We had the very auspicious number of 88 total pairings this semester. Those who completed a final presentation or project are featured below. As always, we’d like to thank our mentors and the department staff.

This was also Ronan O’Gorman’s last semester organizing the DRP, and Alan Goldfarb’s first. Ronan, thank you for all that you’ve done; Alan, welcome to the team!

Mentee	Mentor	Project
Ben Aldaghi	Mitsuki Hanada	Symmetric Functions and the Symmetric Group
Nicholas Ballesteros	Ronan O’Gorman	An Introduction to Affine Schemes
Smokey Child	Spencer Taylor	An Introduction to Algebraic Topology and Differentiable Manifolds
Juan Claver	Yahya Bashandy	Topological $K$-Theory
Ren Colley	Ryan Martinez	Continuous Nonlinear Dynamical Systems
Theo Danial	Etienne Phillips	Monads
Tala De La Cruz	Sai Sanjeev Balakrishnan	Galois Theory Through Cyclotomic Symmetry
Arjun Garg	Etienne Phillips	An Introduction to Quasi-Categories
Omkar Guha	Jake Hofgard	SLE and Universality
Rayan Gummiid	Vilas Winstein	Proof by Brownian Motion
Jesse Hart	Benjamin Daniels	Smooth Manifolds and Lie Groups
Diana Hu	Ruslans Aleksejevs	Turán’s Theorem and Extremal Graph Theory
Carson Humphreys	Yvette Ren	Sacks’ Splitting Theorem through the Finite Injury Priority Method
Spencer Kim	Siming He	Optimal Transport
Elizabeth Lau	Danat Duisenbekov	Mixing Times and Applications
Alexander Lee and Zayan Boghani	Jenna Plute	The Spectral Theorems
Wangyuan (Peter) Li	Amy Dai	Numerical Methods for Stochastic Differential Equations
Austin Lin	Swapnil Garg	Profunctors and Double Categories
Yuhao Liu	Jacob Erlikhman	Grassmannian, Hilbert, and Quot Schemes as Moduli Spaces
Owen McGinty	Sai Sanjeev Balakrishnan	Abelian Extensions of Exponent $n$
Andrew Nguyen	Nilay Patel	Period Three Implies Chaos (on $\mathbb{R}$)
Vignesh Nydhruva	Akash Narayanan	Sheaf Cohomology & Mittag-Leffler
Lucas Park	Harper Wells	Continuum Hypothesis and Forcing in Set Theory
Lia Park	Eric Jankowski	Representation in Dihedral Groups
Milena Safaryan	Isabel Detherage	Spectral Independence
Laney Saye	Ning Tang	Geometrically Constraining Vacuum Spacetime
Justin Shaw	Ethan Ebbighausen	Fractal Geometry
Daniel Song	Etienne Phillips	Enriched Category Theory
Jorge Speck	Leonard Tomczak	Elliptic curves over $\mathbb{C}$, Lattices, and Sum of Squares Theorems
Sofia Torelli	Isabel Detherage	Mixing Times of Markov Chains
Andrew Wang	Benjamin Daniels	De Rham Cohomology of Smooth Manifolds
Jiaxiang Xu	Haoen Li	A Numerical Study of Random Trace Estimators for the Inverse Laplacian
Stephanie Yao	Alex Feiner	Tensoring Exact Sequence
Katherine Yuan	Leonard Tomczak	Sphere Packing and Lattice-Based Cryptography
Tina Zejda	Michael Kielstra	Möbius Automorphisms of the Unit Disk
Zhouling Zhang	Marco David	From Classical Curve Counting to Gromov-Witten Invariants

With 76 pairings, this was another banner semester for the DRP. We are, as always, grateful for the effort of the mentors and department staff who make all this possible. We couldn’t do it without them.

Now for one of my favorite parts of the program: sharing the completed projects! The pairs who submitted a final paper or do a presentation at our end-of-semester conference are featured below. We hope you find their work entertaining and educational — and that it might serve as a resource if you’re considering starting a DRP project yourself.

Mentee	Mentor	Project
Ariana Abel	Akash Narayanan	Connectedness vs Path Connectedness
Nicholas Ballesteros	Alexander Burka	Stone Duality for Boolean Algebras and Cylindric Algebras
Edward Beckon	Leonard Tomczak	Dirichlet’s Theorem on Arithmetic Progressions
Ryan Chan	Zoe McDonald	The Itô Integral
Juan Claver	Yahya Bashandy	Local Kronecker-Weber
Antonio Cortijo	Liza Jacoby	Descriptive Set Theory
Amelia Craciun	Marco David	Enumerative Geometry
Omkar Guha	Jake Hofgard	Combinatorial Upper Bounds for Self-Avoiding Walks on the Integer Lattice
Michelle Holland	Alexander Burka	Lindström’s Theorem
Yang Hu	CJ Dowd	$p$-Adic Hodge Theory
Diana Hu	Ruslans Aleksejevs	Arnold’s Topological Proof of Abel’s Impossibility Theorem
Michael Lan	Katalin Berlow	Nielsen–Schreier theorem
Dylan Lu	John Nolan	Bézout’s Theorem in Algebraic Geometry
Runze Lu	Lewis Pan	Markov Chain Monte Carlo
Shijie Lyu	Ely Sandine	Navier-Stokes Equations for 2-Dimensional Incompressible Slightly Viscous Flow Along Curved Wall
Benedikt Vilji Magnússon	Lizzie Pratt	A Geometric Introduction to Representation Theory
Nathan Mann	Smita Rajan	Algebraic Geometry
Dron Mongia	Thomas Browning	Real Analysis with Filters
Connor Murray	Jerry Yang	De Rham Cohomology
Andrew Nguyen	Nilay Patel	Why Are All Meteorologists Eventually Wrong?
Vignesh Nydhruva	Akash Narayanan	Algebraic Number Theory
Partha Krishna	Swapnil Garg	Verma Modules and the Harish-Chandra Isomorphism
Yifei Qi	Joao Basso	On Matrix Analysis Methods in Quantum Entropy
Xiang Shi	Vinh-Kha Le	The Application of Machine Learning and Deep Reinforcement Learning on Quantum Optimization
Ailsa Sun	Etienne Phillips	Yoneda’s Lemma
Yuxin Tan	Sai Sanjeev Balakrishnan	Primes of the form $x^2+ny^2$
Sofia Torelli	Izzy Detherage	Cheeger’s Inequality
Ayisha Umar	August Noë	A Nonstandard Proof of the Bolzano-Weierstrass Theorem
Ethan Eugene Wynner	Will Fisher	Some Notions in Étale Cohomology
Stephanie Yao	Alex Feiner	Elliptic Curves and Finding Torsion Points
Brian Zhou and Woong Choi	Reed Jacobs	Fundamental Groups

The Directed Reading Program continues to go from strength to strength.  Sure, sometimes I forget to order the pizza, and sure, when that happens, sometimes I call La Val’s in a voice that sounds less like a level-headed DRP organizer and more like a disgruntled wombat, but otherwise the program ran very smoothly. We had 67 pairings, more than either of us can remember having in the past, and by all accounts everyone had fun. Special thanks are due to our supervisor Christian Natividad and to Siti Keo and the rest of the math department staff. Applications for the fall program will open in August, and we look forward to seeing you then.

Pairs who submit a final paper or give a final presentation are featured in our wrap-up post at the end of the semester. It is our privilege to share these projects with you. We hope that you’ll enjoy reading them, and perhaps find inspiration for a DRP of your own!

Mentee	Mentor	Project
Cleo Alexa	Thomas Browning	Solutions of Mordell’s Equation
Aylin Ambriz	Brian Morris	Brouwer’s Fixed-Point Theorem in Two Dimensions
Yuqing Chen	Lizzie Pratt	Matroid Intersection and König’s Theorem
Matthew Corbo	Drew Keisling	Covering Spaces and Galois Theory
Amelia Craciun and Nathan Han	Alois Cerbu	The Max-Flow Min-Cut Theorem
Shrishti Dalal	Sean Gonzales	Expander Codes and Their Construction via Cayley Graphs
Junhe Feng	Robert Bowden	Morse Homology
Xiaohao Feng	Sai Sanjeev Balakrishnan	The Mordell-Weil Theorem over Number Fields and Mazur’s Torsion Theorem
Adrian Fry	Saud Molaib	Combinatorial Species and the Lagrange Inversion Theorem
Crystela Herrera	Jacob Elafandi	Graphs and Homomorphisms
Harutyun Harry Ilanyan	Chan Bae	Hallucination and Calibration in Language Models
Partha Krishna	Amy Dai	Stochastic Differential Equations
Jasmine Latimer	Mitsuki Hanada	The Hook Length Formula
Austin Lin	Jessie Tan	Coloring of the Real Line with the Lovász Local Lemma
Ethan Liu	Luke Triplett	Representative Sampling of High-Dimensional Point Sets
Jihao Liu	Zhongkai Tao	Fourier Transforms, Maximal Functions, Hilbert Transforms, and Singular Integrals
Theo Lysek	Feiyang Lin	Realizable Splittings of Vector Bundles Over P^1 in Exact Sequences
Nathan Mann	Daigo Ito	Algebraic Geometry
Matthew Moon	Eric Jankowski	One Way to Find the Dimensions of Some Varieties
Marc Nguyen	Elliot Kienzle	Euclidean Axiomatization
Vignesh Nydhruva	Akash Narayanan	Connectedness and Path-Connectedness
Serkan Salik	John Nolan	Riemann’s Count for the Dimension of the Moduli Space of Riemann Surfaces of Genus g ≥ 2
Rui Shang	Kabir Kapoor	An Exposition of Basic Mathematical Logic
Abby Smith	Audrey Rosevear	Tischlers Theorem
Ailsa Sun	Gabriel Beiner	Tychonoff’s Theorem
Jack Wang	Liyang Shao	Discrete Ergodic Schrödinger Operators

It’s been a fantastic semester for the DRP! We had almost sixty pairings across all areas of mathematics, pure and applied. The program continues to grow and the department continues to support us ably. We’d like to extend a special thank-you to our supervisor Christian Natividad, to Vicky Lee, and to the rest of the staff who make this possible. Applications for the spring will open in January, and we look forward to seeing you then.

Pairs who submit a final paper or give a talk at our end-of-semester conference have the option to be featured in our wrap-up post. It is our privilege to share these projects with you. We hope they will make for interesting reading — and possibly serve as inspiration for DRPs of your own!

Mentee	Mentor	Project
Keren Bai	Jacob Elafandi	The Fourier Transform
Emily Bell	Yuxi Liu	Dependent Kuramoto Models
Matthew Corbo	Drew Keisling	Number Fields
Jackson Koos Davis	Nancy Mae Eagles	Heegaard Floer Homology of Lens Spaces
Zachary Dell	Robin Huang	Elliptic Curves and Kronecker’s Jugendtraum
Hallgrímur Haraldsson	Pranav Enugandla	Representation Theory of Finite Groups
Chris Hartanto	Charley Hutchison	Cauchy’s Theorem
Dongho Tommy Kim	Connor Halleck-Dube	Representation Theory of GL(2, Fq)
Partha Krishna	Amy Dai	Stochastic Differential Equations
Jingyao Lai and Siyi Huang	Yang Chu	Elementary Financial Mathematics
Jasmine Latimer	Mitsuki Hanada	321-Avoiding Permutations
Mataio Nonaka	Angelos Pelecanos	The Probabilistic Method
Colin Olairez	CJ Dowd	Drinfeld Modules
Alex Olhava	Yifan Chen	Gromov Hausdorff Convergence and Geometric Applications
Karina Palomino	Chan Bae	The Vershik-Okounkov Approach to the Representation of Sn
Ved Pandey	Ronan O’Gorman	Quantifiers as Adjoints
Leo Pedersen	Liza Jacoby	The Baire Category Theorem
Laney Saye	Ning Tang	Wave Decay: Dispersion in the Stationary Phase
Tobin Sudo	Elliot Kienzle	Platonic Solid Projections and Iterated Function Systems
Madison Watts	Reed Jacobs	Number Theory
Emma Yang	Mitsuki Hanada	Labeled Trees and Parking Functions
Haotian Zhao	Jerry Yang	Quadratic Forms and Quadratic Fields

The Spring 2023 DRP was a blast.  We paired about 25 undergraduates with mentors, covering a wide variety of topics.  Check out Herman Malik’s presentation on hyperbolic knot theory or Aren Martinian’s write-up on representations of Lie algebras for examples of the sorts of things our students got up to. Other topics included: The Laplace-Runge-Lenz …

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