Zack Tripp, Vanderbilt University
Location: Stevenson 1320

In “The Pair Correlation of Zeros of the Zeta Function”, Montgomery finds the asymptotics of the pair correlation function in order to give a lower bound on the proportion of zeros that are simple (assuming the Riemann Hypothesis). We will discuss some of the necessary tools to extend his proof to pair correlation for zeros of Dedekind zeta functions of abelian extensions, and as in the Riemann zeta case, we can then use this to obtain results on multiplicities of zeros for these zeta functions. However, we also are able to relate the counts of multiplicities to Cohn-Elkies sphere-packing type bounds, allowing us to use semi-definite programming techniques to obtain better results in lower degree extensions than could be found from a direct analysis. In particular, we are able to conclude that more than 45% of the zeros are distinct for Dedekind zeta functions of quadratic number fields. This is based on joint work with M. Alsharif, D. de Laat, M. Milinovich, L. Rolen, and I. Wagner.

Jonathan Luk, Stanford University
Location: Stevenson 5211

The Einstein equations in general relativity admit explicit black hole solutions which have the disturbing property that global uniqueness fails. As a way out, Penrose proposed the strong cosmic censorship conjecture, which says that this phenomenon of global non-uniqueness is non-generic. We will discuss this conjecture and some recent mathematical progress. This talk is based on joint works with Mihalis Dafermos, Sung-Jin Oh, Jan Sbierski and Yakov Shlapentokh-Rothman. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Jared Speck)

Srivatsav Kunnawalkam Elayavalli, Vanderbilt University
Location: Stevenson 1432

We will report on some results (joint with S. Atkinson) regarding new characterizations of amenability for Connes embeddable II_1 factors. First we introduce the notion of self-tracial stability and discuss immediate connections with amenability. Secondly we will discuss an interesting strengthening of the well known Jung’s tubularity result. The core of this is a technical argument of Kishimoto. There are also some interesting model theoretic questions that come out of this. Finally we build on work of N. Brown and N. Ozawa to answer a question of Popa concerning the “vastness” of the space of embeddings of a non-amenable II_1 factor into the ultraproduct of a given collection of II_1 factors. We will end with interesting questions in the group theory setting. See arxiv: 1907.03359 for submitted paper.

Hayden Jananthan, Vanderbilt University
Location: Stevenson 1206

Rachel Skipper, The Ohio State University
Location: Stevenson 1308

Ben Breen, Dartmouth College
Location: Stevenson 1320

Alexander Olevskii, Tel Aviv University
Location: Stevenson 5211

After a short introduction to Riemann’s uniqueness theory I will discuss the following problem, going back to early 60-s: Can a (non-trivial) trigonometric series
\sum c(n) e^inx ,    c( n) = o(1),
have a subsequence of partial sums which converges to zero everywhere? Joint work with Gady Kozma. Tea at 3:33 pm in SC 1425. (Host: Akram Aldroubi)

Jun Yang, Vanderbilt University
Location: Stevenson 1432

Vu Hoang, The University of Texas at San Antonio
Location: Stevenson 1307

In classical electrodynamics, the self-force of a charged particle on itself is undefined, leading to a major inconsistency in the equations of motion for point particles. The usual solution of the problem involves a renormalization procedure, which is mathematically not well defined. It appears that a successful solution of this problem requires a modification of the Maxwell-Lorentz equations. In this talk, I consider the field equations of higher-order electrodynamics as proposed by Bopp, Lande, Thomas and Podolsky in the 1940’s. This theory modifies the Maxwell Lagrangian by adding terms containing higher-order derivatives of the field tensor, leading to higher-order wave equations that couple charged particles to their radiation fields. Starting with a brief review of the subject’s history, I will focus on recent results by Kiessling-Tahvildar-Zadeh and by M. Radosz and myself, showing that a covariant equation of motion for point particles can be rigorously derived from the postulate of conservation of energy and momentum without any renormalization. I will also present a global existence result for the motion of a single particle under the influence of an external field and its radiation reaction. This is joint work with M. Radosz.

Brian Luczak, Vanderbilt University
Location: Stevenson 1206

Ahram Feigenbaum, Vanderbilt University
Location: Stevenson 1313

The sphere packing problem asks for the densest collection of non-overlapping congruent spheres in $\mathbb{R}^n$. In 2016, Viazovska proved that the $E_8$ lattice is optimal for $n = 8$. Subsequently, she with Cohn, Kumar, Miller, and Radchenko that showed the Leech lattice was optimal for $n =24$. Their proofs relied on the theory of weakly holomorphic and quasi-modular forms to construct Fourier eigenfunctions with prescribed zeros at distances in the $E_8$ and Leech lattices. Similar ideas were applied by Radchenko and Viazovska to obtain interpolation formulas for real Schwartz functions and by Cohn and Gon\c{c}alves to study uncertainty principles in harmonic analysis. In this thesis, we develop a unified approach to the construction of such functions. We show that the weakly holomorphic and weakly quasi-modular forms behind them are uniquely defined by the conditions that they be eigenfunctions of the Fourier transform belonging to the Schwartz class. We construct the Fourier eigenfunctions for all $n$ divisible by 4. We also show an extension of the interpolation formula given by Radchenko and Viazovska in $\mathbb{R}$ to radial functions in $\mathbb{R}^2$ and $\mathbb{R}^3$.

Location: Edwin Warner Park Picnic Shelter 11

Glenn Webb, Vanderbilt University
Location: Stevenson 1206

Jean Pierre Mutanguha, University of Arkansas
Location: Stevenson 1308

Ayla Gafni, University of Mississippi
Location: Stevenson 1320

Mihaela Ifrim, University of Wisconsin Madison
Location: Stevenson 1307

Andy Jarnevic, Vanderbilt University
Location: Stevenson 1206

Daniel Studenmund, Notre Dame
Location: Stevenson 1308

Lauren Ruth and Alex Cameron, Vanderbilt University
Location: Stevenson 1206

Brent Nelson, Michigan State University
Location: Stevenson 1432

Zach Tripp, Vanderbilt University
Location: Stevenson 1206

Lauren Ruth, Vanderbilt University
Location: Stevenson 1308

Bin Gui, Rutgers University
Location: Stevenson 1432

Theodore Drivas, Princeton University
Location: Stevenson 1307

Genevieve Walsh, Tufts University
Location: Stevenson 1308

John Voight, Dartmouth College
Location: Stevenson 1320