Computation and Modeling Projects

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: A Modern Introduction to Dynamical Systems, by Richard Brown

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

Continuous maps of the circle to itself are relatively simple to define and visualize. However, many systems of mathematical or physical interest are modeled extremely well by iterated application of such a map on a point on the circle. For example, asking how an initial point moves under the rotation map by some angle θ will lead us to an investigation of the continued fraction expansion, which is of great interest in number theory. Expanding on this will lead us to define the rotation number of a map and then to the stability of the solar system! On the other hand, the doubling map, which sends a point on the circle to a point with twice its angle from a reference point, is a fantastic example of a chaotic map. Investigation into this map leads to symbolic dynamics, connecting circle maps to linear algebra and information theory. Various connections with other parts of dynamical systems may be discussed, according to interest.

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: Complex Analysis by Theodore Gamelin

Prerequisites: A course in real analysis (e.g. MAT 319/320) and a course in complex analysis (e.g. MAT 342).

Let f_c(z)=z^2+c be a polynomial, with c ∈ C. When studying the dynamics of polynomials, one is interested in the behavior of a point under successful iterations by f. The Mandelbrot set M is defined as the set of c so that the sequence of iterates,{0,c,c^2+c,(c^2+c)^2+c,...}, is bounded. Despite this simple definition, much is still unknown about the Mandelbrot set, and therefore much is still unknown about the iteration of even these simple quadratic polynomials! In this project, we will learn how to study the dynamics of polynomials, with an emphasis on specific examples, trying to avoid any difficult general theory where we can. We will see that we can partition the plane into a stable set called the Fatou set, and a fractal chaotic set called the Julia set. We will see how the dynamics of specific polynomials f_c relates to the Mandelbrot set.

After learning the basics, we can proceed in many directions. One direction is learning how to draw computer images of the Mandelbrot set, and of Julia sets of polynomials. We would also learn the mathematics of why our programs work.

These sets have an interesting fractal structure. Students could also study the theory behind Newton’s method, a technique learned in calculus courses to find the zeros of polynomials. Students could also learn about iterating rational or general entire functions, or learn the proofs and the complex analysis behind some of the several important tools we use in this area.

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: 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.