Mathematical Physics 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: 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: 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: 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.