# Past SPUR Projects

## 2021 Topics

**Enrico Colón** -
Split-multiplicity-free flagged Schur polynomials

Mentor: Yibo Gao

**Preston Cranford and Peter Rowley** -
Finding Bounded Simplicial Sets with Finite Homology

Mentor: Robert Burklund

**Dain Kim and Anqi Li** -
Cubic Goldreich-Levin

Mentor: Jonathan Tidor

**Daishi Kiyohara** -
A new approach to the upper estimate of lattice points on a curve via $\mathscr{l}^2$ decoupling

Mentor: Feng Gui

**Jacob Lerma and Jeffery Opoku-Mensah** -
Positivity of the mass using p—harmonic functions on asymptotically flat 3-manifolds

Mentor: Julius Baldauf

**Yuyuan Luo** -
Explicit Construction of the Hecke Character Associated to Certain CM Elliptic Curves

Mentor: Gefei Dang

**Alan Peng** -
Convolution-exact perverse sheaves on the affine flag variety

Mentor: Calder Morton-Ferguson

**Amanda Ledesma Vanegas** -
A Reduced Swift-Hohenberg Style Model for Vegetation Patterns in Drylands

Mentor: Davis Evans and Alasdair Hastewell

**Frank Wang** -
Toward explicit Hilbert series of quasi-invariant polynomials in characteristic $p$ and $q$-deformed quasi-invariants

Mentor: Calder Morton-Ferguson

## 2020 Topics

**Elisabeth Bullock and Katie Gravel** -
Higher Secondary Polytopes for Two-Dimensional Zonotopes

Mentor: Alexey Balitskiy

**Kevin Chang** -
A v_{1}-banded vanishing line for the mod 2 Moore Spectrum

Mentor: Robert Burklund

**Korina Digalaki** -
Deformed Airy Kernels in Height Function Fluctuations of the Stochastic Six-Vertex Model

Mentor: Roger Van Peski

**Aparna Ajit Gupte and Kerri Lu** -
Fine-Grained Complexity of Sparse Linear Regression

Mentor: Frederic Koehler

**Christina Meng** -
On Configuration Spaces of Higher-Dimensional Analogues of Genus *g* Surfaces

Mentor: Araminta Gwynne

**Dylan Pentland** -
Filtrations on Block Subalgebras of Reduced Universal Enveloping Algebras

Mentor: Andrei Ionov

**Kevin Ren** -
Incidence estimates for $α$-dimensional tubes and $β$-dimensional balls in $\mathbb{R}^2$

Mentor: Yuqiu Fu

**Carl Schildkraut and William Zhao** -
Relating Counts in Different Characteristics for Steiner’s Conic Problem

Mentor: Chun Hong Lo

**Calvin Yost-Wolff** -
A Geometric RSK Correspondence and Relative Positions Within Springer Fibers

Mentor: Oron Propp

**Hung-Hsun Hans Yu** -
Joints of Varieties

Mentor: Jonathan Tidor

## 2019 Topics

**Kevin Beuchot Castellanos** -
Lifting to $\mathbb{Z}/p^2\mathbb{Z}$ and Splittings of the de Rham Complex

Mentor: Robert Burklund

**Andrew Gu** -
Arithmetic of Weighted Catalan Numbers

Mentor: Yibo Gao

**Kaarel Haenni** -
Wilf-equivalence in Weyl Groups and Signed Permutations

Mentor: Yibo Gao

**Alexandra Hoey and Wanyi Xiao** -
A Combinatorial Proof for a Generalized Reciprocity Theorem

Mentor: Pakawut Jiradilok

**Miles Johnson and Natalie Stewart** -
Some Graphical Realizations of Two-Row Specht Modules of Iwahori-Hecke Algebras of the Symmetric Group

Mentor: Oron Propp

**Nelson Niu and Lillian Zhang** -
A Multiplicity-free Macdonald Identity

Mentor: Gurbir Dhillon

**Qiuyu Ren** -
On the Union of Essentially Distinct $\delta$-tubes

Mentor: Yuqiu Fu

**Douglas Stryker** -
Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at -$\infty$

Mentor: Ao Sun

**Christopher Xu** -
Traces of CM Values of Certain Weak Maass Forms

Mentor: Yongyi Chen

**Yuan Yao and Shengtong Zhang** -
Equiangular Lines with a Fixed Angle

Mentor: Jonathan Tidor

## 2018 Topics

**Julius Baldauf-Lenschen** -
Sharp entropy bounds for plane curves and dynamics of the curve shortening flow

Mentor: Ao Sun

**Sanath Devalapurkar** -
Uncompleting *E*-theory and Johnson-Wilson theory at height 2

Mentor: Robert Burklund

**Juan Gil and Joshua Amaniampong** -
Uniform generation of *w*-free strings and *H*-free subgraphs with partial rejection sampling

Mentor: Jake Wellens

**Calvin Hsu** -
Quantitative decay for nonlinear wave equations

Mentor: Ruoxuan Yang

**Laura Koemmpel and Olga Medrano** -
How graph structure and connectivity affects the propagation of cooperation in a network

Mentor: Ashwin Narayan

**Deven Jay Lahoti** -
Torsion subspaces of complex Lie groups

Mentor: Gurbir Dhillon

**Junyao Peng** -
Counting shellings of complete bipartite graphs and trees

Mentor: Yibo Gao

**Nikhil Reddy** -
Generalizing auxeticity to aperiodic networks

Mentor: Vishal Patil

**Dhruv Rohatgi** -
Biclique partitions and off-diagonal ordered Ramsey numbers

Mentor: Jake Wellens

**Julian Wellman** -
Flips in reduced plabic graphs

Mentor: Alexey Balitskiy

**Yunkun Zhou** -
The Sidorenko problem for directed graphs in tournaments

Mentor: Jonathan Tidor

## 2017 Topics

**Brian Chen** -
The Suboptimality of Asymmetric Recursive Reconstruction Algorithms

Mentor: Ashwin Narayan

**Panagiotis Dimakis** -
On a Connection between Affine Springer Fibers and L-U Decomposable Matrices

Mentor: Guangyi Yue

**Alonso Espinosa Domínguez** -
On Kakeya-type Problems For Hyperplanes in $\mathbb{R}$^{d}

Mentor: Paxton Turner

**Hyo Won Kim and Christopher Maldonado** -
On Graphs and the Gotsman-Linial Conjecture for $d = 2$

Mentor: Jake Lee Wellens

**Justin Lim** -
Building Forests in Maker-Breaker Games: Upper and Lower Bounds

Mentor: Frederic Koehler

**Max Murin** -
Parameters for Generalized Hecke Algebras in Type B

Mentor: Seth Shelley-Abrahamson

**Xianglong Ni** -
The Bracket in the Bar Spectral Sequence for a Finite-Fold Loop Space

Mentor: Hood Chatham

**Juan Carlos Ortiz** -
The Diagonal Cohomology Class of Vertical Bundles

Mentor: Jackson Hance

**Luke Sciarappa** -
Model Categories in Equivariant Rational Homotopy Theory

Mentor: Robert Burklund

**Max Vargas and Kimberly Villalobos** -
A Random Walk Formulation of Learning in Restricted Boltzmann Machines

Mentor: Mason Biamonte

**Jianqiao Xia** -
Topology of Two-Row Type Springer Fibers

Mentor: Gus Lonergan

## 2016 Topics

**Kavish Gandhi and Yonah Borns-Weil** -
Bounds on Moment-Based Learning of Mixture Distributions

Mentor: Amelia Perry

**YiYu Zhang** -
A Correspondence between Generic Alcoved Polytopes and Subdivisions of the Root Polytope of Lie Type $A$

Mentor: Pavel Galashin

**Jonathan Tidor and Victor Wang** -
1-color-avoiding Paths, Special Tournaments, and Incidence Geometry

Mentor: Ben Yang

**Tahsin Saffat** -
Linearizable Laurent Phenomenon Sequences

Mentor: Pavel Galashin

**Lingfu Zhang** -
Refinements of 2-dimensional Strichartz Estimate by the Maximum Wave Packet

Mentor: Hong Wang

**Joseph Zurier** -
Symplectic Representation Theory and the Weyl Algebra in Positive Characteristic

Mentor: Augustus Lonergan

**James Rowan ** -
$S$-unit equations and curves of genus $2$ with good reduction away from $3$

Mentor: Borys Kadets

**Hyun Jong Kim** -
Local Conjugacy in $\textrm{GL}_2(\mathbb{Z}/p^2\mathbb{Z})$

Mentor: Atticus Christensen

**Colin Aitken** -
Generators for Maximal Finite Subgroups of the Morava Stabilizer Group

Mentor: Lyuboslav Panchev

**Arkadiy Frasinich** -
An Investigation on an Extension of Mullineux Involution

Mentor: Augustus Lonergan

## 2015 Topics

**Asra Ali** -
Straight-line orthogonal drawings of ternary trees

Mentor: Sylvain Carpentier

**Robert Burklund** -
A classification of formal group laws over torsion-free rings

Mentor: Lyuboslav Panchev

**Adan Medrano Martin del Campo** -
Tropical embeddings of metric graphs

Mentor: Sylvain Carpentier

**Jingwen Chen** -
In search of a Mahler-Type inequality for the external conformal radius

Mentor: Nicholas Strehlke

**Yuchen Fu** -
A family of finite-dimensional representations of generalized double affine Hecke algebras of higher rank

Mentor: Seth Shelley-Abrahamson

**Ofer Grossman** -
An infinite family of non-abelian monogenic number fields

Mentor: Dongkwan Kim

**Zachary Izzo** -
Asymptotic properties of random polytopes

Mentor: Nicholas Strehlke

**Yutao Liu** -
Geometry of minimal surfaces with layered boundary conditions

Mentor: Paul Gallagher

**Mark Sellke and Sammy Luo** -
The Saxl conjecture for 4th powers via the semigroup property

Mentor: Dongkwan Kim

**Anderson Wang** -
Invariants of machines in wiring diagrams

Mentor: Lyuboslav Panchev

**Tianyou Zhou** -
An integral formula for triple linking in hyperbolic space

Mentor: Paul Gallagher

## 2014 Topics

**Dhroova Aiylam and Carlos Cortez** -
Combinatorial interpretations of the trace moment sequences of certain compact Lie subgroups of USp(4)

Mentor: David Corwin

**Aaron Brookner** -
On the Cohen-Macaulayness of certain hyperplane arrangements

Mentor: David Corwin

**Yibo Gao and Yuzhou Gu** -
Fast Algorithms for Solving SDPs on Graphs

Mentor: Adrian Vladu

**Cole Graham** -
Quantitative Bounds for Hurwitz Stable Polynomials under Multiplier Transformations

Mentor: Xuwen Zhu

**Felipe Hernandez** -
Some physical space heuristics for Strichartz estimates

Mentor: Chenjie Fan

**Saarik Kalia** -
Generalizations of the Szemerédi-Trotter Theorem

Mentor: Ben Yang

**Peter Kleinhenz** -
Stationary Solutions of the Nonlinear Schrödinger Equation in Hyperbolic Space

Mentor: Chenjie Fan

**Mrudul Thatte** -
Hausdorff dimension of cycles in d-maps on the circle

Mentor: Xuwen Zhu

**Suhas Vijaykumar** -
Higher Bruhat Orders of the Weyl group B_n

Mentor: Seth Shelley-Abramson

**Georgios Vlachos** -
Approximate distance oracles for sparse graphs

Mentor: Adrian Vladu

## 2013 Topics

**Hyun Sub Hwang** -
Computational power of quantum vs. classical oracles

Mentor: Adam Bouland

**Supanat Kamtue** -
Domino-tiling problem

Mentor: Ben Yang

**Mitchell Lee** -
The complexity class PDQP

Mentor: Adam Bouland

**Zipei Nie and Anthony Wang** -
Points on lines in F^{3}_q

Mentor: Ben Yang

**Sung Gi Park and Qinru Shi** -
New bounds on extremal numbers in acyclic ordered graphs

Mentor: Rik Sengupta

**Yuqi Song and Andi Wang** -
Convergence of eigenvalues to the support of the limiting measure in a beta matrix model

Mentor: Chenjie Fan

**Yanlin Zhao** -
A particle system with interlacing pattern

Mentor: Xin Sun

**Fan Zheng** -
Gauss sphere problem with polynomials

Mentor: Chenjie Fan

## 2012 Topics

**Amol Aggarwal** -
Correlation Functions of the Schur Process Through
Macdonald Difference Operators

Mentor: Guozhen Wang

**Weixin Chen** -
Discrete-Time Anisotropic KPZ Growth of Random Surface in 2+1 Dimensions

Mentor: Xuwen Zhu

**Lynn Chua** -
Psi-Epistemic Theories: The Role of Symmetry

Published in Physical Review A 88:3 (2013).

Mentor: Adam Bouland

**Jiaoyang Huang** -
Spectrum Asymptotics of Airy's Operator under Perturbation

Mentor: Xuwen Zhu

**Yajit Jain** -
Homological Algebra over the Representation Green Functor

Mentor: John Ullman

**Jelena Markovic** -
Least Absolute Deviations Method for Sparse Signal Recovery

Mentor: Ruixun Zhang

**Akin Morrison** -
A Link Theoretic Perspective on the Isotopy Type of Real Algebraic Curves

Mentor: Philip Tynan

**Mark Velednitsky** -
Graph Crossing Number and Isomorphism

Mentor: Adam Bouland

**Praveen Venkataramana** -
On a Z_{4}-Symmetric Operator over a Lemniscatic Elliptic Curve

Mentor: Philip Tynan

**Max Wimberley** -
Rook Placements on Young Diagrams: Towards a q-Analogue of the Harer-Zagier Formula

Mentor: Alejandro Morales

**Tianqi Wu** -
Impedance of Electrochemical Cells

Mentor: Yi Zeng

**Jessie Zhang** -
Periodicity in the Periodic Lambda Algebra

Mentor: Guozhen Wang

## 2011 Topics

**Sergei Bernstein** -
Lagrangian Spheres in *H _{m}* and the Effect of Dehn Twists

Mentor: Ailsa Keating

**Ben Bond** -
Modeling Abelian Networks with Sandpiles

Mentor: Benjamin Iriarte

**Eben Freeman** -
A Doubling Property for the Gradient of a Harmonic Function

Mentor: Lu Wang

**Yunzhi Gao** -
On the Quotient Space of Functions Modulo the Action of Divergence-free Vector Field

Mentor: John Ullman

**Whan Ghang** -
Involutive Power Series and Perfect Squares of Formal Power Series

Mentor: John Ullman

**Brian Hamrick** -
Computational Verification of a Conjecture in Algebraic Geometry

Mentor: Kartik Venkatram

**Travis Hance** -
A New Eulerian Permutation Statistic and Generalizations

Mentor: Nan Li

**Yangzhou Hu** -
On the Number of Fixed-Height Semiorders and Poset Linear Extension Maximization

Mentor: Nan Li

**Gabriel Karpman** -
Application of the Hodge Decomposition to the Harmonic Map Heat Flow

Mentor: Lu Wang

**Minseon Shin** -
A Generalization of an Action of the Braid Group on Upper-Triangular Matrices

Mentor: Ailsa Keating

**Jeffrey Wu** -
Towards Bootstrap-less Homomorphic Encryption

Mentor: Kartik Venkatram

## 2010 Topics

**Aldo Pacchiano Camacho** -
Fast Optical Character Recognition Via Cryptanalysis and Geometry

Mentor: Kartik Venkatram

**Harrison Chen** -
Representation of Rational Cherednik Algebras in Positive Characteristic

Mentor: Martina Balagović

**Andrei Frimu** -
On the Maximal Number of Rational Points on a Curve of Genus 4 Over *F _{11}* and

*F*

_{13}Mentor: Bhairav Singh

**Raghu Mahajan** -
Sweepouts and Level-set Flow

Mentor: Lu Wang

**Christopher Policastro** -
Category $\mathcal{O}$ for the Rational Cherednik Algebra of *G _{12}*

Mentor: Martina Balagović

**Jonathan Schneider** -
Face Numbers and Ehrhart Polynomials

Mentor: Hoda Bhidkori

**XiaoLin Shi** -
Combinatorial Properties and Optimization of a Rooted Graph Polynomia

Mentor: Amanda Redlich

**Umut Varolgunes** -
The Dimension, the Degree, and the Initial Complex of the Secant Ideal of the Second Hypersimple

Mentor: Hoda Bhidkori

**Sally Wolfe** -
Cycle Structure and Pattern Avoidance of *abc*-permutations

Mentor: Amanda Redlich

**Qiaochu Yuan** -
Lattice Paths and the Quadratic Coefficient of Kazhdan-Lusztig Polynomials

Mentor: Bhairav Singh

## 2009 Topics

**Asilata Bapat** -
The Lower Central Series Quotients of a Free Associative Superalgebra

Mentor: David Jordan

**Emily Berger** -
Hurwitz Equivalence in the Dihedral Group

Mentor: James Pascaleff

**Gabriel Bujokas** -
On the Smallness of Laumon's Resolution of Drinfeld's Compactification

Mentor: Qian Lin

**Shaunak Kishore** -
On the Critical Group of Line Graphs and Planar Graphs

Mentor: Hoda Bidkhori

**Qingchun Ren** -
An Exotic Map on Extended Dynkin Diagrams

Mentor: Qian Lin

**Colin Sandon** -
Curves in Tropical Geometry

Mentor: James Pascaleff

**Philip Tynan** -
Explicit Examples of Strebel Differentials

Mentor: David Jackson-Hanen

**Brayden Ware** -
Understanding Shrinking Curves on a Sphere

Mentor: Hoskuldur Halldorsson

**Alexandr Zamorzaev** -
On (Co)Homology of 2-Groups

Mentor: Jennifer French

## 2008 Topics

**Aleksandr Ankhipov** -
On The Pairwise Sums Problem

Mentor: Zuoquin Wang

**Thomas Belulovich** -
Three Problems in Morse Theory

Mentor: Nikhil Savale

**Wes Brown ** -
Antimagic Labelings and the Antimagic Strength of Graphs

Mentor: Hoda Bidkhori

**Peter Caday** -
Investigation into the Original and Dual McKay Correspondences

Mentor: Fucheng Tan

**Rishi Gupta** -
Computing the Zeta Function of Elliptic and Hyperelliptic Curves

Mentor: Ruochuan Liu

**Sungyoon Kim** -
On Dual Stable Grothendieck Polynomials

Mentor: Hoda Bidkhori

**Yuncheng Lin** -
Szemerédi-Type Problems in Finite Fields

Mentor: Julia Wolf

**Cordelia Link** -
On Sets Whose Difference Set Contains No Squares

Mentor: Julia Wolf

**Yinghui Wang** -
Some Study On The Generalized Euler-Maclaurin Formula

Mentor: Zuoquin Wang

**Bohua Zhan** -
Representing Graphs by Linear Subspaces

Mentor: Ruochuan Liu

## 2007 Topics

**Erika Bakse** -
Sumsets and Graph Theory in Additive Combinatorics

Mentor: Catherine Lennon

**Alessandro Chiesa** -
Generalizing Regev's Cryptosystem

Mentor: Amit Jayant Deshpande

**Variants of the Chromatic Number of the Plane** -
Joseph Cooper

Mentor: Luis Rademacher

**Hansheng Diao** -
Freiman-Type Theory For Small Doubling Constant

Mentor: Catherine Lennon

**Rebecca** -
Freund A representation of the type *B _{N}*

*d*AHA

Mentor: Xiaoguang Ma

**Shinnyih Huang** -
Discussion on Sandglass Conjecture and Generalization

Mentor: Hoda Bidkhori

**John Kim** -
On the Basis of $[A_n, A_n]$ / $[[A_n, A_n], A_n]$

Mentor: Xiaoguang Ma

**Khoa Lu Nguyen** -
Some Noncompact Convexity Theorems for Hermitian Matrices

Mentor: Zuoqin Wang

**Anand Rajagopalan** -
Lower bounds for linear locally decodable codes

Mentor: Amit Deshpande

**Xuancheng Shao** -
Minimum Partition into Product Sets

Mentor: Luis Rademacher

**Charmaine Sia** -
Zero-sum problems in finite groups

Mentor: Hoda Bidkhori

## 2006 Topics

**Irida Altman** - Some Irreducible Representations of the Braid Group

**Anand Deopurkar** - Characteristic Classes and Chern-Weil Theory

**Galyna Dobrovolska** - Comparing Products of sl_{n} Characters

**Hyun Soo Kim** - Networks, Surfaces, and Representability

**Raju Krishnamoorthy** - 2-Bottom Schur Functions

**Andriana Lopez** - Factoring Polynomials

**KangHao Lu** - Three Dimensional Rook Theory

**Jason Priestley** - Two Simple Computations Proving the Futaki Invariant

**Dale Winter** - Computing De Rham Instersection Cohomology Groups for Pseudomanifolds