Working together on a senior thesis

I enjoy working with students on undergraduate senior thesis projects! The thesis is a capstone, a culmination of your undergraduate studies. It’s a chance to share your enthusiasm and enjoyment of math, and pass on some inspiration to the next generation of math students. And it’s a chance to show what you’ve learned, and your growth as an independent learner.

What math topics do you love, what are questions you’ve enjoyed exploring? In your project you will get to choose one of those topics, dig in deeply, and explain it in your own words, so that future students will learn what makes the topic so exciting and inspiring for you.

Beyond learning more about your chosen topic and sharing your excitement, you’ll also get to flex your skills as a learner and as a writer. The senior thesis doesn’t just mark the end of your undergraduate studies. It also launches your post-college independence, demonstrating your readiness to self-teach, to independently tackle challenging math subjects, and to produce the highest quality writing that other people can learn from. This is it, this is what college has prepared you for!

Working with you on your senior thesis project, as you go through this fundamental transition from student to mathematician, is a profound honor for me.

If you think you might like to choose me as your senior thesis advisor, I encourage you to email me. But first, please look at the expectations and project ideas on this page and think about whether it would be a good fit for you.

What you can expect if we work together

I direct my senor thesis students to write a thesis paper, a little bit like a term paper or a mini-article. Some examples are on my advising web page.

You can see that the students wrote sort of mini-articles with introductions, explanations, theorems and proofs, and at the end a bibliography.
The senior thesis is usually a chance for you to learn about something, and explain it in your own words. This could be based on readings (learning things that are already published) or it could be based on explorations.

However the senior thesis topic gets chosen, it should be something that excites you, that you want to spend time with, and that you want to share with others. In your senior thesis you should share what you like about math, what makes it exciting for you. Hopefully a future student will see it and also get excited!

It does not have to teach everything about the subject. The idea is more to get a student interested, so that they will be inspired to go and learn more for themselves.
I meet with senior thesis students for 1 hour each week, to guide you in your project, give you feedback, and help with any questions you have. The meeting would be to answer questions, go over things, or I can explain some things if you are stuck.

It will be your project to work on during the week between our meetings. But I will help you!
I want my senior thesis students to type their thesis documents in LaTeX.
Timeline:
1. We will spend a few weeks at the beginning of the semester talking about project ideas, exploring possibilities, and planning what will be in the thesis.
2. Then you’ll spend a few weeks reading and learning. We’ll talk about what you’re learning.
3. Somewhere around mid-semester, you’ll start writing your thesis. This will probably start before you finish all your reading.
4. You’ll write at least one draft before the final version, hopefully two drafts, with multiple rounds of feedback and revision.

Project ideas

The most important thing is to find a project idea that excites you, that you want to learn more about, and that you want to share with others.

What subjects in math do you enjoy? What made you want to be a math major? If your friends or family ask you what you like about math, what do you tell them? What have you enjoyed learning about, and what’s something you’d like to learn more about? This could be something from one of your classes, or something you read or saw in a video online, or just something you’re curious about.

You might like to look through some articles, e.g., Chalkdust Magazine which is written by students, Mathematics Magazine, or other sources that I can suggest. If you see anything there that interests you, it could be a starting point for a senior thesis.

Here are some topics and project ideas that I can suggest:

Topology and number theory

Let $X$ be the $k$ consecutive integers starting from $n$. Take the collection of subsets of $X$ that have a common divisor bigger than $1$. This is a simplicial complex. You will study the homology and cohomology of this simplicial complex, in the limit as $n$ and $k$ go to infinity, as well as other related simplicial complexes defined in terms of similar number theoretic ideas.

This is related to persistent homology and topological data analysis. This project involves some computer programming (in Python or your language of choice), gathering data and identifying patterns, and proving.
Changepoint detection in book readership data

You will gather data from the Seattle Public Library’s open database and perform a statistical changepoint analysis to find reasons why book readership rates change over time. This project will give you experience in gathering and analyzing data, and statistics for time series.

Besides library data, other Seattle city government data is available, and it can be studied too.

Explorations in Number Theory

Number theory topics including:

Every positive number can be written as a sum of four squares. How many triangular numbers are needed?

This leads into research projects such as: How many numbers in a given range can be represented by adding 2, 3, or a limited number of triangular numbers? You will study these questions and explore further: for example, use pentagonal numbers or pyramidal numbers instead of triangular numbers; statistical analysis of sums of random triangular numbers.
Euler’s pentagonal number theorem relates partitions, pentagonal numbers, and generating functions. You will learn about this amazing theorem that connects number theory and combinatorics.

Then you will explore research questions in advanced topics such as two-dimensional partitions (called plane partitions).
Benford’s law is a statistical observation about the leading digits of certain sequences of rapidly growing real numbers. It is a statement about the fractional part of the logarithm of the sequence. In this project you will study the generalization to Gaussian integers, including studying the argument (angle) of a sequence of Gaussian integers.

This has not been studied before. You will be conducting original research on a new topic.

This project is related to tropical geometry and amoeba geometry. (Tropical geometry corresponds to the logarithm of the magnitude as in the “classical” Benford’s law, while the amoeba corresponds to the argument.)
Tensor eigenvectors

Tensors are essential for machine learning and AI. In this project you will research eigenvectors of tensors, focussing on symmetric tensors. This project connects topics of linear algebra, algebraic geometry, calculus, and optimization, using a combination of tools of computation and theory.

Detailed data analysis of research articles

You will develop a new tool to gather and analyze data on research articles in a new level of detail. Current methods only look at the importance of research articles or books. You will develop a tool that breaks down which parts of the articles or books are important. Which chapters, sections, or statements get cited most by other researchers?

This project will use data gathering and analysis, AI, and statistics. It will result in a tool that can be used by researchers.

Gaussian integer valued polynomials

A polynomial $p(x)$ with integer coefficients always has an integer value when the input $x$ is an integer; that is called being integer valued. But a polynomial can be integer valued even if it does not have integer coefficients. For example $p(x) = \frac{1}{2}x(x-1)$ is integer valued because for any integer $x$, $p(x)$ is an integer.

In this project you will study polynomials with Gaussian integer values but not necessarily Gaussian integer coefficients. Once that is solved, you will study Eisenstein numbers and other rings of integers of number fields.

Positive polynomials in modular arithmetic

In modular arithmetic a number is called positive if it is a quadratic residue.

Which polynomials have positive values at all positive inputs?
The quadrant problem is to write the positive quadrant ${(x,y) \in \mathbb{R}^2 : x > 0, y > 0}$ as the image of a polynomial map. It was surprisingly hard to solve.

In this project you will study the quadrant problem in modular arithmetic. The positive quadrant in $(\mathbb{Z}/p\mathbb{Z})^2$ is the set of $(x,y)$ such that both $x$ and $y$ are positive, i.e., nonzero quadratic residues. The project will study whether it is possible to write a polynomial map that outputs this set.

(It would be easy to say $F(a,b) = (a^2,b^2)$, except that this will output points where $x=0$ or $y=0$, if $a=0$ or $b=0$. The challenge is to not output any zeros.)

Other shapes besides quadrants can also be studied, for example the complement of a quadrant, or the intersection of a quadrant with a flipped and shifted quadrant (a “rectangle”).
The same questions in Gaussian integers instead of modular arithmetic.

In any of these versions of the project, you will get serious experience with a real research problem in number theory, with connections to abstract algebra.