# Math Calendar

### Upcoming Events

## Number Theory Seminar

## Explicit Sato-Tate type distribution for a family of K3 surfaces- SC 1312

Hasan Saad- University of Virginia

In the 1960’s, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual Sato-Tate for non-CM elliptic curves). In analogy with Birch’s result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces of a certain family of K3 surfaces 19 is the O(3) distribution. This distribution is quite different from the semicircular distribution. It is supported on [-3,3] and has vertical asymptotes at t=1 and t=-1. Here we make this result explicit by bounding the error term.

As a consequence, we are able to determine when a finite field is large enough for the discrete histograms to reach any given height near t=1. To obtain these results, we make use of the theory of Rankin-Cohen brackets in the theory of harmonic Maass forms.

## Topology & Group Theory Seminar

## Commensurability of lattices in right-angled buildings

Sam Shepherd – Vanderbilt

Given compact length spaces *X* and *Y* with a common universal cover, it is natural to ask whether *X* and *Y* have a common finite cover. In particular, are there properties of *X* and *Y*, or of their fundamental groups, that guarantee the existence of a common finite cover? We will discuss several examples, as well as my new result which concerns the case where the common universal cover is a right-angled building. Examples of right-angled buildings include products of trees and Davis complexes of right-angled Coxeter groups. My new result will be stated in terms of weak commensurability of lattices in the automorphism group of the building.

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## Colloquium

## COLLOQUIUM- A free boundary problem for modeling plaques in the artery -Room-SC 5211

Bei Hu -University of Notre Dame- Math Colloquium to request Zoom Meeting ID and passcode.

Atherosclerosis is a leading cause of death worldwide; it originates from a plaque which builds up in

the artery. We considered a simplified model of plaque growth involving LDL and HDL cholesterols,

macrophages and foam cells, which satisfy a coupled system of PDEs with a free boundary, the interfacebetween the plaque and the blood flow. In an earlier work (with Avner Friedman and Wenrui Hao) of an extremely simplified model, we proved that there exist small radially symmetric stationary plaques andestablished a sharp condition that ensures their stability. In our recent work (with Evelyn Zhao), we look for theexistence of non-radially symmetric stationary solutions. The absent of an explicit radially symmetric stationary solution presents a big challenge to verify the Crandall-Rabinowitz theorem; through asymptotic expansion, weextend the analysis to establish a finite branch of symmetry-breaking stationary solutions which bifurcate fromthe radially symmetric solutions. Since plaque is unlikely to be strictly radially symmetric, our result would beuseful to explain the asymmetric shapes of plaque. Our recent work (with Yaodan Huang, Xiaohong Zha) extends to other possible shapes as well as more realistic modeling efforts.

Host: Glenn Webb (glenn.f.webb@vanderbilt.edu), Xinyue Zhao (xinyue.zhao@vanderbilt.edu)ng,Zhengce Zhang)

## PDE Seminar

## PDE- Nontrivial global solutions to some quasilinear wave equations in three space dimensions- SC 1431

Dongxiao Yu, University of Bonn.- Zoom link: https://vanderbilt.zoom.us/j/97526207493

In this talk, I will present a method to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. Starting from a global solution to the geometric reduced system satisfying several pointwise estimates, we find a matching exact global solution to the original quasilinear wave equations. As an application of this method, we will construct nontrivial global solutions to Fritz John’s counterexample ☐u=u_tu_{tt} and the 3D compressible Euler equations without vorticity for

## Topology & Group Theory Seminar

## Title: Complexity of algorithms in group theory

Vladimir Shpilrain (CUNY)

The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this talk, we address the average-case time complexity of several algorithms in group theory and show that in many cases it is linear and in some cases even constant (with respect to the length of an input). Along the way, we improve several bounds for the worst-case complexity of the word problem in groups of matrices, in particular in nilpotent groups. Most of this talk is based on joint work with A.Yu.Olshanskii.

## Colloquium

## COLLOQUIUM- Yao’s millionaires’ problem and public-key encryption- ROOM-SC 52115211

Vladimir Shpilrain -CUNY- Math Colloquium to request Zoom Meeting ID and passcode.

Yao’s millionaires’ problem is: Alice has a private number a and Bob has a private number b, and the

goal of the two parties is to figure out which number is larger without revealing any information about a or b. We will discuss relations between this fun problem and serious problems like the possibility of secure public-key encryption and the P=NP? problem.

Host: A. Olshanskii (alexander.olshanskiy@vanderbilt.edu)

## PDE Seminar

## Traveling wave solutions to the free boundary Navier-Stokes equations-Stevenson Center 1312

Ian Tice, Carnegie Mellon University.

Consider a layer of viscous incompressible fluid bounded below by a flat rigid boundary and above by a moving boundary. The fluid is subject to gravity, surface tension, and an external stress that is stationary when viewed in a coordinate system moving at a constant velocity parallel to the lower boundary. The latter can model, for instance, a tube blowing air on the fluid while translating across the surface. In this talk we will detail the construction of traveling wave solutions to this problem, which are themselves stationary in the same translating coordinate system. While such traveling wave solutions to the Euler equations are well-known, to the best of our knowledge this is the first construction of such solutions with viscosity. This is joint work with Giovanni Leoni.

## Subfactor Seminar

## Subfactor Seminar- Title -TBD

David Kribs- University of Guelph

Abstract- TBD

## Topology & Group Theory Seminar

## Topology and Group Theory- Title- TBA

Alex Margolis -Vanderbilt

Abstract- TBA

## Colloquium

## Weil-Petersson curves, traveling salesman theorems and minimal surfaces. SC 5211

Christopher Bishop -Stony Brook University

Weil-Petersson curves are a class of rectifiable closed curves in the plane, defined as the closure of the smooth curves with respect to the Weil-Petersson metric defined by Takhtajan and Teo in 2009. Their work solved a problem from string theory by making the space of closed loops into a Hilbert manifold, but the sameclass of curves also arises naturally in complex analysis, geometric measure theory, probability theory, knottheory, computer vision, and other areas. No geometric description of Weil-Petersson curves was known until2019, but there are now more than twenty equivalent conditions. One involves inscribed polygons and can beexplained to a calculus student. Another is a strengthening of Peter Jones’s traveling salesman conditioncharacterizing rectifiable curves. A third says a curve is Weil-Petersson iff it bounds a minimal surface inhyperbolic 3-space that has finite total curvature. I will discuss these and several other characterizations andsketch why they are all equivalent to each other. The lecture will contain many pictures, several definitions, but not too many proofs or technical details.

Host: Dechao Zheng (dechao.zheng@vanderbilt.edu)

## Subfactor Seminar

## Subfactor Seminar- Title – TBD- SC 1432

Michael Davis- University of Iowa

Abstract- TBD

## Topology & Group Theory Seminar

## Topology and Group Theory- Title TBA SC 1312

Ionut Chifan – University of Iowa

Abstract- TBA

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## Subfactor Seminar

## Subfactor Seminar- Title- TBD

Mehrdad Kalantar- University of Houston

Abstract- TBD

## Topology & Group Theory Seminar

## Topology and Group Theory- On subgroups of right-angled Artin groups.

Jone Lopez de Gamiz Zearra -Vanderbilt

In this talk, we will discuss subgroups of right-angled Artin groups (RAAGs for short). Although, in general, subgroups of RAAGs are known to have a wild structure and bad algorithmic behaviour, we will show that under certain conditions they have a tame structure. Firstly, we will discuss finitely generated normal subgroups of RAAGs and show that they are co-(virtually abelian). As a consequence, we deduce that they have decidable algorithmic problems. Secondly, we will recall results of Baumslag-Roseble and Bridson-Howie-Miller-Short on subgroups of direct products of free groups and explain how they generalise to other classes of RAAGs.

## Subfactor Seminar

## Subfactor Seminar- Title- TBD – SC 1432

Corey Jones, North Carolina State University

Asbtract- TBD

## Topology & Group Theory Seminar

## Topology and Group Theory- Title: TBA

Dan Margalit – Georgia Tech

**Abstract: **TBA

## PDE Seminar

## PDE- Title- TBA

Dejan Gajic, Leipzig University- https://vanderbilt.zoom.us/j/992097409

Abstract- TBA

## Subfactor Seminar

## Subfactor Seminar- Title- TBD- SC 1432

Peter Huston

Abstract- TBD

## Topology & Group Theory Seminar

## Topology and Group Theory – Title: TBA- SC1312

Ekaterina Rybak – Vanderbilt

**Abstract: **TBA