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 Recent progresses in the unanimity problem in Majority Dynamics on random graphs 12:00pm -
LOM

Majority Dynamics is a process on a graph, where each vertex starts out with a Red or Blue color, then on each day changes its color to the majority color among its neighbors the previous day. If at some point one color covers every vertex, that color is said to win and such state is called unanimity. Research on unanimity has traditionally focused on the model where initial colors are independently chosen with 1/2 chance, and the graph is generated on them from the G(n, p) model. In this talk, we discuss how recent studies by many authors point out that, fixing the initial colors instead can help answer some of the questions in the traditional model, and pose new interesting conjectures.
Geometric Analysis and Application Translating mean curvature flow with simple end 2:00pm -

Abstract: Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on the joint work with Ao Sun.