Monday, December 5, 2022
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All day |
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3pm |
12/05/2022 - 3:00pm 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) This work is joint with Erik Hiltunen, John Schotland, and Michael Weinstein. Location:
AKW 200
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4pm |
12/05/2022 - 4:00pm 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. Location:
LOM 206
12/05/2022 - 4:30pm 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. Location:
LOM214
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