Spring 2025 Projects

I will be exploring typed lambda calculus and different ways to extend the system in the Lambda Cube, including System F, System F Omega, and hopefully more complex systems.

The pursuit of absolute certainty in mathematics inspired the creation of formal axiomatic systems — frameworks intended to derive all mathematical truths from explicit rules and logical deduction. This ambition was fundamentally challenged in 1931, when Kurt Gödel introduced his incompleteness theorems. The presentation begins with an overview of formal mathematical systems, focusing on the role of axioms and inference rules in systems such as Peano arithmetic. I will introduce  Gödel numbering — a technique for encoding syntactic elements like formulas and proofs as natural numbers — which allows a formal system to internally express statements about its own provability. Using this arithmetization of syntax,  Gödel ingeniously constructed a self referential statement asserting its own unprovability.

I will outline the essential steps in  Gödel’s argument, showing that any consistent, recursively enumerable formal system capable of expressing elementary arithmetic must be incomplete: there exist true mathematical statements that cannot be proven within the system.

Final presentation slides

Our goal this semester is to understand the representation theory of symmetric groups. After reviewing some concepts from Intro to Representation Theory notes by Etingof et. al. such as character theory and induced representations, we will begin reading Bruce Sagan’s textbook on The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions.

Dehn surgery is a fundamental operation in 3-manifold topology that modifies a given 3-manifold by removing a tubular neighborhood of a knot and gluing it back in a different way. Originally introduced by Max Dehn, this technique plays a central role in the classification of 3-manifolds and has deep connections to knot theory, hyperbolic geometry, and the study of geometric structures on manifolds.

This presentation will explore the technique of forcing in set theory with its usage in proving the independence of the Continuum Hypothesis from ZFC.

Final presentation slides

Following Conway, Burgiel, and Goodman-Strass’ book The Symmetries of Things, this project will investigate the group-theoretic tools used to classify planar symmetries (e.g., patterns we can draw in 2D space). We don’t know exactly how far we’re going to get into the book, but I’m hoping to get into the proofs of a few of the key classification theorems, and also look into tiling and color symmetries. A real-life example will be provided by Spain’s Alhambra palace, one of Andalusian Spain’s architectural masterpieces, and which features (arguably) all 17 planar symmetry groups, along with color and tiling symmetry; this problem has drawn interest from a number of algebraists over the years, and provides a concrete tie-in to the abstract material. We will possibly look into how the Alhambra patterns were constructed with ruler and compass, but I don’t know if we’ll get that far — that is, however, another nice historical tie-in, since Muslim mathematicians, taking their cue from the Greek tradition, thought about geometry in terms of what could and could not be done with ruler and compass — which includes the fabulous patterns found in Islamic art and architecture.

This will be a continuation I am going to continue reading “category theory in context.” As of now, I want to do a write up on summarizing what I learned this semester and do a presentation based on that.

We will explore topics in stochastic calculus by reading parts of "Stochastic Calculus and Financial Applications" by Michael Steele. To build a stronger theoretical background, we will first study the foundations of analysis and measure theory by reading select chapters from "Principles of Mathematical Analysis" by Walter Rudin.

Final presentation slides

Over the course of the semester, we hope to survey the literature on cryptocurrency price and volatility modeling, and explore existing research on pairs trading and co-integration between cryptocurrencies. We also hope to apply stochastic methods for valuing cryptocurrency derivatives.

We will study the Étale Fundamental Group and its variants.

We will cover some foundational tools from Fourier Analysis, control theory, and dynamical systems and how they can be used to analyze neural signals and circuits. Next, we will discuss results such as the Wiener-Khinchin theorem and its utility in quantifying excitation-inhibition balance from neural time-series data. Finally, we’ll learn about fractional processes and their connection to scale-free brain dynamics.

We study the construction of Riemann surfaces, functions and integration on Riemann surfaces, while building up to look at algebraic curves and the Riemann-Roch Theorem.

Over the course of the semester, we will rigorously explore how real analysis techniques, such as measure theory, convergence, function spaces, and Lebesgue integration, are used to develop and prove both fundamental and advanced theorems in stochastic calculus. Our focus will be on continuous-time models in mathematical finance, particularly martingales and Brownian motion, as well as important inequalities that help bound expectations, establish convergence, and analyze stochastic behavior. We plan to follow John Michael Steele’s book “Stochastic Calculus and Financial Applications” supplemented by Steven Shreve’s book, “Stochastic Calculus for Finance. Volume II: Continuous-Time Models”.

We will be following Dhruv Ranganathan’s online notes (as well as other resources) on Toric Geometry, aiming to understand the basic correspondence between toric varieties and their associated fans, and studying divisors and line bundles using toric varieties as our primary example where the theory is simpler.

Over the course of the semester, we hope to explore key ideas in algebraic geometry, focusing on the solutions of polynomial equations in affine and projective n-space. This includes studying varieties, the connections between algebra and geometry, and the role of projective space in extending geometric intuition. To build a stronger foundation, we will also learn elements of commutative algebra and topology, using them to better understand the algebraic structures and topological properties that arise in the field. Our primary reference will be Algebraic Geometry by Hartshorne, supplemented by Commutative Algebra by Atiyah and Topology by Munkres.

I have previously worked on a problem finding the distribution of distances between random points on an n-ball, but I will now attempt to apply similar reasoning to much more unwieldy sets: fractals. Motivated by attempting to find the distribution of distances on the Cantor Set, we will try to find concrete solutions to this and similar problems on other fractals. This will likely involve the use of some measure theory and ways of doing calculus outside the normal derivative and integrals we are used to. These interesting constructions may have some surprising results, and will be accompanied by pretty images of fractals!

Final presentation slides 

Our project explores the theoretical foundations and practical design principles of State Space Models (SSMs) as efficient alternatives to transformer architectures in language modeling. While transformers dominate modern AI through self-attention mechanisms, their quadratic computational costs motivate interest in SSMs – rooted in control theory and signal processing which achieve linear scalability via recurrence and convolutional formulations. We analyze how SSMs avoid naive pitfalls through structured linear algebra (e.g. parameterization of state matrices), approximation theory (balancing expressivity with numerical stability), and system discretization techniques. By studying seminal SSM variants (such as S4, Mamba), we will understand critical design choices: selective state transitions, input-dependent gating, and memory-augmented recurrence that bridge performance gaps with transformers. Finally, we consider validating these insights on synthetic sequence tasks or simplified language modeling benchmarks to evaluate SSMs’ practical potential as scalable and effective language modeling frameworks.

We will follow Weibel’s Introduction to Homological Algebra and Maximilian Peroux’s A user’s guide for higher algebra.

We aim to better understand Kolmogorov complexity and explore its connections to fractals. We will experiment with the Point-to-set Principle to estimate dimensionality. The goal is to focus on practical applications and use readings by Stull and Vitanyi/Li as supplement.

There are several notions of dimension in algebra, such as the Krull dimension of a ring, the transcendence degree of the function field of a domain over its fraction field, and the dimension of a Noetherian local ring. For this project, we will explain these definitions, then compare and contrast them by giving conditions which guarantee they coincide.

Final presentation slides