Probability Projects

Book: Fourier Analysis, by T. W. Korner

Prerequisites: A course in real analysis (e.g. MAT 319/320)

In the mid-1800s Joseph Fourier found that periodic functions can be realized as infinite sums of sines and cosines. This realization has made a profound impact on virtually every subfield of science and engineering. The mentee will begin by examining the sense in which these infinite sums converge, through the Dirichlet and Fejer kernels. After this, the student will delve into a number of applications of this theory, including applications to probability, Brownian motion, and dynamical systems.

Book: Dynamic Programming and Optimal Control, Vol. II, 4th Edition: Approximate Dynamic Programming, by Dimitri P. Bertsekas

Prerequisites: A course in linear algebra (e.g. MAT 211). Exposure to real analysis (e.g. MAT 319) is a bonus.

How should rational agents plan for the future? This is the underlying question asked by the Federal Reserve, your neighbor’s Roomba, the electric utility company, and the AlphaZero chess player. Each of these problems features agents interacting with uncertain, evolving environments. Agents are tasked with finding policies that optimize their outcomes with respect to particular performance criteria. In this project, the mentee will explore the theory of Markov Decision Processes, the mathematical model at the heart of this class of optimization problems. They will prove the existence of optimal policies for finite and infinite horizon MDPs under the expected total discounted reward criterion, and study algorithms which directly compute these policies. Depending on the mentee’s interests and maturity, this project can either adopt a more theoretical flavor with analysis of additional performance criteria, or a more applied flavor with reinforcement learning.

Book: Machine Learning: A Probabilistic Perspective, by Kevin P. Murphy

Prerequisites: Linear algebra (e.g. MAT 211). Basic knowledge of probability will be very useful.

Machine Learning (ML) is foremost known as a tool for the industry; however, the theoretical aspects behind the tools of ML also encompass a very active area of research. Most of these foundations are better understood under the lens of probability.

In this project, we will learn enough probability to understand the whys and hows behind some of the basic algorithms of ML such as k-Nearest neighbors, Gaussian models, or Neural networks. Specific topics will depend on the interest of the student. If time permits it, and the student wishes to pursue doing so, we will implement (i.e. code) some working algorithm.

Book: Random Matrices, by M. L. Mehta

Prerequisites: Linear algebra (e.g. MAT 211), probability and statistics (e.g. AMS 310). This project can be tailored to applications to physics, in which case statistical mechanics (PHY 306) is necessary.

Random Matrix Theory (frequently abbreviated as RMT) is an active research area of mathematics with numerous applications to theoretical physics, number theory, combinatorics, statistics, financial mathematics, mathematical biology, engineering telecommunications, and many other fields.

Random matrices are matrices with their entries drawn from various probability distributions, which are called random matrix ensembles. Our goal is to study the eigenvalues of such matrices, which oftentimes have a rich mathematical structure when the matrices are large. For example, the spacing between eigenvalues are described by universal laws which are also found to describe the nontrivial zeroes of the Riemann Zeta Function.

During this project, we will cover a number of applications of RMT to mathematics or physics, depending on the interests of the student.

Book: Random Walk and the Heat Equation, by Greg Lawler

Prerequisites: Linear algebra (e.g. MAT 310), real analysis (e.g. MAT 319/320), and some knowledge of probability (at the level of AMS 311). A student interested in studying Brownian motion as part of this project must have taken or be enrolled in a course in measure theory (e.g. MAT 324).

There are two models for looking at the diffusion of heat inside of some medium. The heat equation is a deterministic partial differential equation which describes this diffusion. However, on the physical level, the diffusion of heat may be described in terms of “random” collisions of particles. The path of one such particle can be thought of as a random walk or Brownian motion. Such random processes have applications in several areas of science, engineering, and finance.

A discrete random walk can be thought of as a particle moving in the integer lattice, choosing a direction randomly at each stage. In this project, we will study discrete random walks in the plane and general euclidean space, answering questions such as how many times does a random walker return back to their starting point? We will see how this probabilistic point of view can enrich the study of a discrete version of the heat equation along with other areas of analysis. Depending on the student’s interests, we may branch off into further topics, such as more detailed analysis of discrete random walks, continuous random walks (Brownian motion), and other advanced topics in probability, such as martingales.