# Search Research Symposium Abstracts

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## Degree $d$ points on plane curves

Given a plane curve $C$ defined over $\mathbb{Q}$, when the genus of the curve is greater than one, Faltings' theorem tells us that the set of rational points on the curve is finite. It is then natural to consider higher degree points, that is, points on this curve defined over fields of degree $d$ over $\mathbb{Q}$. We ask for which natural numbers $d$ [Read More...]

**Presenter:**Lea Beneish, University of North Texas

**Authors:**Lea Beneish and Andrew Granville

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**September 30, 2023; 9:45 am

## Computational Aspects of Bianchi Modular Forms

In this talk, I will introduce a techniques for computing Bianchi module forms and demonstrate a few interesting examples found from an [Read More...]

**Presenter:**Kalani Thalagoda, Tulane university

**Authors:**Dan Yasaki

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**September 30, 2023; 10:10 am

## A $p$-adic analytic Brauer group

The Brauer group of a local field plays a central role in class field theory. On one hand, the Brauer group of a field $K$ classifies central simple algebras, up to Morita equivalence. On the other hand, the Brauer group $\mathrm{Br}(K)$ is identified with the Galois cohomology $H^2(\mathrm{Gal}_K, \overline{K}^\times)$. In this talk, we introduce a [Read More...]

**Presenter:**Deewang Bhamidipati, UC Santa Cruz

**Authors:**Deewang Bhamidipati

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**September 30, 2023; 10:35 am

## On extensions of Drinfeld modules

Drinfeld introduced what are now called Drinfeld modules, which can be regarded as a function field analogue of elliptic curves. By Anderson’s work, there are higher dimensional generalizations of Drinfeld module called $t$-modules. As they are modules, if we consider two Drinfeld modules $E$ and $F$, we can study the properties of $Ext^1(E,F)$. For [Read More...]

**Presenter:**Changningphaabi Namoijam, Colby College

**Authors:**Yen-Tsung Chen and Changningphaabi Namoijam

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**September 30, 2023; 11:00 am

## Rethinking Research Workshops:

Rethinking Number Theory is a virtual research workshop series that began in the early remote-working days of 2020. The goals of the workshop are for participants to learn new math, get to know colleagues, and have a joyful, affirming research experience. Our "professional development" sessions are meetings where we discuss ways to make our [Read More...]

**Presenter:**Heidi Goodson, Brooklyn College, City University of New York

**Authors:**Heidi Goodson, Allechar Serrano López, Mckenzie West

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**October 1, 2023; 9:45 am

## Thinking Session: Joyful Collaboration

An important part of the Rethinking Number Theory workshops is providing the space, opportunity, and conversational prompts to discuss ways in which the mathematical community can be made more inclusive. We ask for examples of positive experiences alongside conversations that challenge the norms of the mathematical community. In this talk, we will hold a [Read More...]

**Presenter:**Mckenzie West, University of WIsconsin-Eau Claire

**Authors:**Heidi Goodson, Allechar Serrano López, Mckenzie West

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**October 1, 2023; 10:10 am

## Automorphic forms and string scattering amplitudes

A significant obstacle for quantum field theory is incorporating gravity, and a proposed solution to this challenge is to use oscillating strings to model gravitons. In this framework, understanding graviton behaviour involves analyzing certain functions on moduli spaces. These functions often arise as automorphic forms. For example, when calculating string [Read More...]

**Presenter:**Maryam Khaqan, University of Toronto

**Authors:**Maryam Khaqan, Holley Friedlander, Kim Klinger-Logan, Manish Pandey, and Runqiu Xu.

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**October 1, 2023; 10:35 am

## Intersection of Ekedahl-Oort strata with the supersingular locus in unitary Shimura varieties of sgn(q-2,2)

Unitary Shimura varieties are moduli spaces of abelian varieties in characteristic $p$ with certain extra structures, including CM by an imaginary quadratic field of signature $(q-2,2)$. A fruitful way to understand them is by stratifying these spaces. We focus on two such stratifications: the Ekedahl-Oort (E-O) stratification, defined with respect to the [Read More...]

**Presenter:**Sandra Nair, Colorado State University

**Authors:**Deewang Bhamidipati, Maria Fox, Heidi Goodson, Steven Groen, Sandra Nair* and Emerald Stacy.

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**October 1, 2023; 11:00 am

## Groups associated to Frobenius polynomials

The Honda--Tate theorem gives a precise correspondence between isogeny classes of abelian varieties defined over finite fields and their corresponding Frobenius polynomials. In this talk, we explore certain groups attached to these isogeny classes: (1) the Galois group of the Frobenius polynomial, and (2) the Serre-Frobenius group, studied in recent joint [Read More...]

**Presenter:**Santiago Arangp-Piñeros, Emory University

**Authors:**Santiago Arangp-Piñeros

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**October 1, 2023; 2:00 pm

## A graph-theoretic approach to analyzing decoding failures of BIKE

We present experimental findings on the decoding failure rate (DFR) of BIKE, a fourth-round candidate in the NIST Post-Quantum Standardization process, at the 20-bit security level using graph-theoretic approaches. We select parameters according to BIKE design principles and conduct a series of experiments using Rust to generate significantly more decoding [Read More...]

**Presenter:**Angela Robinson, PhD, NIST

**Authors:**Sarah Arpin, Tyler Raven Billingsley, Daniel Rayor Hast, Jun Bo Lau, Ray Perlner, Angela Robinson

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**October 1, 2023; 2:25 pm

## Stacky Curves and Generalized Fermat Equations

Solutions to many Diophantine equations in number theory can be studied using the theory of algebraic stacks. In this talk, I will describe a collection of "generalized Fermat equations" whose integer solutions can be found using an appropriate family of stacky curves. These curves are interesting in their own right: they provide potential [Read More...]

**Presenter:**Andrew Kobin, Emory University

**Authors:**Juanita Duque-Rosero, Chris Keyes, Andrew Kobin, Manami Roy, Soumya Sankar, Yidi Wang

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**October 1, 2023; 2:50 pm

## Using Trees to Visualize Solutions to Some Diophantine Equations

Visualizations often make complicated mathematical structures easier to comprehend. In this presentation, I will show how our team used the structure of an infinite tree to understand the solutions to a family of Diophantine equations of the form $x^s+D=2^t y$ for a fixed constant $D$, with solutions where $y$ is odd, $s\in\{2,3\}$, and [Read More...]

**Presenter:**Eva Goedhart, Franklin & Marshall College

**Authors:**Maila Brucal-Hallare, Ryan Max Riley, Vaishavi Sharma, Bianca Thompson

**Symposium Year:**2023

**Session:**Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]

**Presentation Time:**October 1, 2023; 3:15 pm

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