Applied Mathematics Projects

Book: Clifford Algebras: an Introduction, D. J. H. Garling

Prerequisites: A course in linear algebra (e.g. MAT 310). A course in abstract algebra (MAT 313, MAT 314 encouraged)

The Clifford algebra associated to a given vector space with a quadratic form (like Euclidean space) is a very useful object to use to study the geometry of that quadratic form. Another way to say that this is a useful concept is to see that it comes up in multiple different contexts, like algebra, geometry, topology and physics. Our goal in this project is to get acquainted with this concept and its usefulness. We will aim at classifying the real and complex Clifford algebras, introduce the spin groups and study their relevance to the representation theory of orthogonal groups. This last fact comes up in physics, as discovered by Dirac, which we will also discuss.

Book: Convex Optimization, by Stephen Boyd and Lieven Vandenberghe

Prerequisites: A course in linear algebra (e.g. MAT 211).

Convexity is the secret sauce behind the success of modern optimization techniques, which have in turn generated myriad applications from circuit design to machine learning. Despite its widespread diffusion across the disciplines of engineering, finance, and computer science, convex optimization possesses at its center a startlingly elegant theory. Convexity is an algebraic property of Euclidean spaces which manages to interact cleanly with their topological structure. Duality of convex programs leads not only to deeper insights into the geometry of the problems but also to consequential optimization algorithms. In this project, the mentee will develop an acquaintance with convexity and learn to recognize problems which are convex in nature. They will study Lagrange duality and its many interpretations. They will also explore different algorithms for unconstrained and constrained optimization problems. Depending on the mentee’s interests and maturity, this project can either adopt a more theoretical flavor with an excursion into subdifferential analysis, or a more applied flavor with “hands on” computational problems.

Book: Nonlinear Dynamics and Chaos, by Steven Strogatz

Prerequisites: Linear algebra (e.g. MAT 211). Exposure to differential equations in some sense.

Biological systems are, more often than not, complex, interconnected, and non-linear. In addition, limited experimental data make the complete understanding of a given system difficult-to-impossible. With this in mind, many scientists turn to modeling of these systems as a way to understand them more fully. A prominent class of these models fall under dynamical systems, taking the form of a differential equation.

The purpose of this project will be to explore the field of continuous-time dynamical systems, using biological models as a guide throughout the process. We will look at phase-plane analysis, bifurcations, and introduce the concept of chaos. Supplementary topics that may be discussed, depending on the interests of the student (common modeling assumptions, integration techniques, interpretations of system characteristics in an intuitive way, etc).

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: Mathematical Foundations of Neuroscience, by Ermentrout and Terman

Prerequisites: Calculus I-IV (e.g. MAT 203, 303).

In the 1950's, Hodgkin and Huxley experimentally derived a system of non-linear ordinary differential equations that, to this day, people use to understand the behaviors of neurons. The goals of this DRP will be to understand how these equations come about and the ways in which people understand the solutions to this equation. The former will take us through a tour of how differential equations are used in chemistry and physics. The latter will take us on a tour of some concepts in dynamical systems. (These are covered in Chapters 1 and 3 in the book). Students will then simulate a solution of the Hodgkin and Huxley equations by implementing a numerical ODE solver in MATLAB/Octave. Time permitting, based on student interest, additional chapters from the book can be explored.

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.

Book: Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDE, by Alex Kasman

Prerequisites: A background in partial differential equations (e.g. MAT 341)

When we imagine a particle in motion, we envision a single object with zero size which does not change as it moves, but rather holds itself together in a consistent way. However, when a wave hits the shoreline, it is not a single particle that arrives. Waves have nonzero size. Moreover, they constantly encounter disturbances around them, from uneven ground below to animals swimming, yet still hold their fixed shape without error. If we modeled these waves by a linear partial differential equation, such a stable solution – a “soliton” - would not be possible. However, for a certain class of nonlinear partial differential equations, these solitons are known to exist. Most famous among these is the Korteweg-de Vries equation. The explanation for this curious stability of solutions comes down to properties of this equation related to algebraic geometry. This project will investigate this shocking relationship between nonlinear PDE and algebraic geometry. To get to the core of this, we will introduce elliptic curves and the algebra of differential operators. This will bring us to a discussion of the Lax pair corresponding to this equation and, time permitting, the geometry of Grassmannian spaces. We will simultaneously learn how to use Mathematica to visualize and explore this subject.

Book: A Brief Introduction to Spectral Graph Theory, by Bogdan Nica or Spectral Graph Theory, by Fan R. K. Chung

Prerequisites: A course in linear algebra (e.g. MAT 211)

Graphs are mathematical objects used to represent societies and networks. The aim of spectral graph theory is to understand certain properties of graphs, by examining certain related matrices and their eigenvalues. For example, a classic theorem proved using spectral graph theory is the following: In a society where every two people have exactly one mutual friend, there is one person who is friends with everybody. This project can take on a more algebraic or a more analytic flavor depending on the interests of the student.

Book: Topology, Geometry, and Gauge Fields: Foundations, by G. Naber

Prerequisites: A course in linear algebra (e.g. MAT 310). A course in real analysis (e.g. MAT 319/320).

This project can adapt itself to the student’s interests and background. We will start with some topology: introduction to topological spaces, homotopy and homology, and depending on the student this can be enough for the whole reading project. On the other hand, we can also move on to differentiable manifolds and Lie groups. The chosen book is especially approachable, because these are introduced in a more explicit, hands-on, way. The book then moves on to the application of all of these to the study of Gauge Fields, which lie at the heart of modern physics, especially particle physics, and which became a stage for deep collaborations between geometers and theoretical physicists.

Book: Variational Calculus and Optimal Control, by Troutman

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

Consider a function with multiple inputs and one output. For instance a function that takes in a point on the xy plane and outputs a single real number. In Calculus 3, presumably, we learned how to find minima of this equation: assuming our function is "nice" we check the partial derivatives. Imagine now we have an infinite dimensional space as an input (still only a single real number as output). The theory of the calculus of variations gives us tools to sensibly think about finding minima of these functions in some special cases. Troutman introduces the reader to this theory via its many applications (e.g. control, physics, minimal surfaces). In this DRP, we follow Troutman through Chapter 6 - which he estimates is an introduction to the basics. Topics include: examples of problems that may be addressed, appropriate notions of derivatives in this setting, a discussion of the restrictions we require of functions in this setting, and analysis of some necessary and sufficient conditions for finding minima. Time permitting, we may take more detailed looks at applications to mechanics and/or optimal control.

Book: The Mathematics of Voting and Elections: A Hands-On Approach, by Jonathan Hodge and Richard Klima

Prerequisites: Calculus and Proofs

Currently, debates are raging about fairness in existing democratic systems. How powerful are the permanent members of the UN Security Council? Is gerrymandering and redistricting abuse an existential threat to democracy? Should election victors be declared based on plurality or are there other better methods to decide who won an election? Many of these extremely complicated questions can actually be addressed by mathematics! This project will discuss some of the ways to interpret these questions mathematically and will discuss some basic results. Depending on interest, discussions may involve linear algebra, probability, geometry, or optimization theory. Some classical results, such as the famous Arrow Impossibility Theorem,and some new topics, such as the efficiency gap (recently discussed at the US Supreme Court), may be included.