a real space model of a scalar electromagnetic field coupled to a continuum collection of two level atoms. From this we will obtain a pair of nonlocal partial differential equations describing the energy eigenstates that have at most one photon present in the field. Next, We will consider the case that the density of atoms is periodic with respect to a general lattice and describe a structure theorem for the spectral bands of the one photon Hamiltonian. Finally we will discuss the existence of linear crossings of the spectral bands (Dirac Points) in two cases: 1. Atomic densities which are weak perturbations of a constant. (Low contrast) 2. Atomic densities which are sums of shifted scaled atomic inclusions. (High contrast). This work is joint with Erik Hiltunen, John Schotland, and Michael Weinstein.
Group Actions, Geometry and Dynamics	Intersection number and intersection points of closed geodesics on hyperbolic surfaces	4:00pm - LOM 206	In this talk, I will talk about the (geometric) intersection number between closed geodesics on finite volume hyperbolic surfaces. Specifically, I will discuss the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics.
Geometry, Symmetry and Physics	Cohomological Non Abelian Hodge Theorem for curves in positive characteristic	4:30pm - LOM214	The Non Abelian Hodge Theory (NAHT) of Simpson, Corlette, et al. yields canonical diffeomorphisms between the moduli spaces of Higgs bundles, flat connections, and representations of the fundamental group of a curve. In positive characteristic, there is a "twisted" version of NAHT, but it is not clear how to extract geometric information from it. I will report on joint work with graduating student Siqing Zhang, where we prove a cohomological version of NAHT, i.e., we exhibit a canonical isomorphism between the etale cohomology rings of the moduli of Higgs bundles and of flat connections. I will explain how this works via vanishing cycle theory. TIme permitting, I will also discuss some perhaps unexpected corollaries relating cohomology rings of different moduli spaces, in equal and in mixed characteristic.

December 6, 2022
Geometry & Topology	Geometry of geodesic currents	3:00pm - LOM 202	The space of projective, filling currents PFC(S) contains many structures relating to a closed, genus g surface S. For example, it contains the set of all closed curves on S, as well as an embedded copy of Teichmuller space, and many other spaces of metrics on S. We show that the symmetrized Thurston metric on Teichmuller space naturally extends to a complete, proper metric on PFC(S). We will then discuss the geometry of PFC(S) under this metric.
Walter Feit Memorial Lecture	Equivariant birational geometry	4:30pm -	Abstract: The classification of finite groups and their regular actions on projective spaces have occupied generations of mathematicians. Actions on fields, especially, within the framework of Galois theory, are also of major interest. Recently, there have been interesting developments concerning the classification of actions on function fields of algebraic varieties. New techniques, based on ideas of motivic integration, led to the introduction of new invariants of such actions. I will discuss their construction and geometric applications (joint work with B. Hassett and A. Kresch).

December 7, 2022
Colloquium	Braid varieties	4:15pm -	I will introduce and discuss a remarkable class of algebraic varieties, called braid varieties. These include all open Richardson and positroid varieties, and are closely related to augmentation varieties for Legendrian links. The topology of braid varieties is related to various link invariants such as HOMFLY polynomial and Khovanov-Rozansky homology, while their coordinate ring has a cluster structure. The talk is based on joint works with Roger Casals, Mikhail Gorsky, Ian Le, Linhui Shen and Jose Simental.

December 9, 2022
Friday Morning Seminar	Friday Morning Seminar	10:30am - LOM 205	A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!
Graduate Student Seminar	TBD	12:00pm - LOM
Geometric Analysis and Application	TBA	2:00pm - TBA	TBA