Introduction

The purpose of this piece is to share some reflections on my experience teaching MATH 416 (Applied Mathematics) in Fall 2025. This was my first time teaching the course, and it is important to state that at the outset because, in a sense, I was going in somewhat “blind.” I had a vision for the course, but there is always a difference between one’s initial vision and the actual reality that unfolds during the semester. In what follows, I describe my journey teaching this course, some of the things I learned, what I believe went well, and what I think did not go as well and should be improved.

Anchoring the Course in Heil’s Text

I chose to anchor the course around Christopher Heil’s textbook. Heil is an excellent and very clear writer, and although the material in that text is largely beyond the standard undergraduate level, I felt that—because of his clarity—I could draw inspiration from it and bring some of the core ideas down to an advanced undergraduate audience.

There were genuine successes in doing so, but there were also some areas where the level and pace created difficulties. The tension between maintaining mathematical depth and ensuring accessibility for undergraduates became one of the central themes of my reflection.

Writing Within the Major

A central goal of the course was to help students “write within the major.” The phrase is somewhat loosely defined, but my interpretation was that students should be exposed to the customary ways mathematicians communicate results. When mathematicians obtain a new theorem, there is a standard process by which they formulate definitions, state the result, and present a rigorous proof.

I could not simply assign an undergraduate research–style problem and ask students to begin writing at that level immediately. They first needed exposure to the content so that they could acquire the necessary skills, concepts, and understanding. To implement this, I designed the homework so that it contained embedded opportunities for mathematical writing.

For me, “writing in mathematics” is not the same as writing in the humanities. An essential part of mathematical writing begins with careful reading, followed by the ability to write clear and logically structured proofs that substantiate mathematical statements. There is also algorithmic writing—describing procedures and methods—which is distinct from proof writing. In addition, there is code writing, using computational tools to implement ideas.

Computational Tools: Mathematica and Large Language Models

In this course, the primary software environment was Mathematica, a symbolic computer algebra system written with mathematicians in mind. It is relatively easy for mathematics majors to pick up the syntax, it is fairly forgiving, and it comes with extensive libraries and examples that students can explore.

At the same time, we are in the age of large language models, such as ChatGPT and others, which can now produce computer code and assist with many aspects of problem solving. It is, in my view, important for students to learn how to use all available tools responsibly. For that reason, I tried to “bake” all of this into the structure of the course: students would write within the major, use a computer algebra system, and (where appropriate) learn how to leverage large language models as part of a modern mathematical workflow—in a way that supports, rather than replaces, their own understanding.

Foundational Material: Complex Numbers and Hilbert Spaces

I began the course by reviewing the complex numbers, since I planned to move relatively quickly into Hilbert spaces, which are typically complex vector spaces endowed with an inner product. From there, we examined a collection of key concepts: the Fourier transform, Fourier series, periodic signals, square-integrable functions, square-summable sequences, and the basic translation and modulation operators.

All of these topics can be studied at a very deep and abstract level, but my intention here was more of a general survey. I wanted students to be exposed to some very important and powerful ideas, and then to give them access to a powerful computer algebra system that they could use to solve interesting problems.

Fourier Theory and Applications

After covering basic aspects of Fourier theory, I presented some problems that show how these tools can be applied. One such theme involved random walks, where students examined probabilities for a random walk on the integer lattice with respect to a fixed probability distribution.

Along the way, I highlighted some adjacent results—interesting theorems that illuminate the theory—but did not require students to fully internalize every technical detail. The aim was to show how one moves from conceptual, highly theoretical ideas to computations and applications that live in the realm of practicality.

We also used our more superficial—but still meaningful—understanding of Fourier theory to solve certain partial differential equations. In particular, we used the Fourier transform and its inversion formula to derive solutions to some first-order ODEs and to the heat equation in simple settings.

From Fourier Series to Frames

This naturally led us into the world of bases and frames. The idea was to show how the notions of Fourier series and Fourier transforms can be generalized in a natural way to other transforms built from more flexible “basis-like” objects.

In this context, I introduced the concept of Parseval’s identity and explained how the definition of a frame can be interpreted as a perturbed or generalized version of it. We discussed frame bounds, analysis operators, synthesis operators, and frame operators, and we saw how the eigenvalues of the frame operator are connected to the frame bounds.

We then studied Naimark’s dilation theorem (often referred to informally as Naimark’s duality), which tells us that one way to construct a Parseval frame for a Hilbert space is to project an orthonormal basis from a larger Hilbert space. We used that framework together with ideas from multivariable calculus to construct examples of frames.

Linking to Group Representations and Algebra

I also made a point of linking these constructions with group theory and abstract algebra, to show students how different branches of mathematics interact in applied settings. For example, we considered a representation of the dihedral group acting on a three-dimensional vector space in order to construct finite frames in finite-dimensional spaces. This gave students practice computing analysis operators, synthesis operators, frame operators, and so on.

Having Mathematica at our disposal was extremely helpful here; it allowed us to carry out computations that would otherwise be quite tedious by hand. From my perspective, this integration of theory, computation, and algebraic structure was one of the most successful aspects of the course, and some students reported that they found this combination very beneficial.

Challenges with Pace and Accessibility

However, in order to cover all of this material, the course had to move rather quickly. This is the part that leaves me somewhat uneasy. At various points I had the sense that while some students were comfortable keeping up, others were struggling. Eventually I had to slow down substantially, especially when we arrived at Gabor bases and frames.

Here we entered the world of infinite-dimensional Hilbert spaces, using infinite collections of translates and modulations to construct classical Gabor systems. We first constructed a Gabor basis using the indicator function of the unit interval and an integer lattice in the time–frequency plane. This provided a concrete example but also revealed an important tension between pure and applied mathematics.

On the one hand, in pure mathematics we can construct beautiful, exotic objects that are theoretically sound. On the other hand, when we attempt actual computations—such as writing series expansions for specific functions using such a Gabor basis—we can encounter serious numerical and computational challenges. Even a powerful tool like Mathematica can struggle with certain series or integrals arising from such constructions.

Many students had good intuitions at this stage, but this was clearly one of the points where we needed to pause, reflect, and slow down. I deliberately slowed the pace for over a week, giving students time to catch up on homework and to work on their midterm project.

Midterm Project: From Homework to Proposal

The midterm project asked students to submit a proposal for a final paper, together with a preliminary outline of their ideas. They were encouraged to use the weekly labs and homework as inspiration for a topic. Each student chose a topic of personal interest connected to the course themes, and then outlined the structure of their paper. I provided guidance throughout this process, and they are still working on these projects.

Final Phase: Gabor-Like Frames and Wavelets

In the final phase of the module, my plan is to show students how to construct other types of Gabor-like frames that do not rely strictly on integer lattices, and to move toward wavelet theory. This will require revisiting prior concepts, especially orthonormal bases and the use of dilations to move from one orthonormal basis to another.

The goal is to construct window functions that decay rapidly, which can improve numerical behavior and make computations smoother, thereby avoiding some of the difficulties we encountered with discontinuous or slowly decaying generators.

The last part of the course (scheduled for December) is devoted to translations and dilations, wavelets, Shannon wavelets, and wavelet sets. My intention is to present these ideas as gently as possible. They can, of course, be unpacked at a very deep theoretical level, but that is not our aim here. Our goal is to understand the concepts at a solid conceptual level and then leverage computational tools to carry out meaningful calculations.

Conclusion and Future Directions

In summary, this has been my first time teaching this course, and it has been both challenging and rewarding. It has been a very enjoyable course for me personally, and I believe for many of the students as well, although it has certainly been demanding for others. I now feel that I have a concrete starting point. If I teach this course again, there is clear room to refine the structure, adjust the pacing, and build an even more coherent and supportive experience that remains rigorous while being more accessible.

Finally, in the age of large language models, I am convinced that we must think carefully about how to pair our traditional computational engines (such as Mathematica) with AI tools so that we can handle large and complex calculations both efficiently and transparently. This pairing allows students to see the power of mathematics in real time and in realistic contexts, which I believe is beneficial both for their learning and for the broader perception of mathematics itself.