# Search Research Symposium Abstracts

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## Mean-field games: generalization to manifold and a flexible algorithm with convergence analysis in Euclidean space

Conventional mean-field games study the behavior of a large number of rational agents moving in Euclidean space. In this work, we explore the mean-field games with agents on a manifold. Regarding the Nash Equilibrium of a mean-field game on manifold, we study its PDE formulation and variational formulation, and establish the equivalence between the PDE [Read More...]

**Presenter:**Jiajia Yu, Rensselaer Polytechnic Institute

**Authors:**Jiajia Yu, Rongjie Lai, Wuchen Li, Stanley Osher

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 17, 2022; 2:45 pm

## Hitting probabilities for searchers with frequent stochastic resetting

Many systems in physics, chemistry, and biology are modeled in terms of stochastically resetting searchers looking for targets. One useful statistic in this setting is the hitting probability of a target, which is the probability that a searcher finds that target before finding another target. In this work, we study hitting probabilities for a stochastic [Read More...]

**Presenter:**Samantha Linn, University of Utah

**Authors:**Samantha Linn, Sean D Lawley

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 17, 2022; 2:45 pm

## Coefficient Inverse Problem in a Stochastic Partial Differential Equation

We consider an inverse problem of identifying the coefficient function in a stochastic partial differential equation. An example of the application of this problem is the estimation of the transmissivity (or hydraulic conductivity) in groundwater models from the measurement(s) of the hydraulic head. A projected-gradient-type stochastic approximation scheme [Read More...]

**Presenter:**Yidan Yang, Rochester Institute of Technology

**Authors:**Yidan Yang, Baasansuren Jadamba

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 17, 2022; 2:45 pm

## Spear and Shield: Coding to Thwart Adversarial Aggression

In coding theory, we study methods to send information more effectively, for example, in telecommunications. Authentication is important when sending information to detect any potential interference. Our research is built on the recent results on authentication with multiple users. We explore some limitations for constructing good codebooks with certain [Read More...]

**Presenter:**Ariel Liu, University of Wisconsin-Eau Claire

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 17, 2022; 2:45 pm

## Integers that are sums of two rational sixth powers

We prove that 164634913 is the smallest positive integer that is a sum of two rational sixth powers but not a sum of two integer sixth powers. If $C_k$ is the curve $x^6+y^6=k$, we use the existence of morphisms from $C_k$ to elliptic curves, together with the Mordell-Weil sieve, to rule out the existence of rational points on $C_k$ for various [Read More...]

**Presenter:**Alexis N Newton, Emory University

**Authors:**Alexis Newton, Jeremy Rouse

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 17, 2022; 2:45 pm

## Representations of Integers by Powers of Two Primes

In number theory, a common topic of discussion is how many ways we can represent integers as sums or products of other integers. Given primes $p$ and $q$, we use finiteness results from solutions to the $S$-unit equation and the Catalan conjecture to find that there are only a finite number of integers that have more than one representation of the form [Read More...]

**Presenter:**Natalie Wijesinghe, UW - Eau Claire

**Authors:**Mckenzie West and Natalie Wijesinghe

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 17, 2022; 2:45 pm

## Crime and the Community

There exists a long history of mathematical tools being applied to the problem of reducing crime. In 2020, the mathematical community called for mathematicians to boycott collaborations with police departments due to concerns that such work could lead to racially biased policing. We acknowledge that concern but believe that math may still have a role in [Read More...]

**Presenter:**Alissa Whiteley, Clarkson University

**Authors:**Alissa Whiteley, Dr. Joseph Skufca

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 18, 2022; 2:00 pm

## Maximization of the second Laplacian eigenvalue on the sphere

Isoperimetric problem for eigenvalues of the Laplacian on a given manifold is concerned with finding an upper bound and understands for which metrics the upper bound is attained. On the sphere, the first nonzero eigenvalue can be maximized by the standard “round” metric. For 2-dimensional sphere, the second eigenvalue becomes maximal as the surface [Read More...]

**Presenter:**Hanna Kim, University of Illinois at Urbana Champaign

**Authors:**Hanna N. Kim

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 18, 2022; 2:00 pm

## Growth of Torsion over Imaginary Quadratic Fields of Class Number 1

Let $K$ be a noncyclotomic imaginary quadratic field of class number one and $E$ an elliptic curve defined over $K.$ We study the relation of $E(L)_{\text{tor}}$ with $E(K)_{\text{tor}}$ where $L$ is a quadratic extension of $K.$ If $E(K)_{\text{tor}} \subsetneq E(L)_{\text{tor}},$ we say that torsion grows. If $E$ satisfies one of the following cases: [Read More...]

**Presenter:**Irmak Balcik, University of Southern California

**Authors:**Irmak Balcik

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 18, 2022; 2:00 pm

## Graph Universal Cycles

Graph universal cycles are a graph analogue to universal cycles introduced in 2010. We demonstrate connections between universal cycles and graph universal cycles for threshold graphs and permutation graphs. Additionally, we introduce graph universal partial cycles, a more compact representation of graph classes, which use “do not know” edges. We then [Read More...]

**Presenter:**Elizabeth Sprangel, Iowa State University

**Authors:**Rachel Kirsch, Clare Sibley, and Elizabeth Sprangel

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 18, 2022; 2:00 pm

## Arithmetic Dynamics on Character Varieties

For a reductive algebraic group over integers, the G- character variety of a finitely presented group F parametrizes the set of closed conjugation orbits in Hom(F,G). The group of automorphisms of F acts naturally on Hom(F,G) by precomposition. This leads to a natural action of the group of outer automorphisms, Out(F), on the character variety. The dynamics [Read More...]

**Presenter:**Cigole Thomas, George Mason University

**Authors:**Cigole Thomas

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 18, 2022; 2:00 pm

## Minimal skew semistandard Young tableaus and the Hillman-Grassl correspondence

Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula as a positive sum over excited diagrams of products of hook-lengths. In 2018, Morales, Pak, and Panova gave a $q$-analogue of Naruse's formula for semi-standard tableaux of skew [Read More...]

**Presenter:**GaYee Park, University of Massachusetts Amherst

**Authors:**GaYee Park, Alejandro Morales, Greta Panova

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 18, 2022; 2:00 pm

## Optimal multiclass classification and generalized prevalence estimation with applications to SARS-CoV-2 antibody assays

An accurate classification strategy is crucial to interpreting diagnostic test results and includes problems with more than two classes. This process is complicated when the relative fraction of the population in each class, or generalized prevalence, is unknown. We develop a prevalence estimation method independent of classification and an associated [Read More...]

**Presenter:**Rayanne Luke, Johns Hopkins University, National Institute of Standards and Technology

**Authors:**Rayanne A. Luke, Paul N. Patrone, Anthony J. Kearsley

**Symposium Year:**2022

**Session:**Poster Presentation

**Presentation Time:**June 18, 2022; 2:00 pm

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