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# Events for 04/16/2021 from all calendars

## Teaching Online Departmental Open Forum

## Noncommutative Geometry Seminar

## Mathematical Physics and Harmonic Analysis Seminar

## Algebra and Combinatorics Seminar

## Full Committee L & I Meeting

**Time: ** 12:00PM - 1:00PM

**Location: ** Zoom

**Speaker: **Vanessa Coffelt & Justin Cantu, Texas A&M University

**Description: **All forums will be
from 12:00 pm to 1:00 pm. The topic will be emailed to faculty the week
prior to the forum. We will use the following Meeting ID: 979 6560
9771, and will send out the password in the emails.

**Time: ** 1:00PM - 2:00PM

**Location: ** Zoom 951 5490 42

**Speaker: **Yang Liu, SISSA

**Title: ***Hypergeometric Functions and Heat Coefficients on Noncommutative Tori*

**Abstract: **As the counterpart of conformal geometry, modular geometry on noncommutative manifolds explores the basic notions such as metric and curvature in Riemannian geometry (e.g. noncommutative tori) in a purely spectral framework. It was initiated by Connes-Tretkoff’s Gauss-Bonnet theorem on noncommutative two tori. Another milestone is the construction of modular Gaussian curvature due to Connes-Moscovici, which is derived from variation of the second heat coefficient of some Laplacian type operator. In this talk, I would like to report a few observations on the general structures of those heat coefficients. The word “modular” refers to the new ingredient of the coefficients, arising from the interaction between modular automorphisms associated to the volume state and the underlying smooth structure of the noncommutative manifolds. More precisely, one has to upgrade coefficients of local differential expressions from scalars to so-called rearrangement operators that fix various issues caused by the noncommutativity between metric coordinates and their derivatives. Like the notion of genus to a characteristic class, the spectral functions behind the rearrangement operators turn out to be intriguing. That is where hypergeometric functions come into play. The main result is the explicit formula of the second heat coefficient of a more general Laplacian type operator (beyond conformal perturbations studied in the literature). The talk is based on my recent preprint arxiv:2004.05714.

**URL: ***Event link*

**Time: ** 2:00PM - 3:00PM

**Location: ** Zoom

**Speaker: **Cosmas Kravaris, Texas A&M University

**Title: ***On the density of eigenvalues on discrete periodic graphs*

**Abstract: **Using the Floquet-Bloch transform, we show that Zd-periodic graphs have finitely many finite support eigenfunctions up to translations and linear combinations and show that this can be used to calculate the density of eigenvalues. We study the Kagome lattice to illustrate these techniques and generalize the claims to amenable quasi-homogeneous graphs whose acting group has Noetherian group algebra (this includes all virtually polycyclic groups). Finally, we provide a formula for the von Neumann dimension (i.e. density) of eigenvalues on Zd-periodic graphs using syzygy modules.

**Time: ** 3:00PM - 4:00PM

**Location: ** Zoom

**Speaker: **Songling Shan, Illinois State University

**Title: ***Chromatic index of dense quasirandom graphs*

**Abstract: **Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $\Delta(G)>n/3$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. Glock, K\"{u}hn, and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. We show that the conjecture of Glock, K\"{u}hn, and Osthus is affirmative.

**Time: ** 3:00PM - 5:00PM

**Location: ** Zoom