Text: Carlsson and Vejdemo-Johansson, Topological data analysis with applications
Mentee: Lior Ben-David
Mentor: Miika Tuominen
Geometry of groups
Mentee: Anthony Doll
Mentor: Mark Pengitore
From spaces and continuity to categories and functoriality
Text: Bradley, Bryson, and Terilla, Topology: a categorical approach
Mentee: Yejun Kim
Mentor: Eleftherios Chatzitheodoridis
On primes and irreducibles: aren’t they the same?
Text: Ireland and Rosen, A classical introduction to modern number theory
Mentee: Jake Fulford
Mentor: Alejandro de las Peñas Castaño
The central limit theorem
Text: Lesigne, Heads or tails: an introduction to limit theorems in probability
Mentee: Charlie Wang
Mentor: Yaolong Shen
Fall 2022
Naturality of the double dual evaluation map
Text: Serge Lang, Linear algebra & Harold Simmons, An introduction to category theory
Mentee: John Layne
Mentor: Eleftherios Chatzitheodoridis
Measure theory: the foundations of probability
Text: Sydney Resnick, A probability path
Mentee: Zoe Zerui Wang
Mentor: Daniel Slonim
Hilbert’s 10th problem
Text: Murty and Fodden, Hilbert’s tenth problem: an introduction to logic, number theory, and computability
Mentee: Trevor Millard
Mentor: Michael Wills
The fast Fourier transform
Text: M. C. Pereyra and L. A. Ward, Harmonic analysis: from Fourier to wavelets
Mentee: Zihan Mei
Mentor: Joe Webster
Foundations of machine learning
Text: Shai Shalev-Shwartz and Shai Ben-David, Understanding machine learning: from theory to algorithms
Mentee: Qingyang Wang
Mentor: Mojdeh Tarighat Feller
Fractal geometry
Text: Falconer, Fractal Geometry
Mentee: Maahi Vyas
Mentor: Miika Tuominen
Quadratic reciprocity
Mentee: Bliss Bodawala
Mentor: Kian Cheong Aik
Partitions and representation theory of Sn
Text: Howe, An invitation to representation theory
Mentee: Grace Guinan
Mentor: Eleanor McSpirit
Lattice-based cryptography
Text: Hoffstein, Pipher, and Silverman, An introduction to mathematical cryptography
Mentee: Jack Liu
Mentor: Alejandro de las Peñas Castaño
Spring 2022
Introduction to Knot Theory
Text: The Knot Book, Adams
Mentee: Faith Ellis
Mentor: Louisa Liles
Reasoning and Mathematical Elegance
Text: Proofs from THE BOOK, Aigner
Mentee: Blair Stilley
Mentor: Walker Stern
Categories and Arrows
Text: Category Theory in Context, Riehl
Mentee: Declan Stacy
Mentor: Walker Stern
Complex Analysis
Text: Functions of One Complex Variable, Conway
Mentee: David Bass
Mentor: Shunyu Wan
Elliptic Curves and Cryptography
Rational Points on Elliptic Curves, Silverman
Mentee: Said Kaili
Mentor: Will Craig
Introduction to Machine Learning
Text: Machine Learning: A Probabilistic Perspective, Murphy
Mentee: Mina Kim
Mentor: Valentina Zapata Castro
General Topology and an Application
Text: Topology, Munkres
Mentee: Zacary Abusmaies
Mentor: Tanner Carawan
p-adic Numbers and Interpolation of Special Zeta Values
Text: p-adic Numbers, p-adic Analysis, and Zeta-Functions. Koblitz
Mentee: Tanner Leonard
Mentor: Joe Webster
Fractals and Hausdorff Dimension
Text: Fractal Geometry: Mathematical Foundations and Applications, Falconer
Mentee: Ivory Tang
Mentor: Gennady Uraltsev
Introduction to Fourier Analysis
Text: Fourier Analysis: An Introduction, Stein, Shakarchi
Mentee: Ningxin Zhang
Mentor: Gennady Uraltsev
Fall 2021
Geometric Group Theory
Text: Office Hours with a Geometric Group Theorist, Clay and Margalit
Mentee: Olivia Goodrich
Mentor: Alec Traaseth
The Congruent Number Problem
Text: Introduction to Elliptic Curves and Modular Forms, Koblitz
Mentee: Catherine Cossaboom
Mentor: Will Craig
Brownian Motion and Stochastic Differential Equations
Text: Probability: Theory and Examples, Durrett
Mentee: Jeremy Nachison
Mentor: Don Phillips
Point Set Topology
Text: Introduction to Topology, Baker
Mentee: Liran Li
Mentor: Shunyu Wan
A Survey of Galois Theory
Text: A Book of Abstract Algebra, Pinter
Mentee: Yunsheng Lu
Mentor: Matt McCarthy
Finite Dimensional Lie algebras and their Root Systems
Text: Lie Algebras of Finite and Affine Type, Carter
Mentee: Alex Emmert
Mentor: Weinan Zhang
Möbius Inversion and Cyclotomic Polynomials
Text: Generating Functionology, Wilf
Mentee: Michael Walker
Mentor: Joe Webster
The Fundamental Group of a Knot Complement
Texts: The Knot Book, Adams
Mentee: Donovan Cates
Mentor: Louisa Liles
Cryptographic Hash Functions from Expander Graphs
Text: Introduction to Modern Cryptography, Katz and Lindell
Mentee: Eli Kin
Mentor: Neelav Dutta
Spring 2021
Hilbert’s Nullstellensatz
Text: Abstract Algebra, Dummit and Foote
Mentee: Omar Alsalihi
Mentor: Alec Traaseth
Isometries of the Hyperbolic Plane
Text: Notes of Caroline Series
Mentee: William MacMillan
Mentor: Ian Runnels
Exploring the Robinson-Schensted-Knuth Algorithm
Text: Young Tableaux With Applications to Representation Theory and Geometry, William Fulton
Mentee: Katherine Cadzow
Mentor: Mojdeh Tarighat
Grid Homology
Text: Grid Homology for Knots and Links, Ozsváth, Stipsicz, and Szabó
Mentee: Nick Cooney
Mentor: Peter Johnson
Fall 2020
Galois Theory and its Applications
Text: A Book of Abstract Algebra, Charles Pinter
Mentee: Bilal Khan
Mentee: Casia Siegel
Mentor: Will Craig
Group Theory and Rubik’s Cube
Text: Notes of Janet Chen
Mentee: Sabrina Fuller
Mentor: Alec Traaseth
Classification of Surfaces
Text: A Guide to the Classification Theorem for Compact Surfaces, Gallier and Xu
Mentee: Harsh Padhye
Mentee: Zhijun Cao
Mentor: Alec Traaseth
Analyzing Spaces
Text: Functional Analysis, Bachman and Narici
Mentee: Emma Graham
Mentor: Bruno Braga
Synthetic Topology
Text: Synthetic Topology of Data Types and Classical Spaces, Martin Escardo
Mentee: Joey Rudek
Mentor: Neelav Dutta
Spring 2020
Homological proof of the Brouwer fixed point theorem
Text: Algebraic Topology, Hatcher
Mentee: Thomas Blue
Mentor: Chris Lloyd
Fall 2019
The Geometries of Surfaces
Text: Low-Dimensional Geometry, Bonahon
Mentee: Haoyu Li
Mentor: Ian Runnels
Point-set Topology
Text: Topology, Munkres
Mentee: Dean Sublett
Mentor: Shunyu Wan
Fibered Categories
Text: Categorical Logic and Type Theory, Jacobs
Mentee: Nijat Khanbabayev
Mentor: John Harnois
Character Theory
Text: Linear Representations of Finite Groups, Serre
Mentee: Greg Conneen
Mentee: Spencer Martin
Mentor: Chris Lloyd
Alexander’s Theorem
Text: The Knot Book, Adams
Mentee: Noah Brenny
Mentor: Rostislav Akhmechet
Spring 2019
Gröbner bases and applications in characteristic p
Text: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Cox, O’Shea, and Little
Mentee: Abrar Majidi Idrissi
Mentor: Huy Dang
Classification of the Simple Lie Algebras
Text: Introduction to Lie Algebras, Erdmann and Wildon
Mentee: Hisham M Assana
Mentor: Arun Kannan
The Geometry of Free Groups
Text: Geometric Group Theory: An Introduction, Loeh
Mentee: Katherine Andrews
Mentor: Ian Runnels
Hamiltonian ODEs: Deterministic and probabilistic studies
Text: ODEs and Dynamical Systems, Teschl
Mentee: Yunlu Li
Mentor: Mouhamadou Sy
The Yoneda Embedding and its Applications
Text: Categories for the Working Computer Scientist, Barr and Wells
Mentee: Joseph Snitzer
Mentor: Chris Chung
Real Analysis and Probability Theory
Text: Real Analysis, Folland
Mentee: Chris Pufko
Mentor: Ethan Zell
Representation Theory of Finite Groups
Text: Representations and Characters of Groups, James and Liebeck
Mentee: Jonah Newman
Mentor: Liron Speyer
Exploring Schottky Groups
Text: Indra’s Pearls: The Vision of Felix Klein, Mumford, Series, Wright
Mentee: Aaron Christian
Mentor: Jiajun Yan
Fall 2018
Support vector machines and kernel methods
Text: Foundations of Machine Learning, Mohri, Rostamizadeh, and Talwalkar
Mentee: Dylan Hunt
Mentor: George Seelinger
All Groups can be Realized as a Fundamental Group
Text: Algebraic Topology, Hatcher
Mentee: Kyle Hoffmann
Mentor: John Harnois
Topological Proof of the Nielsen-Schreier Theorem (a subgroup of a
free group is free)
Text: Algebraic Topology, Hatcher
Mentee: Mia Shaker
Mentor: Chris Lloyd
Difference Between Algebraic Bases and Schauder Bases
Text: Real Analysis: Modern Techniques and Their Applications, Folland
Mentee: Yifei Yang
Mentor: Joseph Eisner
Wirtinger Presentations of Fundamental Groups of Knot Complements
Text: Algebraic Topology, Hatcher
Mentee: Zach Baugher
Mentor: Rostislav Akhmechet
Theorem: Let G be a finitely generated group with two different
finite generating sets S and S’. Then G with the word metric induced
by S is quasi-isometric to G with the word metric induced by S’
Text: Office Hours with a Geometric Group Theorist, Clay and Margalit
Mentee: Sam Dulin
Mentor: Ian Runnels
Constructing Hyperbolic Structures on Closed Surfaces (of genus g)
Text: Low-dimensional geometry: from euclidean surfaces to hyperbolic knots, Bonahon
Mentee: Andrew Zazzera
Mentor: Jiajun Yan
Brownian Motion and its Applications
Text: Brownian Motion, Peter Morters and Yuval Peres
Mentee: Xinru Cheng
Mentor: Ethan Zell
Summer 2018
An Introduction to Computability Theory and ‘Universal Programs’
Text: Computability: An introduction to recursive function theory, N.J. Cutland