In a few days I’ll be starting my sixth semester as a mathematics Ph.D. student at the University of Tennessee, roughly marking the halfway point until I’m out with a degree in hand and [hopefully] with a job. I’ve been giving some thought to the background I’ve developed through my coursework to get to the point I’m at with research. Some courses ended up being a lot more or less useful than I anticipated; some I really wanted to take, while others my advisors felt strongly that I sign up for. With that in mind, my goals of this article are as follows:

- Look back at all the courses I’ve taken up to this point, and some that I plan to take in the future. Which ones were the most useful/most interesting?
- Think forward to other things I want to learn about or explore before I graduate.
- Create a reference that other math graduate students can refer to (both at my home institution and elsewhere) as an example of what courses might be taken over the course of a few years. However, I wouldn’t follow this guide exactly since my course plan was to a large extent catered to what background I’ve needed for research. On top of that not all the courses I listed might be offered at your home instituition.

Before I begin listing courses, I should provide a bit of context on my research: when I came to the University of Tennessee in 2019, my primary research interest was the theoretical aspects of partial differential equations (PDE), particularly existence and uniqueness theory. As I began taking courses and getting a sense of what areas of research were actively being worked in, my interest expanded to include some things more accurately referred to as “applied mathematics.” In particular, the research I’m working on concerns optimal control problems, where the functions being minimized come from models in peridynamics, a relatively modern and developing area of solid mechanics. There is also much to be done from a numerical analysis standpoint for these problems. On the other hand, quite a bit of theoretical mathematics, such as analysis and existence-uniqueness theory for PDE. The end result is a project that is a nice mix of theory and computation, which is ultimately what I was looking for when I made my choice of advisors (see the last section of my older article for more info on how I made this decision).

Also, I will list the courses by their course number, rather than in the order I took the courses. This is because many of the courses I took didn’t explicitly depend on each other, besides those that belong to the same sequence. You may wish to take such courses in a different order.

Course name: Partial Differential Equations I

Course number: MATH 535

When I took it: Fall 2019

Overview of content: method of characteristics for first-order PDEs; Fourier series; separation of variables

How useful was it? I haven’t touched the material since I passed my preliminary examination in PDEs

Course name: Partial Differential Equations II

Course number: MATH 536

When I took it: Spring 2020

Overview of content: Laplace, heat, and wave equation; maximum principles; energy methods; properties of harmonic functions

How useful was it? I see similar results appear in recent research literature in slightly different contexts, but the material from the course itself hasn’t been crucial yet.

Course name: Real Analysis

Course number: MATH 545

When I took it: Fall 2019

Overview of content: Lebesgue integration theory; measurable sets and functions; integral-limit theorems; integral inequalities; L^p spaces

How useful was it? The course covered a lot of important material quickly. I use tools such as Fatou’s Lemma and the Dominated Convergence Theorem over and over. Also this is a great course for building up one’s mathematical maturity, and I’m of the opinion every math Ph.D. student should take a course covering this material.

Course name: Complex Analysis

Course number: MATH 546

When I took it: Spring 2020

Overview of content: complex differentiability; holomorphic functions; finding zeros of complex-valued functions; power series; conformal mappings; topological transformations of curves; Liouville Theorem

How useful was it? I used some of the power series theory I learned for a side project in analytic combinatorics. Otherwise this course hasn’t been of much use for me.

Course name: Applied Linear Analysis

Course number: MATH 547

When I took it: Spring 2021

Overview of content: the notion of a bounded linear operator; convexity; Banach and Hilbert spaces; reflexivity; compactness arguments

How useful was it? Easily the single most useful course I’ve taken in graduate school, it acts as an all-in-one-package of analysis needed for people studying PDEs and numerical analysis. It also was helpful for giving me more background in topology since a lot of the material covered had different “versions” used in analysis and topology, that we proved to be equivalent to each other. On top of that the professor was excellent (see also my comments on MATH 679).

Course name: Riemannian Geometry I

Course number: MATH 567

When I took it: Spring 2020

Overview of content: this was a course that covered some basic topology and differential geometry and used it for an in-depth treatment of relativity theory

How useful was it? Seeing the differential geometry notation has been situationally useful…I took this course when I was more seriously considering doing projects in geometric PDE and mean-curvature flow, and it helped me decide not to pursue this research direction. The course itself was excellently taught, though, so it’s kind of a shame.

Course name: Numerical Mathematics I

Course number: MATH 571

When I took it: Fall 2020

Overview of content: eigenvalues and eigenvectors; matrix manipulations; diagonalization; solving linear systems (including algorithms and convergence rates); Banach Fixed Point theory; also see my December 2020 article where I talked about what this course taught me in more detail, including some content on proof logic

How useful was it? Getting a ton of practice with linear algebra made some of my subsequent analysis courses easier; on the flip side, having taken Real Analysis before this course was helpful since I was better able to handle very long technical arguments that often involved a lot of inequalities. There was also some programming in Python which is always a useful skill to practice.

Course name: Numerical Mathematics II

Course number: MATH 572

When I took it: Spring 2021

Overview of content: finding roots of nonlinear equations; solving initial value problems for ordinary differential equations (ODE); solving initial value problems for partial differential equations (PDE), including Lax-Milgram Theory; basics of finite element methods (FEM)

How useful was it? The actual implementation schemes we talked about have not been terribly useful for my research to date, but the Lax-Milgram theory and FEM introduction was a nice segway into MATH 574 (see below)

Course name: Finite Element Methods

Course number: MATH 574

When I took it: Fall 2021

Overview of content: The third in the saga of courses taught by one of my co-advisors Abner Salgado, we spent much of the first part of the course talking about PDE theory, including Lax-Milgram and Sobolev Spaces. This, along with an introduction to the abstract theory of finite elements and finite exterior calculus, let us study some specific finite element solutions to simple problems from PDE theory. Error estimates were also abundant in this course; that is, we saw that approximated solutions were close to the true solutions guaranteed to exist by PDE theory.

The instructor gave us a choice of what to do to earn our course grade: a series of homework problems, some implementation of some finite element methods in a programming language of our choice, or a presentation. I opted to do a presentation, and as a result got to more deeply study the connection between finite element methods and calculus of variations through an abstract tool known as gamma-convergence.

How useful was it? Working through the material for my presentation, including the meetings I had with Abner to prepare for it and ask questions, was by far the most useful aspect of the course since I was using gamma-convergence in my research around the same time. I also anticipate needing to write out and implement finite element schemes in the near future for my research, so surely the background will continue to come in handy.

A word of caution about this course: while the prerequisites were fairly minimal, I would urge against taking a course in finite elements at the very beginning of a Ph.D. program. It took me two years to get three courses in PDE, two in numerical math, and the real and applied linear analysis courses. All of this background was very important for getting the most out of this course (notably, though, complex analysis was useless here).

Course name: Optimization

Course number: MATH 577

When I took it: Fall 2021

Overview of content: first and second derivative tests in abstract spaces; gradient descent; projected gradient method; Uzawa’s method; linear programming; duality techniques

How useful was it? The theory wasn’t all that illuminating given its overlap with some of the courses I took earlier, but the programming assignments let me build some proficiency in MATLAB, which was much appreciated. In addition, optimization is very similar in spirit to optimal control, which is one of the centerpieces that my thesis research revolves around.

Course name: Advanced Partial Differential Equations I

Course number: MATH 635

When I took it: Fall 2020

Overview of content: Fourier transforms and an in-depth treatment of the theory of Sobolev Spaces including properties of the spaces, embeddings, compactness, weak solution theory for elliptic PDEs

How useful was it? Extremely useful material to see as many times as I can, as it’s such a well-developed theory, and pieces of it have been reproduced elsewhere, such as in nonlocal problems, where the underlying norms have singularities.

Course name: Advanced Partial Differential Equations II

Course number: MATH 636

When I took it: I’m taking it in Spring 2022

Overview of content: Yet to be seen precisely but it’s supposedly going to be about the theory of wave and Schrödinger equations, using quite a bit of harmonic analysis

How useful do I think it will be? Remember how I said I initially wanted to study existence-uniqueness theory for solutions to PDEs? Well I expect I’ll be coming full circle with this course and be learning about things I’ve been curious about for a long time. The actual usefulness, however, will depend on where my thesis project goes; it will likely be beneficial background to have if I start studying time-dependent problems. While I think I can have a thorough thesis without even touching these, it’s definitely something I’m open to.

Course name: Advanced Analysis I

Course number: MATH 645

When I took it: Fall 2020

Overview of content: introduction to functional analysis; C* algebras; normal operators; Kadison-Singer Conjecture

How useful was it? There is a world where I come back to my notes and textbook from the course at a later date to look for deeper connections between the theory of nonlocal integral operators and C* algebras, but up until now I haven’t seen or needed the course’s material much since I took the course. I would have considered working with the professor for this class on a n analysis project, but he was set to retire, and in fact this was the last class he ever taught. As I write this I think back fondly to the courses I took with Dr. Carl Sundberg (this one plus MATH 545/546).

Course name: Advanced Analysis II

Course number: MATH 646

When I took it: Spring 2022

Overview of content: it’s supposed to be a sampling of topics from geometric measure theory of interest to people studying topology, analysis, and PDE

How useful do I think it will be? I expect this course will be pretty situational as far as my thesis research is concerned; I described wanting to take it to my advisors as a “chance to indulge the pure mathematician in me.” In other words, I’m taking it mostly for fun. It’s not most peoples’ definition of fun, I know.

Course name: Advanced Topics in Numerical Mathematics

Course number: MATH 679

When I took it: Fall 2021

Overview of content: more than any other course the content tends to vary depending on who is teaching the course; this time around the offering was a mathematically robust theory on neural networks and machine learning. While we didn’t do any algorithm implementation, we got a tour of the ongoing research in the mathematics of machine learning. Along the way, we saw some numerical analysis, optimization, linear algebra, PDEs, and stochastics. If a student was on the fence between focusing on pure mathematics or applied mathematics for their Ph.D., this course would make a very strong case for the applied math camp. It took a lot of careful planning on the professor’s part to orchestrate such a series of lectures.

How useful was it? While I don’t plan on incorporating any machine learning into my research anytime soon, just seeing all of mathematics wrapped up together in a tight package was quite the spectacle in of itself. One of the main reasons I took this course was because the professor was someone from whom I had taken courses previously and very much enjoyed. Building a working relationship with this professor was also important because he served on the committee for my oral specialty exam this past November.

After this coming semester, I expect to have four more semesters before graduating (not including summers). While I don’t know what exactly what will be offered, what will run, or what I will take, here are some things I see as possibilities.

Advanced Topics in Numerical PDE I/II (MATH 673/MATH 674): the topics vary but usually involve some advanced treatment of finite element methods, relying a lot on PDE theory and functional analysis; probably a perfect fit for me

Continuum Mechanics (MATH 537): since continuum mechanics is largely regarded as the precursor to peridynamics, this would be a useful course for further appreciating the physics and mechanics behind the models I have looked at and read about; this is a relatively niche course and probably isn’t of use for everybody

Mechanics of Materials (MSE 512): I imagine this would be similar to continuum mechanics but aimed more so at engineering students; I mostly likely won’t end up taking this one but I want to keep it in the back of my mind.

Functional Analysis I/II (MATH 641/MATH 642): this would be similar in content to Applied Linear Analysis, but more in the context of pure mathematics than in numerics and PDEs; I imagine this course won’t be absolutely essential but it can’t hurt if I feel like I’d have time for it

Calculus of Variations (MATH 534): a course all about techniques for minimizing integral functionals (where the input is a function belonging to a certain function space, and then it is plugged into an integral); the course has not run since before I came to UT because the department has struggled to get enough interested students for the course to run; I don’t think this is a huge loss if I can’t take it, if only because I’ve self-studied a decent portion of the course’s typical material

And that’s all there is to it! I feel like I’ve taken a lot of courses and spent a lot of time on problem sets, exam prep, and coding assignments, but for the most part it was worth it. My future self will [probably] thank me.