Events for 10/29/2021 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:40PM

Location: Zoom

Speaker: Burak Hatinoglu, UC Santa Cruz

Title: Spectral properties of the periodic 4th order Schrodinger operator on the hexagonal lattice

Abstract: This talk will focus on the 4th order Schr\"{o}dinger operator $d^4/dx^4 + q(x)$ on the hexagonal lattice (graphene) and its geometric perturbations with self-adjoint vertex conditions and a real periodic symmetric potential $q$. I will consider the following spectral properties of this Hamiltonian on graphene: absolutely continuous, singular continuous and pure-point spectra, dispersion relation, singular Dirac points and energy levels of reducible Fermi surfaces. I will also discuss some of these spectral properties for the same operator on lattices in the geometric neighborhood of graphene. This talk is based on a joint work with Mahmood Ettehad (University of Minnesota).

Noncommutative Geometry Seminar

Time: 2:00PM - 3:00PM

Location: ZOOM

Speaker: Marc Rieffel , University of California at Berkeley

Title: Dirac Operators for Matrix Algebras Converging to Coadjoint Orbits

Abstract: In the high-energy physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general precise setting for understanding such statements, in which the matrix algebras are viewed as C*-metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. But physicists want even more to treat structures on spheres (and other spaces like coadjoint orbits), such as vector bundles, Yang-Mills functionals, Dirac operators, etc., and they want to approximate these by corresponding structures on matrix algebras. I will sketch a somewhat unified construction of Dirac operators on coadjoint orbits and on the matrix algebras that converge to them. As Connes showed us, from Dirac operators we may obtain C*-metrics. Our unified construction enables us to prove our main theorem, whose content is that, for the C*-metric-space structures determined by the Dirac operators that we construct, the matrix algebras do indeed converge to the coadjoint orbits, for a quite strong version of quantum Gromov-Hausdorff distance. This is a long story, but I will sketch how it works.

URL: Event link

Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

Location: Zoom

Speaker: Elizabeth Grimm, Illinois State University

Title: Hamiltonicity of 3-tough (P2 ∪3P1)-free graphs

Abstract: Chv ́atal conjectured in 1973 the existence of some constant t such that all t-tough graphs with at least three vertices are Hamiltonian. While the conjecture has been proven for some special classes of graphs, it remains open in general. We say that a graph is (P2 ∪3P1)-free if it contains no induced subgraph isomorphic to P2 ∪3P1, where P2 ∪3P1 is the disjoint union of an edge and three isolated vertices. In this talk, we show that every 3-tough (P2 ∪3P1)-free graph with at least three vertices is Hamiltonian.

Geometry Seminar

Time: 4:00PM - 4:50PM

Location: zoom

Speaker: Prasit Bhattacharya, University of Notre Dame

Title: Equivariant Steenrod Operations

Abstract: The classical Steenrod algebra is one of the most fundamental algebraic gadgets in stable homotopy theory. It led to the theory of characteristic classes, which is key to some of the most celebrated applications of homotopy theory to geometry. The G-equivariant Steenrod algebra is not known beyond the group of order 2. In this talk, I will recall a geometric construction of the classical Steenrod algebra and generalize it to construct G-equivariant Steenrod operations. Time permitting, I will discuss potential applications to equivariant geometry.