Analytic Number Theory
It is said that Dirichlet's theorem on the infinitude of primes in arithmetic progressions invented analytic number theory. In this project, we will go over a proof of this theorem, which provides a good excuse to study Fourier analysis. Time permitting, we will also explore how analytic methods can be applied in other ostensibly discrete problems, for example prime number theorem and Lagrange's four square theorem.
Chromatic Symmetric and Quasisymmetric Functions
We will study symmetric function theory and its applications to graph colorings. Every graph has an associated chromatic symmetric function, which is the generating function of the graph's proper colorings. Much of the work on chromatic symmetric functions is motivated by a long standing open problem that concerns graphs constructed from partially ordered sets and when the resulting chromatic symmetric function has positive coefficients in a certain basis. We will first follow Stanley's Enumerative Combinatorics Vol. 2 before reading Shareshian and Wachs' 2014 paper "Chromatic quasisymmetric functions".
We will look at Lagrangian and Hamiltonian formalisms for classical mechanics following the books of Landau and Lifshitz, and Arnold.
We plan to study complexity theory and its intersection with mathematics. After learning the basics of complexity theory, we will focus on more specific problems of interest.
Elliptic Curve Cryptography
The purpose of this project is to explore the basics of elliptic curves (focusing on the theory over finite fields) and to ultimately study their applications to cryptography (an approach known as Elliptic-curve cryptography). The main references are portions of Silverman-Tate's "Rational Points on Elliptic Curves", and portions of Silverman's "An Introduction to Mathematical Cryptography", among other scattered resources.
Elliptic Functions and Modular Forms
Man Cheung Tsui
After briefly looking at how to analytically continue the zeta function and proving the prime number theorem, we investigate doubly periodic functions on the complex plane, modular forms, and modular curves. Our goal is to use the resulting theory to get formulas computing the number of ways a positive integer can be written as a sum of k squares. References: Stein and Shakarchi, Complex Analysis, Chapters 6, 7, 9; Diamond and Shurman, A First Course in Modular Forms, Chapters 1, 2, 3, (4).
Group Theory and Applications
We will discuss group theory from basic principles, including set theory, subgroups, and quotient groups. After this we will discuss particular interesting examples, which may include groups of matrices and their applications in linear algebra, the braid group, and the fundamental group of a topological space.
Information Theory with Applications to Machine Learning
Over the course of the semester, we hope to study some introductory topics in the field of information theory as well as their applications. This study will begin with a survey of some basic elements of the field such as entropy, data compression, and noisy-channel coding. We will spend the rest of the semester examining applications of information theory to statistical inference, with the goal being to reinterpret neural networks from an information theoretic perspective. We plan to follow David MacKay’s book "Information Theory, Inference, and Learning Algorithms," supplemented by Shannon's landmark paper "A Mathematical Theory of Communication."
Lie Groups and Control Theory
We will focus on Lie groups and Lie algebras with the motivation of exploring the intersection of abstract algebra and differential geometry. In addition, we will discuss its general applications to control theory, particularly regarding bilinear systems.
Quantum Theory and Representations of Groups
Our goal will be to understand 1-dimensional quantum systems, such as the quantum free particle, the hydrogen atom, and the harmonic oscillator, as well as develop some of the foundations needed for higher dimensional quantum field theories. The approach will focus on developing the necessary mathematical foundations on representation theory and analysis.
Main reference: Peter Woit - Quantum Theory, Groups and representations: An Introduction.
Topology from the Differentiable Viewpoint
Artur B. Saturnino
The degree of a smooth map is a powerful concept that is central to differential topology. In this project we will define this invariant and study how it is used to show the Fundamental Theorem of Algebra, Browder's Fixed Point Theorem, the Borsuk-Ulam theorem and, if we have time, the Poincare-Hopf and Hopf's Theorem. This project is based on Milnor's book with the same name.