Spring 2024 Projects

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.

Final presentation slides

We have been currently studying types as a way to define stability. After we have built the foundation through quantifier elimination and types, we will learn more about stability of different theories in model theory.

An elliptic curve over a finite field takes the form y² = x³ + ax + b. Equipped with an identity element (a “point at infinity”) and an addition operation that, given points P₁ and P₂, 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 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.

Final presentation slides 

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."

Final presentation slides

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.

Final presentation slides

In this talk, I briefly describe my readings on variational calculus, a field that extends beyond traditional calculus to solve problems involving the optimization of functionals. My talk begins with the simple problem of finding the shortest distance between two points. To answer this problem, I go through the derivation of the Euler-Lagrange equations and discuss their power in helping us solve our problem. Then, I discuss how Calculus of variations was borne from another type of variational problem.

Over the course of the semester, we explore aspects of machine learning beyond supervised learning. We learn about Variational Autoencoders (VAE) and generative models. We also look at dimensionality reduction techniques such as MDS, which are commonly used for visualizing high-dimensional data. We plan to follow and reference “Probabilistic Machine Learning: Advanced Topics” by Kevin Patrick Murphy, supplemented by various papers related to deep learning and high-dimensional data.

Over the course of the semester, we plan to study some introductory topics in algebraic graph theory and their applications. The project will begin with exploring incidence matrices, adjacency matrices, and distance matrices of trees. Later in the semester, we will examine positive definite completion problems, graph-based matrix games, matching, and walks. The ultimate goal is to understand how more concrete problems are approached using these ideas. We plan to follow R.B. Bapat’s “Graphs and Matrices” which will be supplemented as needed.

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.

Final presentation slides

I will work through “The Rising Sea” by Ravi Vakil to learn about schemes and algebraic geometry.

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.

Final presentation slides

Over the course of the semester, we hope to study Physics-Informed Machine Learning (PIML), which represents a burgeoning field at the intersection of physics, machine learning, and computational science. It focuses on incorporating laws of physics into machine learning algorithms to improve their accuracy, efficiency, and generalizability, especially when dealing with complex physical systems or when data are sparse, noisy, or incomplete. This integration is achieved by embedding physical principles (such as conservation laws, dynamics, and boundary conditions) directly into the learning process, either through the architecture of the neural networks, the training process, or the loss functions used for optimization. In particular, we’re interested in its application to the complex biological systems.

Through the semester, we plan to approach time series analysis from a variety of lenses, including traditional statistical models and modern machine learning techniques. We rigorously study autoregressive models, state-space models, stochastic processes in a variety of domains, and volatility to develop strong motivation for the advancement of forecasting and big-data driven methods. From there, we study deep neural networks and the intricacies of time series data to understand how sequential data can be learned. With special interest, we study advances in reinforcement learning and policy optimization toward forecasting. We refer to a variety of references from the statistics, machine learning, and financial economics literature.

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.

Final presentation slides

We plan to read "Shape of Space" by Jeffery Weeks.

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.

Final presentation slides

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!

Final presentation (desmos)

Over the course of the semester, we hope to study some introductory concepts and examples in the world of stacks and moduli spaces. We will begin with motivating examples in topological tacks and classification problems. Then, we will spend the rest of the semester getting into the formal language of sites, sheaves, and stacks. Finally, if time permits, we will look into the moduli space of stable curves. We plan to follow K. Behrend’s notes “Introduction to Algebraic Stacks,” supplemented by Jarod Alper book “Stacks and Moduli.”

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.

Final presentation slides

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.

Final write-up