### Summer 2023

**Universal and Homogeneous Median Algebras**

Author: Ruohan Hu

Mentors: Ruiyuan Chen

Funding: C339240 - Mathematics Department Special

**Towards a Superparticle Population Density Method for Simulating Systems fo Coupled Oscillators**

Author: Andy Chen

Mentors: Daniel Forger

Funding: G014390 - James Van Loo Applied Mathematics and Physics Undergraduate Support Fund

**Degree Spectra of Functions on ω on a Cone**

Author: Jad Damaj

Mentor: Matthew Harris-Trainor

Funding: F064311 - 007432-Restricted Sub of F064254-Information Coded in Mathematical Structur

**Generating Tetrahedral Mesh for Solving Ion Channel Simulation Problems Using Finite Element Methods**

Author: Yueyang Ding

Mentor: Zhen Chao

Funding: U038823 - Research Incentive Program

**A Computability-Theoretic Characterization of [0,1] up to Homeomorphism Among Polish Spaces**

Author: Eissa Haydar

Mentor: Matthew Harrison-Trainor

Funding: F064311 - 007432-Restricted Sub of F064254-Information Coded in Mathematical Structur

**Optimal Connectivity Results for Spheres in the Curve Graph of Low and Medium Complexity Surfaces**

Authors: Helena Heinonen, Roshan Klein-Seetharaman, Minghan Sun

Mentor: Alex Wright

Funding: F063892 - 006974-CAREER: Moduli spaces of surfaces

**Convergence of the Deep Glerkin Methos for the Mean Field Control Problem**

Author: Jake Hofgard

Mentor: Asaf Cohen

Funding: F061124 - 004122-Restricted Sub of F057577-Asymptotic Analysis of Queueing Systems Un

**The Minimal Fibering Degree of a Toric Variety Equals the Lattice Width of its Polytope**

Author: Audric Lebovitz

Mentor: David Stapelton

Funding: F060536 - 003463-Restricted Sub of F060519-Commutative Algebra: Extremal Singularitie

**Ekedahl-oort Types of Artin-Schreier Curves**

Author: Nianchen Liu

Mentor: Patrick Daniels

Funding: U038823 - Research Incentive Program

**Modeling 1D Cold Electrostatic Plasma with a Lagrangian Particle Method**

Author: Horacio Moreno Montanes

Mentor: Robert Krasny

Funding: F061672 - 004690-Restricted Sub of F061623-Collaborative Research: Computational Tool

**Investigating Interpretability of Gender**

Author: Ashvin Pai

Mentor: Martin Strauss

Funding: F050847 - 095128-Restricted Sub of F050810-A Stud of Wave Patterns

**Small x Asymptotics for Special Function Solutions of Painlevé III Equation**

Author: Hao Pan

Mentor: Andrei Prokhorov

Funding: N/A

**Isolated Singularities of a Monge-Ampére Metric in Dimension Two**

Authors: Neil Patram and Ben Scott

Mentor: Mattias Jonsson

Funding: F064310 - 007431-Restricted Sub of F064252-A study of problems in complex analysis; F054129 - 098637-sub of F054071-A study of problems in complex analysis, dynamics

**Signature-based Model in Variance Swap Pricing**

Authors: Zuyuan Han and Haisu Ding

Mentors: Qi Feng and Bingyan Han

Funding: U038823 - Research Incentive Program

**Inhomogeneous Diophantine Approcimation: Winning Property and Beyond**

Author: Liyang Shao

Mentor: Shreyasi Datta

Funding: C339240 - Mathematics Department Special

**Soliton Solutions to the Korteweg-de Vries Equation**

Author: Haoyan Shi

Mentor: Elliot Blackstone

Funding: C339240 - Mathematics Department Special

**The Gromov Boundary of Hyperbolic Groups and Finite State Automata**

Author: Abhi Shukul

Mentor: Theodore Weisman

Funding: F057224 - 027960-Restricted Sub of F057220-Rigidity Properties in Dynamics and Geometry

**Survey: Explorations in Theorems of Schur, Schur-Horn for Hermitian Matrices**

Author: Marcin A. Sobotka

Mentor: Anthony Bloch

Funding: F061081 - 004079-Restricted Sub of F061041-Dynamics, Integrability and Control of Mec

Author: Congyan (Cruise) Song

Mentor: Dao Nyguen

Funding: C339240 - Mathematics Department Special

**Extension of Post's Lattice to Countable-Borel Clones**

Author: Ilir Ziba

Mentor: Ruiyuan Chen

Funding: F066381 - 009584-Restricted Sub of F064457-Restricted Sub of F064457-Participant Support

Authors: Nir Elber, Hahn Lheem

Mentors: Jialiang Zou and Elad Zeligher

Funding: F053987 - 098498-Restricted Sub of F053986-RTG: Number Theory and Representation Theory

### Summer 2022

**Encoding Cohomology, Classifying Extensions, and Explicit Galois Gerbs**

Author: Nir Elber

Mentors: Alexander Bertonloni Meli Patrick Daniels; Nicholas Rome

Funding: F053986-RTG: Number Theory and Representation Theory

**Christoffel Transformations of Legendre Polynomials**

Author: Andrew Fischer-Garbutt

Mentors: Ahmad Barhoumi

Funding: F056921-Random matrices, spin glass, and interactions

**Differentially Private Algorithms for Graph Sequences**

Author: Yiwei Fu

Mentor: Martin Strauss

Funding: RIP-Martin Strauss

**Modular Principal Series Representation of GL_{2}_{ }Over Finite Rings**

Authors: Gahl Shemy

Mentor: Charlotte Chan

Funding: F061315-Geometric methods in representation theory

**Analyzing the Growth Dynamics of Bladder Cancer Using Ordinary Differential Equation Modeling**

Authors: Joel Greenfield

Mentor: Daniel Bergman

Funding: F054152-Collaborative Research: Nonsmooth maps, co

**Existence of Uncountably Many Singular Vectors in R**

Authors: Riley Guyett

^{n}for Higher Dimensional Generalizations of Dani CorrespondenceMentor: Nattalie Tamam

Funding: F057220-Rigidity Properties in Dynamics and Geometry

**Federated Learning On Medical Applications**

Authors: Sizhuang He

Mentor: Maria Han Veiga

Funding: RES INCENTIVE PROG - MATHEMATICS

**Exmaples of Real Conic Bundles with Quartic Descriminant Curve**

Authors: Mattie Ji

Mentor: Lena Ji

Funding: F060536Commutative Algebra: Extremal Singularitiex

**The Linear Complimentarity Problem and its Applications in Physics Based Animation**

Authors: Rohan Kodati

Mentor: Shravan Veerapaneni

Funding: F050847 - A Stud of Wave Patterns

**The Logic of Choiceless Cardinality Comparison
**Authors: Dhruv Kulshreshtha

Mentor: Matthew Harrison-Trainor

Funding: Chair Discretionary Funds - Math

**Circular-Kernel-Based Curvature-Motion Algorithms**

Authors: David Li

Mentor: Selim Esedoglu

Funding: F058006 - High Order Schemes for Gradient Flows

**Aritmetic of Infinite Dedekind-finite Sets**

Authors: Jing-Yi Liu

Mentor: Andreas Blass

Funding: Mathematics Department Special

**Metric Diophantine Approximation on Non-Degenerate Manifolds**

Author: Justin Liu

Mentor: Shreyasi Datta

Funding: F057220-Rigidity Properties in Dynamics and Geometry

**Incorporating Domain Knowledge in Neural Networks**

Authors: Zhongqi Ma

Mentors: Maria Han Veiga, Dao Nguyen

Funding: Mathematics Department Special

**Computable Model Theory and Infinitary Logic
**Authors: Miles Kretschmer

Mentor: Matthew Harrison-Trainor

Funding: F057220-Rigidity Properties in Dynamics and Geometry

**Mechanics of Nonholonomic Hybrid Lagrangian Systems
**Authors: Evan Ortiz

Mentor: Anthony Bloch

Funding: F061041-Dynamics, Integrability and Control of Mechanics

**Deep Galerkin Method for Mean Field Control Problems**

Authors: Jingruo Sun

Mentors: Asaf Cohen

Funding: Mathematics Department Special

**Numerical Methods for the Heat Equation
**Authors: Sofia Taylor

Mentor: Robert Krasny

Funding: F061623-Collaborative Research: Computational Tool

**Overdamped Langevin Dynamics on Lie Groups**

Authors: Xinchang Wang

Mentor: Qi Feng

Funding: U038823 - Research Incentive Program

**Hardy-Littlewood Circle Method**

Authors: Xun Wang

Mentor: Nicholas Rome

Funding: U038823 - Research Incentive Program

**Predicting Gas Temperature in Protoplanetary**** ^{
}**Authors: Jason Zheng

Mentors: Maria Han Veiga and Arthur Bosman

Funding: G014390 - James Van Loo Applied Mathematics and Physics Undergraduate Support Fund

### Summer 2021

**Computing the Constant in the Left-tail Asymptotic of Maximum Eigenvalue Distribution of Finite GUE**

Author: Xiaoqi Peng

Mentors: Andrei Prokhorov & Fred Adams

Funding: G014390 - James Van Loo Applied Mathematics and Physics Undergraduate Support Fund

**Bijections of Strictly Convex Sets & Closed Convexprojective Surfaces that Preserve Complet Geodesics**

Author: Drimik Chowdhury

Mentors: Ralf Spatzier & Giuseppe Matrone

Funding: U038823 - Research Incentive Program

**Convex Relations for a Generalization of Mumford-Shah**

Author:Andrew Fischer-Garbutt

Mentor: Selim Esedoglu

Funding: F047060-Sub of F046971-Computational Tools & F058006-Sub of F058005-High Order Schemes

Using Mathematics and Computing to Understand How Bees UsePhysics to Keep Their Hives Cool

Authors: Jiashu Han

Research Advisor: Robert Krasny

Funding: U038823 - Research Incentive Program

**Polychromatic Ramsey Properties of Ideals**

Authors: Jonah Nan

Research Advisor: Zach Norwood

Funding: U048325 - Chair Discretionary Funds

**Attempts to Approximate the Partition Function of Hamiltonian Cycles
**Authors: James Walrad & Lily Wang

Research Advisor: Alexander Barvinok

Funding: F054167-Sub of F054157-FRG: Collaborative Research

**QNM Expansion to Solutions of Maxwell's Equation in Open Domains**

Authors: Kevin Ke

Research Advisor: Jorn Zimmerling

Funding: F050847-Sub of F050810

**Differential Closure of Ideals**

Authors: Lillian McPherson, Monroe Stephenson, & Fuxiang Yang

Research Advisor: Jennifer Kenkel & Daniel Smolkin

Funding: F040585-Sub of F040563-CAREER & F050391-Sub of F050378-Participant Support-Commutative Algebra & F054167-Sub of F054157-FRG: Collaborative Research

**Good-Deal Bounds in Asset Pricing**

Authors: Mingxuan Ge

Research Advisor: Prakash Chakraborty

Funding: U038823 - Research Incentive Program

**Modeling Vibrating Strings
**Authors: Pengrun Huang & Ella Palacios

Research Advisor: Alejandro Uribe

Funding: U038823 - Research Incentive Program & F050847-Sub of F050810

**Gender Prediction Using Email Data**

Authors: Peihan Liu

Research Advisor: Martin Strauss

Funding: U029433 - RIP - Martin Strauss

**Right tail of two-point distribution of the KPZ fixed point**

Authors: Ron Nissim

Research Advisor: Jinho Baik

Funding: F056922-Sub of F056921-Random matrices

**Using Infinite Games with Perfect Information to Characterize the Product Perfect Set Property**

Authors: Terry Guan

Research Advisor: Iian Smythe

Funding: U031348 - Undergrad program funding

**Wave Kinetic Equation and Kolmogorov-Zakharov Cascade Spectra**

Authors: Yubing Cui & Joshua Messing

Research Advisors: Zaher Hani & Ricardo Grande Izquierdo

Funding: U038823 - Research Incentive Program & F054244-Sub of F054209-CAREER

**Carnot Algebras with Small Graded Automorphism
**Authors: Ethan Cohen

Research Advisor: Thang Nguyen

Funding: F057224-Sub of F057220-Rigidity Properties

**Efficient Sliding Locomotion of Three-Link Bodies with Inertia
**Authors: Adam Earnest

Research Advisor: Silas Alben

Funding: F051415-Sub of F051324-Participant Support

**Ratios for Spanning Tree Polynomials of Covering Graphs**

Authors: Jeremy Flics and Brendan Nell

Research Advisors: Sunita Chepuri

Funding:F053989-Sub of F053988-Geometry of Cluster Algebras & F057224-Sub of F057220-Rigidity Properties

**Faster Dynamic Algorithms and Sensitivity Oracles for Graph Connectivity Problems
**Authors: Jingyi Gao

Research Advisors: Thatchaphol Saranurak and Michael Zieve

Funding: G014390 - James Van Loo Applied Mathematics and Physics Undergraduate Support Fund

**Extensions of Representations of Division Algebras Over Non-Archimedean Local Fields **

Authors: Andrew Keisling and Dylan Pentland

Research Advisors: Tasho Kaletha and Karol Koziol

Funding: F050174-Sub of F050164-Participant Support

**High-Wavenumber Steady Solutions of Two-Dimensional Rayleigh-Benard Convection Between Stress-Free Boundaries
**Authors: Alexander Takla

Research Advisors: Baole Wen

Funding: F040585 - 085629-Restricted Sub of F040563-CAREER

**Singular Vectors in F _{q}((T^{-1}))^{n
}**Authors: Yewei Xu

Research Advisors: Shreyasi Datta

Funding: U031348 - Undergrad program funding

### Summer 2020

**The Copy Lemma and Non-Shannon Information Inequalities
**Authors: Michael Tang

Research Advisor: Andreas Blass

Funding: F054167-Sub of F054157-FRG

**Stochastic Volatility Modeling: From Rough Heston Model to Affine Volterra Equatio
**Authors: Zixuan Wang & Yuanchun Ye

Research Advisor: Shuoqing Deng

Funding: U038823-Research Incentive Program & U053975-AMS-Simons Travel Award

**Quiver Polynomials and Bumpless Pipe Dreams**

Authors: Alexander Vidinas & Zixuan Xu

Research Advisor: Anna Weigandt

Funding: F042421-Sub of F042420-Combinatorics of Cluster Varieties & F050391-Sub of F050378-Participant Support

**Income Stability of Pooled Annuity**

Authors: Ge Qu

Research Advisor: Thomas Bernhardt

Funding: U038823 - Research Incentive Program

**Generating a Fractal Tree**

Authors: Jingyi Gao

Research Advisor: Mitchell Newberry

Funding: G014390-James Van Loo Applied Mathematics and Physics Undergraduate Support Fund

**Strongly Obtuse Rational Lattice Triangles
**Authors: Anne Larsen

Research Advisors: Chaya Norton & Alex Wright

Funding: F054083-Sub of F054069-GL+(2,R) orbit closures

**Optimal Stopping of exotic American option by Deep Learning
**Authors: Qinghao Lin

Research Advisor: Shuoqing Deng

Funding: U048325 - Chair Discretionary Funds-Math

**Extensions of irreducible representations of quaternion algebras
**Authors: Alex Wang

Research Advisor: Karol Koziol

Funding: F053987-Sub of F053986-RTG

**Periodic points on the regular n-gon**

Authors: Rafael Saavedra

Research Advisor: Chaya Norton & Alex Wright

Funding: F054083-Sub of F054069-GL+(2,R) orbit closures

**The Relativistic Vlasov-Maxwell System in Two Space Dimensions
**Authors: Advika Rajapakse

Research Advisor: Neel Patel

Funding: F040585-Sub of F040563-CAREER

**Medium Scale Curvature for Hyperbolic Groups**

Authors: Andrew Keisling

Research Advisor: Thang Nguyen

Funding: F057224-Sub of F057220-Rigidity Properties

**Ramsey Theory in Models of Set Theory Where the Axiom of Choice Fails
**Author: Zekun Jia

Advisor: Zach Norwood

Funding: U038823 - Research Incentive Program

**Coding Theory & Grassmannian Varieties**

Author: Pengrun Huang

Advisor: Eric Canton

Funding: U053975 - AMS-Simons Travel Award

**A Low Dimensional Model for Mouse Circadian Rythms
**Author: Carrie Fulton

Advisor: Victoria Booth

Funding: F054171-Sub of F054152-Collaborative Research

**Hilbert Functions of Square-Free Monomial Ideals**

Author: Marianne Debrito & Meixuan Sun

Advisor: Janet Page & Timothy Ryan

Funding: F050391-Restricted Sub of F050378-Participant Support & U038823 - Research Incentive Program

**Thurston’s fibered faces for non-orientable 3-manifolds and an application to minimal stretch factors**

Author: Caleb Partin

Advisor: Becca Winarski

Funding: F054332-Sub of F054307-Logarithmic gauged linear sigma models

### Summer 2019

**More on the Magnus Representation of the Torelli Group
**Authors: Mira Wattal

Research Advisor: Becca Winarski

Funding:

**The Faces of Toric Arrangements**

Authors: Junshan Chen & Amanda Yao

Research Advisor: Christin Bibby

**Symbolic Powers of Defining Ideals of Veronese Rings**

Authors: Fangu Chen & Alan Tang

Research Advisors: Eric Canton, Eloisa Grifo, & Jack Jeffries

**Children's Drawings and the Riemann-Hilbert Problem**

Author: Drimik Roy Chowdhury

Research Advisor: Caleb Ashley

**Fast Phase Retrieval from Masked Fourier Measurements**

Author: Cyril Cordor

Research Advisors: Aditya Viswanathan, Yulia Hristova, Alejandro Uribe-Ahumada

**Local Cohomoology of L (1,1,...,1) Module**

Author: Zhou Fang

Research Advisor: Andrew Snowden

**A Classification of Certain Symmetric Ideals**

Author: Vignesh Jagathese

Research Advisor: Andrew Snowden

**Low Energy Band-Limited Surfaces**

Author: Stephen Jasina

Research Advisor: Ian Tobasco

**The Hofer-Zehnder Capacity of Symplectic Toric Manifolds**

Author: Yichen Liu

Research Advisor: Alejandro Uribe

**Efficient Sliding Locomotion of a Two-Link Scallop**

Author: Connor Puritz

Research Advisor: Silas Alben

**One-Dimensional Domains of Large Type**

Author: Amanda Schwartz

Research Advisor: Mel Hochster

**Heronian Friezes**

Author: Linus Setiabrata

Research Advisor: Sergey Fomin

**Global Decay for Medium Intiial Data in the Inhomogeneous Muskat Problem**

Author: Nikhil Shankar

Research Advisor: Neel Patel

**Form-Function Divergence in Induced Trisomy-7**

Author: Roy Siegelmann

Research Advisor: Indika Rajapakse

**Weak Formulation of Discrete Time Principal Agent Problem**

Author: Xinjie Sun & Zhenxi Wu

Research Advisor: Nicolas Hernandez & Alexandros Saplaouras

**Visualizing Fluid Flows Using Line Integral Convolution Method**

Author: Shuyang Wang

Research Advisors: Shravan Veerapaneni & Ruowen Liu

**The Uniformization Theorem
**Author: Zachary Halberstam

Research Advisor: Caleb Ashley

**Duplicitous Permutations and Bumpless Pipe Dreams**

Author: Rainie Heck

Research Advisor: Anna Weigandt

### Summer 2018

Some Ramsey-theoretic statements without the Axiom of Choice

Authors: Joshua Brot and Mengyang Cao

Advisor: David Fernández-Breton

Gauged Linear Sigma Model Spaces

Author: Felipe Castellano-Macias

Advisor: Felix Janda

Identifiability of linear compartmental models of infectious disease transmission

Author: Jeremy D'Silva

Advisor: Marisa Eisenberg

Energetics of Elastic Curves

Authors: Jaeyoon Kim and Charles Devlin

Advisor: Ian Tobasco

A decision-theory based model of HPV vaccination and infection dynamics

Author: Alison Elgass

Advisor: Marisa Eisenberg

Min-Max Method and Gromov's Waist Inequality

Authors: Shiliang Gao and Juntai Zhou

Advisor: Daniel Burns

A Population-Based Model of the Effects of Spinal Cord Stimulation on Pain Processing

Author: Ariella Goldberg

Advisor: Sofia Piltz

Minimal Triangulations of Sphere Products

Author: Jonathon Jacobs

Advisor: Hailun Zheng

Mobius-invariant Lorentz geometry of the space of circles on S^{2}

Author: Ben Johnsrude

Advisor: David Barrett

Symplectic Geometry Through Polytopes

Authors: Colby Kelln

Advisor: Daniel Burns

Syzgies of the Serge Embedding via Simplicial Complexes

Authors: Luke Kiernan and Ryan Capouellez

Advisor: Andrew Snowden

Balanced Triangulations of Sphere Products with Minimal Vertices

Author: Alexander Wang

Advisor: Hailun Zheng

The Scaling Function Connection Problem in the Two-Dimensional Ising Model

Author: William Warner

Advisor: Thomas Bothner

Galois Representations Valued in Reductive Groups and Their Centralizers

Authors: Nikolay Grantcharov and Wyatt Reeves

Advisor: Tasho Kaletha

### Summer 2017

Symmetric and Asymmetric Centroid Bodies of the Regular

Author: Ryan Britton

Advisor: Beatrice-Helen Vritsiou

Reduced Words of (p,q)-Clans

Author: Brian Burks

Advisor: Brendan Pawlowski

Constructing a Simple Hybrid Model of Perineural Invasion

Author: Jeremy D'Silva

Advisor: Marisa Eisenberg

The (4,3) Property in Families of Fat Sets in the Plane

Author: Shiliang Gao

Advisor: Shira Zerbib Gelaki

Obstructing Embeddings of Finite R-Partite Right-Angled Artin Groups into Mapping Class Groups

Author: Ben Gould

Advisor: Wouter Van Limbeek

Permanents of Positive Semi-Definite Hermitian Matrices

Author: Wu Han and Roi Orzach

Advisor: Alexander Barvinok

Random Walks on the Fundamental Group of the OncePunctured Torus

Author: Adam Holeman and Jennifer Jones-Baro

Advisor: Caleb Ashley and Harrison Bray

Numerical Solution of Nonlinear Poisson Boltzmann Equation

Author: Jingzhen Hu

Advisor: Robert Krasny

A Topological Characterization of the Knotted and Linked Products Arising from Site-Specific Recombination on *T*(2, n)#*C*(2, r) DNA Substrates

Author: Jordan Katz

Advisor: Diana Hubbard

Hindman's Theorem in Abelian Groups

Author: Sung Hyup Lee

Advisor: David Fernandez-Breton

Complexity in Predator Prey Dynamics:Population Increase via Reciprocal Predation

Author: Xuenan Li

Advisor: Charles Doering and David Goluskin

Invariant Differential Operators Associated with the Pick's Matrices

Author: Wanxing Liu

Advisor: Daniel Burns

Ellipsoid Collision as a Linear Complementarity Problem

Author: Izak Oltman and Jaewon Hur

Advisor: Eduardo Corona

Thurston Norm of 2 Generator, 1 Relator Groups

Author: Natalia M. Pacheco-Tallaj

Advisor: Kevin Schreve and Nicholas Vlamis

Long-Time Simulations of Vortex Sheet Dynamics

Author: Daniel Solano

Advisor: Silas Alben and Ling Xu

Coherent Conguration on Infinite Set

Author: Jincheng Wang

Advisor: Andreas Blass

Distributions for TASEP models

Author: Han Wu

Advisor: Jinho Baik

Calyxes and Corollas

Author: E. Dean Young

Advisor: Harm Derksen

### Summer 2016

The Complex Roots and Approximation of Permanents

Authors: Han Wu & Max Kontorovich

Advisor: Alexander Barvinok

Cyclic Sieving of Matchings

Authors: Grant Bowling & Qingzhong Liang

Advisor: Thomas Lam

Understanding Sleep-Wake Dynamics with Two Process Model and BDB Model

Author: Diya Sashidhar

Advisor: Victoria Booth

Isometries And Equivalences Between Point Configurations, Extended To ε-diffeomorphism

Author: Neophytos Charalambides

Mentors: David Speyer

Almost-Isometries for Point Cloud and Image Registration

Author: Cyrus Anderson

Mentors: Steven Damelin & David Speyer

Labeling Isometric and Almost Isometric n-Point Configurations in R^{D}

Author: Michael Lu

Mentors: David Speyer

Green's function method, pKa calculation, and Poisson-

Boltzmann Equation

Author: Jingzhen Hu

Mentor: Robert Krazny

Computing examples of Hurwitz correspondences in cyclic and non-cyclic portraits.

Author: Julio Soldevilla

Mentor: David Speyer

On the Metastatic Mechanisms of Ovarian Cancer

Author: Ali Farhat

Mentor: Trachette Jackson

A Polynomial Time Deterministic Algorithm for Identity Testing Read-Once Polynomials

Author: Daniel Minahan

Mentor: Ilya Volkovich & Jeffrey Lagarias

First Koszul Homology over a Local Artinian Ring

Author: Nina Pande

Mentor: Mel Hochster

Weak Choice Principles and Forcing Axioms

Author: Elizabeth Lauri

Mentor: David Fernandez Breton

On the Pseudospctrum of One-Dimensional Schrödinger Operators with Complex Potential

Author: Xiyu Zhai

Mentor: Alejandro Uribe

Parameter Optimization and Data Fitting for the Flip-Flop Model of Sleep-Wake Regulation

Author: Michael Buiteweg

Mentor: Victoria Booth

The Hadwiger Conjecture for Unconditional Convex Bodies

Author: Daniel Barg

Mentors: Beatrice Vritsiou & Boaz Slomka

Billiard Dynamics on Surfaces of Revolution

Author: Isaac Garfinkle

Mentor: Daniel Visscher

Degree Bounds in the Nonmodular Case Invariant Theory

Author: Moise Mouyebe

Mentor: Harm Derksen

Curve Complexes of Non-Orientable Surfaces

Author: Alex Pieloch

Mentor: Richard Canary

Entropy of Triangular Lattices

Author: Shangnan Zhou

Mentors: Xiaoming Mao & Stephen DeBacker

Partial Recovery on Random Regular Graphs

Authors: Dylan Airey & Steven Damelin

Mentors: Steven Damelin & David Speyer

The Fundamental Theorem of Affine Geometry on Tori

Authors: Jacob Shulkin & Wouter Van Limbeek

Mentor: Wouter Van Limbeek

Algorithms for the Evolution of Surfaces

Author: Matthew Stone

Mentor: Selim Esedoglu

A Polarization Operation for Psuedomonomial Ideals

Author: Jeffrey Sun

Mentors: Sema Gunturkun & Jack Jeffries

Understanding ɛ-Distorded Diffeomorphisms

Author: Sean Kelly & Steven Damelin

Mentors: Steven Damelin & David Speyer

### Summer 2015

Volume Preserving Threshold Dynamics for Grain Networks

Author: Jing An

Advised by: Prof. Selim Esedoglu

The Problem of Constructing Efficient Universal Sets of Quantum Gates

Authors: Qingzhong Liang and Jessica Thompson

Advised by: Dr. Steven Damelin and Prof. Martin Strauss

A Finiteness Property of Torus Invariants

Authors: Stella Gastineau and Samuel Tenka

Advisor: Prof. Andrew Snowden

Stability of Junctions in Grain Networks

Authors: Scott Ginebaugh

Advisor: Prof. Selim Esedoglu

New Examples of Dimension Zero Categories and a Computer Program for Calculations in Representation Theory

Author: Andrew Gitlin

Advisors: John Wiltshire-Gordon and Prof. David Speyer

Study of A Particular Combinatorial Game

Authors: Zifan Li and Yuan Chen

Advisor: Prof. Andrew Snowden

A Mathematical Model of a Rod Photoreceptor Population

Author: Louis Joslyn

Advisor: Prof. Adam Stinchcombe

Degree Bounds on the Ring of Invariants

Author: Moise Mouyebe

Advisor: Harm Derksen

Numerical Ring Invariants and the Alternating Sums of Graded Betti Numbers

Author: Michael Phelan

Advisor: Prof. Sarah Kitchen

Perturbation analysis on HJB equations

Author: Guanting Chen

Advisor: Prof. Virginia Young

A One Loop Test of AdS/CFT on AdS4 x S7=Zk

Author: Wenli Zhao

Advisors: Prof. James Liu and Prof. David Speyer

An Analytic Approach to the Problem of Matroid Representibility

Authors: D. Capodilupo, S. Freedman, M. Hua, and J. Sun

Advisor: Dr. Steven Damelin and Prof J. Lagarias

Investigating Torus Orbit Closures in the Hilbert Scheme of Points in the Plane

Author: Kristofer Siy

Advisor: Prof. Jenna Rajchgot

Space Complexity of Infinite Turing Machines

Author: Dylan Stephano-Shachter

Advisor: Prof. Andreas Blass

Passive Elasticity in a Model of Snake Locomotion

Author: Angelia Wang

Advisors: Prof. Xiaolin Wang and Prof. Silas Alben

Lower Bound Cluster Algebras: Presentations, Cohen-Macaulayness and Normality

Author: Bradley Zykoski

Advisors: Prof. Greg Muller and Prof. Jenna Rajchgot

Properties of a Chain-Like Linkage

Author: Shangnan Zhou

Advisors: Prof. Xiaoming Mao and Prof. Stephen DeBacker

Multiscale Modeling of the Sleep/Wake Systems

Author: Elena Malloy

Advised by: Prof. Victoria Booth

The Conditional Distribution of the Eigenvalues of the Ginibre Ensemble

Authors: Hwanwoo Kim & Eugene Siegel

Mentors: Alon Nishry & Diego Ayala

### Summer 2014

Tropical Constructions and Lifts

Author: Hunter Ash

Advised by: Prof. Tyler Foster

Modeling Type One Diabetes with a Linear Response Function Based Model

Authors: Matthew Bauerle and Xinyan Han

Advised by: Prof. Yaniv Plan and Prof. Roman Vershynin

Biases in statistical inference: measles as a case study

Author: Alexander Becker

Advisor: Prof. Aaron King

Invariant Distributions on *l*-spaces

Author: Ben Brady

Advisor: Prof. Stephen DeBacker

A Permutation Model with Finite Partitions of the Set of Atoms as Supports

Author: Benjamin Bruce

Advisor: Prof. Andreas Blass

Substitution Polynomials and the Decomposition of Rational Functions

Author: Eric Levin

Advisor: Prof. Danny Neftin

Relatively Prime and Squarefree Integers

Author: Nora Farouk

Advisor: Prof. Patrick Boland

Exploring the Geometric Model of Riffle Shuffling

Authors: Jieun Lee, Janine Lewandoski, and Kevin Reifor

Advisor: Prof. Mark Conger

Elliptic Curves and Their Torsion

Author: Siyan “Daniel” Li

Advisor: Prof. Ian Shipman

Effect of Transaction Tax on Optimal Portfolio Liquidation and Predatory Trading

Author: Enkhjargal Lkhagvajav

Advisor: Prof. Sergey Nadtochiy

On Robust Arm-Acquiring Bandit Problems

Author: Shiqing Yu

Advisor: Prof. Xiang Yu

Sleep-wake cycles and jet lag

Author: Channa Schramm

Advisor: Prof. Victoria Booth

Hyperbolic geometry and moonshine groups

Author: Bazyli Szymanski

Advisor: Prof. Pat Boland

The Number of Generators of the First Koszul Homology of an Artianian Ring

Author: Alex Zhongyi Zhang

Advisor: Prof. Mel Hochster

Fund Manager's Portfolio Choice

Author: Zhiqing Zhang

Advisor: Prof. Gu Wang

### Summer 2013

Monte Carlo Valuation of High Dimensional American Options

Author: Michael Angileri

Advised by: Dr. Arash Fahim

A Fast Algorithm to the RadiativeTransport Equation and Implementation of

Theory Into an Applet

Author: Mihai V. Avram

Advised by: Manabu Machida

Monomial Valuations, Cusp Singularities, and Continued Fractions

Author: David Bruce and Molly Logue

Extended Vamos Matroids and the Half-Plane Property

Author: Sam Burton and Yewon Youm

Numerical Methods for the One Way Water Wave Equation

Author: Sean Carney

Advised by: Smadar Karni

Data Assimilation using Optimal Control and Kalman Filtering

Author: Wei Chen and Evelyn Lunasin

Investigating Hyperbolic Geometry

Author: Sean Douglas and Caleb Springer

An Alternative Model for Card Shuffling

Author: Christopher Drouillard

Central Simple Sugeralgebras and the Spin Group

Author: Benjamin Gammage

Binary Matrix Completion and its Application to Restaurant Recommendation System

Author: Chenrui Gao

Advisors: Prof. Yaniv Plan, Prof. Roman Vershynin

Fundamental Domain and Hyperbolic Geometry

Author: Yubo Guo

Staircase ribbons in Sn

Author: Colin Holmes

Seeking Mechanisms Behind the Formation of Synchronized Behavior in

Cellularly Diverse Neural Networks

Author: Scott Knudstrup

Advisors: Victoria Booth

Multicategory and Monoidal Category

Author: Luhang Lai

Weakly Circle-Preserving Maps in Inversive Geometry

Author: Joel C. Gibbons and Yusheng Luo

Optimizing Snake Locomotion

Author: Matthew Osborne

Advisor: Professor Silas Alben

Time Series of Implied Density

Author: Wonho Rhee

Advisor: Prof. Sergey Nadtochiy

Polynomial Identities on Eigenforms

Author: Joseph Richey and Noah Shutty

Mathematical Model and Analysis of Cellular Diffusion

Author: Dylan Corey Sena

Modeling Retinal Neurons

Author: Jonathan Timkovich

A Bound on Length Isospectral Families and Applications

Author: Weston Ungemach

Discrete Groups of Isometries and Fundamental Domains

Author: Yutong Wang

Advisor: Patrick Boland, Lizhen Ji

On Multivariate Path-Dependent Pricing

Author: Zhiyuan Wang

Advisor: Xiang Yu

Computations in the Lie Algebra of F4 and a Parameterization of Nilpotent Orbits

Author: Karl Winsor

Advisor: Stephen DeBacker

### Summer 2012

Chaotic Planetary Dynamics in Post Main-Sequence Stellar Systems

Kassandra Anderson

Advisors: Fred Adams & Anthony Bloch

Random Matrices with Prescribed Row and Column Sums

Dangxing Chen

Advisor: Alexander Barvinok

Towards a Szego Limit Theorem for Berezin-Toeplitz Operators with Singular Symbols

Tyler Friedman

Advisor: Alejandro Uribe

Calculations on a Lattice L Spanned by a Set of Conformal Vectors

Todd Gaugler

Advisor: Robert Griess

Commutators in the Metaplectic Group

Daniel Hast

Advisor: Dmitriy Boyarchenko

Non-consecutive Congruences of Dedekind-finite Sets

Author: John Holler and Matthew Tanzer

Advised by: Andreas Blass

Examining the Sample Complexity of the Maximum Likelihood Estimator in

Phylogenetic Tree Reconstruction

Sangwon Hyun

Advisor: Jose Alcala Burgos

Fundamental Domains of Discrete Groups Acting on Euclidean Space

John Kilgore

Advisor: Lizhen Ji

Quiver Representations

Molly Logue

Advisor: Harm Derksen

L1 Norm of the Valle-Poussin Kernel

Harsh Mehta

Advisor: Hugh Montgomery

Forbenius Manifold of K3 LG-model

Non-symplectic Automorphisms of Prime Order on K3-Surfaces

Rachel Suggs

Advisor: Yongbin Ruan

The Kreiss Stability Condition for Initial Boundary Value Problems

Michaela Wood

Advisor: Divakar Viswanath

### Summer 2011

Summer 2011

Dispersive Quantization

Sudarshan Balakrishnan

Advisors: Professor Benson Muite

Numerical Integration of Black Hole Solutions to the Einstein/Yang-Mills Equations

Rebekah Bartlett

Advisor: Professor Joel Smoller

A Multivariate Median in Banach Spaces and Applications to Robust PCA

David Bruce

Advisor: Professor Roman Vershynin

F-Pure Thresholds

Sifat Rahman & Joseph Billian

Advisor: Professor Bhargav Bhatt

Explorations in Genetic Oscillators

Christopher Chung

Advisor: Professor Richard Yamada

Face Vectors of Sphere Products

Kit Clement

Advisor: Patrick Boland

Modeling Human Sleep Patterns

Rebecca Gleit

Advisor: Professor Victoria Booth

Analysis of Mutual Predation in an Age-Structure System

Nathan Golovich

Advisor: Professor Charles Doering

Exploring KGH Bounds in t = 4

John Holler and Nick Wasylyshyn

Advisor: Professor Brett Hemenway

Zeta Functions, the Tate Conjecture, and Catanese Surfaces

Paul Lewis

Advisor: Professor Christopher Lyons

The Evolution of One Dimensional Crystal Surfaces in Two Configurations

Mark Liu

Advisor: Professor Hala Al Hajj Shehadeh

Calculation Process for a Star with Magnetic Field

Xiaotong Suo McClure

Advisor: Professor Joel Smoller

An Analysis of Position Probability Distributions of Trilateration

and Triangulation for Extremely Deep Space Navigation

David McMillon

Advisor: Professor Mark Conger

Geodesics of Periodic Metrics on R2

Daniel Smolkin

Advisor: Professor Ralf Spatzier

Convergence and Error Bounds of the Trotter-Kato

Scheme for the HJB equation in a two-factor model

Howard Xu

Advisor: Professor Erhan Bayraktar0

Establishing Conditions for REM Cycling in Different Networks

Aparna Ananthasubramaniam

Advisors: Professors Cecilia Diniz Behn and Victoria Booth

A quantitative study of the triangulated molecular surfaces perturbations caused by small displacements of the atomic centers

Mo Bai

Advisor: Professor Weihua Geng

Dark Halo Orbits and Tidal Stripping With the Spirographic Approximation

Rebekah Bartlett

Advisor: Professor Anthony Bloch

Tumor Angiogenesis

James Brunner

Advisor: Professor Trachette Jackson

Mathematically modeling the light responses of ganglion-cell photoreceptors

Samuel Faught

Advisor: Professor Daniel Forger

Investigations of Coupled Oscillators

Kara Fulton

Advisor: Professor Richard Yamada

Computation of Equilibrium Measures and Other Quantities of Interest for Random Matricies

Asad Lodhia

Advisor: Professor Jinho Baik

Pattern Avoidence on Arch Connected Graphs

Benjamin Kornacki

Advisor: Professor Pavlo Pylyavskyy

Sums of Isometric Pairs of Lattices

Paul Lewis

Advisor: Professor Robert Griess

Impulse Control of MultiDimensional Jump Diffusions with Infinite Variation

Christopher Link

Advisor: Professor Erhan Bayraktar

Checking For Isomorphism in Circulant Graphs

Augustus Odena

Advisor: Professor Michael Zieve

REU Paper 2010

Claudia Raithel

Advisor: Professor Anthony Bloch

Characterization of the full degree 2 del Pezzo Surface over the field of nine elements

Daniel Smolkin and Erin Kelly

Advisor: Professor Amanda Knecht

On the 2D Orientation Preserving Crystallographic Groups

Kyle Sinclair and Pearce Washabaugh

Advisor: Professors Christopher Mooney and Ralf Spatzier

An Algebraic Approach to Conformal Eld Theory

Yang Xiu

Advisor: Professor Igor Kriz

### Summer 2009

Affine Alternating Sign Matrices

Michael Bennett

Advisor: Professor Pavlo Pylyavskyy

A Universal Degree Bound for Rings of Invariants of n Point Configurations Modulo Torus Actions

Jonathan Brito

Advisor: Professor Benjamin Howard

A Mathematical Model of Tumor Growth and Angiogenesis

Akshay Desai

Advisor: Professor Trace Jackson

A Model of Suprachiasmatic Nucleus Firing Rate in the Rodent

Michelle Fleshner

Advisor: Cecilia Diniz Behn

Long Time Behavior of a Modified Becker-D¨oring System: Initial Conditions Without Compact Support

Lee Gunderson

Advisor: Professor Peter Smereka

Singularities of Logarithmic Potentials at Analytic Corners

Boaz Haberman

Advisor: Professor Divakar Viswanath

Algorithms for Threat Detection: An Inverse Boundary Value Problem

Casandra Hall

Advisor: Professor Selim Esedoglu

Analyzing Tori in Split Groups over Quasifinite Fields via Bruhat-Tits theory

Wade Hindes

Advisor: Professor Stephen DeBacker

Developing Stochastic and Deterministic Models of Quorum Sensing

Tyler Johnson

Advisor: Richard Yamada

The Recursive Computatibility of Hyperelliptic Hodge Integrals with at Least One 1 Class

Andrew Kiluk

Advisor: Professor Renzo Cavalieri

The Effects of Varying Connectivity Strength on the Frequency of Inhibitory Neurons in Several Network Types

Khalid Kunji

Advisor: Professor Victoria Booth

Reconstruction and prediction of chaotic signals

Xuan Liang

Advisor: Professor Divakar Viswanath

Improving the Upper Bound on Dirichilet L-Functions Induced by a

Character Modulo a High Power of a Prime

Christopher Link

Advisor: Professor Djordje Milicevic

Heegaard Splittings and Bounds for Closed Hyperbolic 3-Manifolds

Brian Mann

Advisors: Professors Richard Canary and William Breslin

Linear Factors and the Pollard Rho Method

Brian Nixon

Advisor: Professor Harm Derksen

The 100% Off Auction:

Daniel Russo

Advisor: Professor Tilman Borgers

Using periodic orbits to predict trajectories on the Lorenz attractor

Kirill Serkh

Advisor: Professor Divakar Viswanath

Approximating Dendrimer Bonding Site Locations

Stephen VandenBrink

Advisor: Professor Jack Waddell

The Thermodynamics of Solid Tumor Growth: A Continuum Mechanics Approach

Sarah Verner

Advisor: Professor Krishna Garikipati

The Caldero degenerations of the Grassmannian of 3-planes in 6-space and the positive tropical G(3,6)

Seth Wolbert

Advisor: Professor Benjamin Howard

### Summer 2008

The Kernel of a Derivation on the Field of Rational Functions and the Zariski Closure of Local Integral Curves

Wade Hindes

Advisor: Professor Harm Derksen

Mathematical Models for Quorum Sensing in Vibrio harveyi and Pseudomonas aeruginosa

Kaytlin Brinker

Advisor: Professor Jack Waddell

Model of an Orexin Neuron

Katie Williams

Advisor: Cecilia Diniz Behn

Tropical Intersections and Shadow Points

Alex Larson

Advisor: Renzo Cavilieri

Coexistence of Multiple Parasitoids on a Common Host in a Globally Connected Metapopulation

Benjamin Berman

Advisor: Aaron King

Total Promotion

Kevin Dilks

Advisor: Kyle Peterson

Properties of Steady State Solutions for Critical Two-Layer Shallow Water Flows

Alfredo Wetzel

Advisor: Smadar Karni

Synchronization of Signaling Systems

Hasan Cheema

Advisor: Daniel Forger

An Equivalent Condition for the Reducibility of Tropical Curves

Brian Mann

Advisors: Hannah Markwig and Renzo Cavalieri

Generalising Cut and Join and Recursions of Hurwitz Numbers

Andrew Kiluk

Advisor: Renzo Cavalieri

Equidistribution of Descent-Like Statistics for the Hyperoctahedral Group

Laurie Lai

Advisor: Kyle Peterson

### Summer 2007

Sub-game Perfect Equilibria of the Simultaneous Ascending Auction

Brian Baisa

Advisor: Professor Tilman Börgers

Steinberg Complexes and Affine Descents

Kevin Dilks

Advisor: Professors Kyle Petersen and John Stembridge

Action Potential Behavior in Hodgkin-Huxley and FitzHugh-Nagumo Models and Networks of O and I-cells

Stephen Gao

Advisor: Professor Victoria Booth

Optimization of Hydroelectric Pumped Storage: An Extension of Optimal Switching

Anthony Kuehne

Advisor: Professor Michael Ludkovski

Level Sets of the Takagi Function and Hausdorff Dimension

Zachary Maddock

Advisor: Professor Jeffrey Lagarias

Working with Maple

Anna M. Musial, Sam L. Rosenbaum

Advisor: Professor Arthur Wasserman

Two New Exceptional Quivers of Finite Mutation Type

Theodore Owen

Advisor: Professor Harm Derksen

Hyperelliptic Hodge Integrals

Peter Troyan

Advisor: Professor Renzo Cavalieri

Changes in Betti numbers of Hessenberg varieties on restricted tableaux

Derek Van Farowe

Advisor: Professor Julianna Tymoczko

Properties of Steady State Solutions for Shallow Water Flows

Alfredo Wetzel

Advisor: Professors Smadar Karni and Jorge Balbas

Ternary Basic Digit Sets

Christopher J. Young

Advisor: Professor Jeffrey Lagarias

### Summer 2006 - 2003

**Summer 2006**

Ex-Post Equilibria in Games of Incomplete Information

Timothy McQuade

Advisor: Professor Tilman Borgers

The Gap Size Distribution of Parked Cars and the Coulomb Gas Model

Anthony Fader

Advisor: Professor Jinho Baik

Properties of the Takagi Function

Zachary Maddock

Advisor: Professor Jeffrey Lagarias

Virtual Cathode Simulation in 1D using a Grid-Free Poisson Solver

Ying Ki Yim

Advisor: Professor Lyudmyla Barannyk

Probabilistic Interpretations of Variables in Transient M/G/1 Random Walks

Drew Bollinger and David Wheeler

Advisor: Professor B.A. Taylor

**Summer 2005**

A Generalization of Conway Number Games to Multiple Players

Christopher Cunningham

Advisor: Professor Igor Kriz

Efficient And Accurate Unstructured Mesh Generation

Matthew Elsey

Advisor: Professor James Rossmanith

A Transformation Of A Standard Random Walk

Jason Goldstick

Advisor: Professor Joseph Conlon

Using Lattice Packings to Create Extended Codebooks

Daniel Sikora

Advisor: Professor Anna Gilbert

Directed Last-Passage Site Percolation with Bernoulli Site Distribution

Robert Hines

Advisor: Professor Jinho Baik

Dynamics of Interacting Point Charges and Vortices within a Constant Magnetic Field

Stephen Marin

Advisor: Professor Andrew Christlieb and Robert Krasny

Summer 2004

Momentum Matching for the Pendulum on Cart Problem

Esha Krishnaswamy

Advisor: Professor David Schneider

Linear Perturbations of a Class of Berezin-Toeplitz Operators

Adam Bendorf

Advisor: Professor Alejandro Uribe

New Results in Extremal Number Theory

Randolph Pistor

Advisor: Professor Kannan Soundararajan

Homeomorphism Groups of the Sierpinski Carpet and Sierpinski Gasket

Joel Louwsma

Advisor: Professor Bruce Kleiner

Computing Traveling-Wave Front Solutions in a Diffusive Predator-Prey Model

Brian Kim

Advisor: Professor Gregory Lyng

Inversions Within Restricted Fillings of Young Tableaux

Sarah Iveson

Advisor: Professor Julianna Tymoczko

Stabilization of Inverted Pendulum Systems

David Aristoff

Advisor: Professor Tony Bloch

An Overview of Green's Functions on Metrized Graphs

Douglas Li

Advisor: Professor Mattias Jonsson

Some R-tree Results

Jeremy Cook

Advisor: Professor Elmas Irmak

On a conjecture concerning non-intersecting congruence classes

Robert Hines

Advisor: Professor Kannan Soundararajan

The Problem of Type

Charles Crissman

Advisor: Professor Mario Bonk

The Likelihood of Cyclic Results in Pairwise Elections

Greg Malivuk

Advisor: Professor Divakar Viswanath

**Summer 2003**

Inverted Double Planar Pendulum on a Cart: Feedback Stabilization Using the Method of Controlled Lagrangians

David G. Aristoff

Advisor: Professor Anthony Bloch

A Mathematical Model of Receptor-Mediated Apoptosis: Fas Trimerization and FADD Recruitment

Ronald Lai

Advisor: Professor Trachette Jackson

The Temperly-Lieb Functions on Totally Non-Negative Matrices

Brendon Rhoades

Advisor: Professor Mark Skandera

Schur Positivity Conjectures about Temperly Lieb functions on Jacobi Trudi Matrices

Sean Gerrish

Advisor: Professor Mark Skandera

Dynamical Systems and Circle Maps

Ronit Slyper

Advisor: Professor Nikola Petrov

RELATIONS IN N[X] LEADING TO FINITE DIMENSIONAL POLYNOMIAL RIGS AND

EPIMORPHIC IMAGES OF N

Bertrand J. Guillou

Advisor: Professor Andreas Blass