Below is a list of the DRP projects from Spring 2024, sorted alphabetically by the mentee’s last name. Click on an entry to see the full project description.

Mathematical Logic

**Mentee:** Nohman Akhtari

**Mentor:** Jin Wei

We followed Joseph Mileti’s Book on Mathematical Logic and started by introducing Propositional and First-Order Logic (Soundness, Completeness, Compactness, Loewenheim-Skolem Theorems). After combining semantics and syntax, we had a short discussion on Model Theory and Satisfiability before moving on to Computability Theory, specifically the Church-Turing Thesis and Universal Machines. In a last installation, we introduced the Gödel Coding and combined logic with computation to prove Gödel’s fundamental Incompleteness Theorem.

Model Theory and Stability

**Mentee:** One An

**Mentor:** Krishan Canzius

Elliptic Curve Cryptography

**Mentee:** Avi Bagchi

**Mentor:** Andrew Kwon

An elliptic curve over a finite field takes the form *y ^{2} = x^{3} + ax + b*. Equipped with an identity element (a “point at infinity”) and an addition operation that, given points

*P*and

_{1}*P*, produces a third point, elliptic curves can mimic the structure of an abelian group. Elliptic curve cryptosystems (ECC) harness this notion of “point doubling,” as ECC relies on the hardness of the discrete log problem: determining a

_{2}*k*such that

*kP = Q*, where

*P*and

*Q*are points on the elliptic curve. One algorithm for the discrete log problem is Pollard’s Rho attack, which uses an iterative function to generate a pseudorandom sequence of points — collisions along this random walk imply the existence of a cycle whose length suggests the value for

*k*. Although it is an improvement from brute-force, Pollard’s Rho attack is far less efficient than quantum attacks. Exploiting the recursive nature of modular exponentiation, Shor’s Algorithm uses a quantum Fourier transform to efficiently determine periodicity to quickly solve the discrete log problem. Thus, in a quantum world, ECC proves largely futile, urging the need for efficient, quantum-safe cryptosystems.

Algebraic Number Theory

**Mentee:** Molly Bradley

**Mentor:** Deependra Singh

This semester, we are planning to study algebraic number theory. We will begin with equations over finite fields and the *p*-adic integers and later progress to learning about quadratic forms and *L*-functions. We will be following Serre's "A Course in Arithmetic."

Domain Theory and the Lambda Calculus

**Mentee:** Tanner Duve

**Mentor:** Oualid Merzouga

This semester we have studied domain theory, a branch of mathematics which studies particular kinds of partially-ordered sets called domains, and is based heavily on category theory and universal algebra. Our study of domains is motivated by its applications to the theory of programming languages, in particular its use for providing a denotational semantics for the untyped lambda calculus. We began by reviewing the lambda calculus and studying some background concepts from category theory and universal algebra. We then spent substantial time studying the category of omega-complete partial orders, and relevant advanced topics in category theory and order theory. We then went into how to solve domain equations, and ultimately how to build a model of the lambda calculus. We also explored some connections to topology and probability/probabilistic computation.

Theory and Applications of Calculus of Variations

**Mentee:** Isabella Goosen

**Mentor:** Jacob van Hook

Advanced Topics in Probabilistic Machine Learning

**Mentee:** Iris Horng

**Mentor:** Leonardo Ferreira Guilhoto

Explorations in Algebraic Graph Theory

**Mentee:** Rachel Incollingo

**Mentor:** Nikita Borisov

Symplectic Geometry and its Role in Classical Mechanics

**Mentee:** Kason Kunkelmann

**Mentor:** Tianyue Liu

In this project, I got in introduction to some topics in symplectic and differential geometry (like smooth manifolds, differential forms, and Lie algebras) and their applications into better understanding/formulating classical (specifically Hamiltonian) mechanics. To this end, I read parts of "Mathematical Methods of Classical Mechanics'' by Arnol'd and "Classical Mechanics and Symplectic Geometry'' by Maxim Jeffs.

Algebraic Geometry

**Mentee:** Eric Myzelev

**Mentor:** Marc Muhleisen

Markov Chains in Baseball

**Mentee:** Dylan Perlstein

**Mentor:** Ryan Brill

Since baseball is a game of finite states, I am aiming to use past data to predict a players next at bat using the matrix of their historical probabilities of each outcome given a certain game state, as well as taking into account the pitcher they are facing. This can be used to simulate games, and therefore the whole season to predict outcomes. I plan to build upon one of the early articles that introduced this concept to baseball, "A Markov chain Approach to Baseball" by Bruce Bukiet, Elliotte Rusty Harold, and Jose Luis Palacios.

Physics informed machine learning

**Mentee:** Estelle Shen

**Mentor:** Shyam Sankaran

Studies in Time Series, Forecasting, and Learning

**Mentee:** Arnab Sircar

**Mentor:** Vicente Gonzalez Bosca

An Introduction to Schemes

**Mentee:** Ethan Soloway

**Mentor:** Avik Chakravarty

We will be studying schemes, a fundamental object in Algebraic Geometry serving as a generalization of algebraic varieties. We will study their construction, topological properties, sheaf structure, and other uses. We are largely following Ueno's Book "From Algebraic Varieties to Schemes," while occasionally using other resources like Mumford's "Little Red Book," Liu's "Arithmetic Curves, and of course the dreaded Hartshorne.

Shape of Space

**Mentee:** Sophia Szczepek

**Mentor:** Ellis Buckminster

A Logic For Language: Exploring the Lambek Calculus

**Mentee:** Eric Tao

**Mentor:** Eben Blaisdell

What determines the order of words in a sentence? How do the meanings of individual words combine together to form the meaning of an entire sentence? In this talk, we'll see how a logical system called the Lambek calculus provides the perfect framework for answering these questions. While these questions naturally arise in the study of language, they also provide the perfect excuse to introduce one of the most beautiful theorems in mathematical logic: the Curry–Howard isomorphism, which gives a correspondence between mathematical proofs and computer programs.

Buffon's Needle, Noodle, and Other Problems

**Mentee:** Nicholas Terry

**Mentor:** Kyle Poe

Among all of the ways to calculate π, one would not imagine that simply throwing needles at the ground is one of them. However, even a buffoon could achieve such a feat just by dropping needles randomly on a tiled floor, and counting the percentage that cross the borders between two tiles. In this talk, we will see some uses of geometric probability, as well as generalizations of Buffon's Needle Problem to noodles, differently tiled floors, and maybe more!

Stacks and Moduli

**Mentee:** Chenglu Wang

**Mentor:** Yidi Wang

Foundations of Algebraic Geometry

**Mentee:** Eric Yu

**Mentor:** Benjamin Keigwin and Marc Muhleisen

In this project, we will read sections of Atiyah and Macdonald's "Introduction to Commutative Algebra", Riehl's "Category Theory in Context", and Vakil's "The Rising Sea: Foundations of Algebraic Geometry" in order to work towards understanding some examples of duality between algebra and geometry, including Stone duality and Hilbert’s Nullstellensatz.

The Distinct Distances Problem

**Mentee:** Darren Zheng

**Mentor:** Jae Ho Choi

Suppose you were given n points to place in a plane. What is the minimum size of the set of distances between each pair of points? Despite the simple set-up, the bounding of this and related distance set problems, such as Falconer's conjecture, are generally difficult. We seek to understand some of the tools used to tackle them by reading through a paper "On the Erdős distinct distances problem in the plane" by Guth & Katz and assorted texts to get the necessary background.