Fall 2024 Projects

We will work on formalizing methods to solve polynomial equations in lean.

We will go through the fundamentals of computability theory including the primitive recursive functions, general recursive functions, models of computation, and decidability results.

Mapping class groups are a essential object to understanding the topology and geometry of surfaces and 3-manifolds. We’re going to follow the textbook A Primer on Mapping Class Groups by Benson Farb and Dan Margalit to gain insights on the tools needed to study low dimensional topology and geometric group theory.

We will study Laplacian and adjacency matrices and their eigenvalues, chromatic polynomials, and random walks on graphs.

Over the course of the Directed Reading Program, we hope to mimic a one-semester functional analysis course. We will be reading through "Kreyszig Functional Analysis with Applications" and working on weekly problem sets. Topics covered will include: Banach and Hilbert Spaces; uniform boundedness, open mapping, and closed graph theorems; spectral theory: operators on normed spaces, compact and self-adjoint operators.

We will explore fractal geometry through its applications in Iterated Function Systems (with an emphasis on chaotic systems and attractors), Brownian Motion, and Partial Differential Equations.

We seek to develop a strong understanding of lattice-based cryptographic techniques. This implies first studying lattices and their computational complexity, before delving into the uses of lattice problems in a cryptographic context. This study will be guided by Daniele Micciancio's "Lattices Algorithms and Applications" course, which covers interesting applications of lattice problems, such as identity based encryption and fully homomorphic encryption. The specific direction of the project through these various topics will take shape as we develop a better understanding of the foundations and the forms of each of the applications proposed in the course.

We will learn basic concepts of category theory and problem-solving through "Topoi: the categorical analysis of logic" by Goldblatt. We aim to explore how category theory provides an abstract framework for other fields of math.

Throughout the semester, we will explore the theory behind Brownian motion, a type of stochastic process, and learn about its uses in applied mathematical modeling. We first read Chapter 5 of the book by Gould and Pemantle, titled “The Brownian Zoo,” to get an overview of Brownian motion. Afterwards, we plan to follow parts of the books “Introduction to Stochastic Processes” by Gregory Lawler and “Stochastic Calculus and Financial Applications” by Michael Steele to learn about Brownian motion's properties, variations, and applications.

We will be working through the introduction to Topology with an emphasis on Algebraic Topology. We currently do not have an end goal but the idea is to eventually look at homology groups and other algebraic methods of analyzing different topologies.

We will study algebraic number theory following “Cohomology of Number Fields” by Neukirch, Schmidt and Wingberg.

We study Gauge theory using Hamilton’s ”Mathematical Gauge Theory” as a reference. We study types of fiber bundles and connections on fiber bundles, learning relevant topics about Lie groups and Lie algebras as necessary. We follow with some applications of gauge theory to physics.

This project looks at NCAA basketball players to project how good of a shooter they will be in the NBA. Using tracking data, we are going to show how this data is better than traditional box score data by comparing a baseline regression model to a complex hierarchical model using individual player shot probabilities given a court x and y coordinate.

We will be studying complex manifolds, an analogous structure to smooth manifolds with analytic coordinate maps. We will aim to understand these objects both as manifolds but also as complex algebraic varieties, such as in the classical case of Riemann Surfaces. We will largely be following “Principles of Algebraic Geometry” by Griffiths and Harris, but also using other texts on manifolds and complex varieties to fill in the gaps.

Throughout the semester, we will explore key concepts in combinatorial matrix theory, following the framework laid out in “A Combinatorial Approach to Matrix Theory and its Applications” by Dragoš M. Cvetković and Richard A. Brualdi. As we progress, we will examine practical applications of combinatorial matrix theory in finance, focusing on how combinatorial matrix techniques can enhance understanding and solutions in this field.

Over the course of the semester, we will explore the general concepts of topology by following James Munkres’ “Topology.” After covering point-set topology, we may delve into other areas of topology, such as algebraic topology with Allen Hatcher’s book “Algebraic Topology.”

Nakajima quiver variety is a geometric construction that plays a central role in representation theory and mathematical physics. We plan to follow Kirillov’s Quiver Representation and Quiver Varieties.

We will have a deep dive into the probability space, properties of Martingale and Random Walk.

I hope to explore the the world of information theory, and over time, develop and understanding of how information theory influences decision making and coding theory. We will first begin my gaining a broad overview on the field of information theory in order to gain familiarity with the core concepts and ideas. We will focus on understanding how information theory intersects with other fields that I am interesting in, particularly probability/statistics, algorithmic learning, and network theory. After gaining a wide overview, we will narrow down the scope of the project based on one intersection that we find interesting to explore further. I suspect it will likely be a computer science or statistics relevant topic, such as decision making/probabilities or algorithmic learning. The goal is to explore practical applications of information theory and how it influences other fields.

Following Isaac’s “Finite Group Theory”, we will work through the proofs of several landmark results including the Schur-Zassenhaus theorem, Hall’s C-, D-, and E-theorems, and Burnside’s p^a q^b theorem. If time permits, we will also look at the character-theoretic proof of Burnside’s theorem presented in Robinson’s “A Course In The Theory Of Groups”, which is more in the spirit of Burnside’s original argument.

Geometric measure theory is concerned with the multiple ways of measuring the size of mathematical objects, most commonly sets, and the relationship between different methods. Suppose you have a set S of points in R^d large in Hausdorff measure. Define its distance set as D(S), which is the set of all distances between pairs of points in S. How big must S be in Lebesgue measure? The goal is to study this problem called Falconer’s conjecture, and similar problems, throughout the semester.