# Math Calendar

### Upcoming Events

## SSP ?= RC

Bogdan Chornomaz, Vanderbilt University

Location: Stevenson 1312

In this talk we will address the SSP ?= RC conjecture, which we were discussing last semester. SSP is a class of finite lattices satisfying Sauer-Shelah-Perles, that is, lattices in which every set of elements shatters at least as many elements as it has. The conjecture says that those are exactly finite relatively complemented lattices, and, while it is almost trivial to show that SSP implies RC, the opposite direction does not seem to be easy. We will briefly recall the basic properties of RC lattices and a sufficient condition for SSP. Then we present recent advancements towards the proof.

## Finiteness Properties for Simple Groups

Rachel Skipper, The Ohio State University

Location: Stevenson 1308

A group is said to be of type $F_n$ if it admits a classifying space with compact $n$-skeleton. We will consider the class of R\”{o}ver-Nekrachevych groups, a class of groups built out of self-similar groups and Higman-Thompson groups, and use them to produce a simple group of type $F_{n-1}$ but not $F_n$ for each $n$. These are the first known examples for $n\geq 3$. As a consequence, we find the second known infinite family of quasi-isometry classes of finitely presented simple groups.

## Heuristics for Abelian Fields: Totally Positive Units and Narrow Class Groups

Ben Breen, Dartmouth College

Location: Stevenson 1320

We describe heuristics in the style of Cohen-Lenstra for narrow class groups and units in abelian extensions of odd degree. These results stem from a model for the 2-Selmer group of a number field. We conclude with computational evidence for cyclic extensions of degree n = 3,5,7.

## Around Cantor Uniqueness Theorem

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)

## Examining Gene Expression Profiles in the National Longitudinal Study of Adolescent to Adult Health

Lauren Gaydosh, Vanderbilt University

Location: Garland Hall 101

## Motzkin Algebra and the Fusion Rule A_n of Its Bimodules

Jun Yang, Vanderbilt University

Location: Stevenson 1432

There is a family of algebras associated with the Motzkin numbers and a parameter d. We will prove that one can construct a hyperfinite $II_1$ factor from them if and only if d is $2cos(\pi/n)+1, (n\geq 3)$ or no less than 3. We will construct a sequence of bimodules over it. Then we will discuss the irreducibility, dimensions and get a fusion rule of $A_n$. This is joint work with Vaughan Jones.

## The Mathematical Physics of the Relativistic Radiation Reaction

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.

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Brian Luczak, Vanderbilt University

Location: Stevenson 1206

## Thesis Defense: Applications of Modular Forms to Geometry and Interpolation Problems

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 $\mathbb{R}^3$.

## Super-Teichmueller Spaces and Related Structures

Anton M. Zeitlin, Louisiana State University

Location: Stevenson 1432

The Teichmueller space parametrizes Riemann surfaces of fixed topological type and is fundamental in various contexts of mathematics and physics. It can be defined as a component of the moduli space of flat G=PSL(2,R) connections on the surface. Higher Teichmueller space extends these notions to appropriate higher rank classical Lie groups G, and N=1 super-Teichmueller space likewise studies the extension to the super Lie group G=OSp(1|2). In this talk, I will discuss the solution to the long-standing problem of giving Penner-type coordinates on super-Teichmueller space and its higher analogues and will also talk about several applications of this theory including the recent generalization of the McShane identity to the super case.

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Hayden Jananthan, Vanderbilt University

Location: Stevenson 1206

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Glenn Webb, Vanderbilt University

Location: Stevenson 1206

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Jean Pierre Mutanguha, University of Arkansas

Location: Stevenson 1308

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Ayla Gafni, University of Mississippi

Location: Stevenson 1320

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Mihaela Ifrim, University of Wisconsin Madison

Location: Stevenson 1307

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Andy Jarnevic, Vanderbilt University

Location: Stevenson 1206

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Daniel Studenmund, Notre Dame

Location: Stevenson 1308

Lauren Ruth and Alex Cameron, Vanderbilt University

Location: Stevenson 1206

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Lauren Ruth, Vanderbilt University

Location: Stevenson 1308

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Brent Nelson, Michigan State University

Location: Stevenson 1432

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Zach Tripp, Vanderbilt University

Location: Stevenson 1206

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Jason Behrstock, CUNY

Location: Stevenson 1308

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Bin Gui, Rutgers University

Location: Stevenson 1432

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Theodore Drivas, Princeton University

Location: Stevenson 1307

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Genevieve Walsh, Tufts University

Location: Stevenson 1308