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:

Partially ordered sets (posets)

The height of a poset is the maximum length of a chain, a subset $x_1 < x_2 < \dotsb < x_n$; the height of an element is the maximum length of a chain ending at that element. The width of a poset is the maximum size of an antichain, a subset in which no two elements are comparable. A poset has the Sperner property if the largest size antichain is given by the set of elements at some fixed height. What posets have the Sperner property?

Some of my favorite posets are given by sets of monomials like ${1,x,y,x^2,xy,y^2,x^3,x^2y,y^3}$, ordered by divisibility. I would like to learn more about Sperner property, and other properties, of algebraic posets like these.

Other interesting posets are given by sets of integers such as ${1,2,3,\dotsc,12}$, again ordered by divisibility. What happens if we take $k$ consecutive integers starting from $n$, and then take a sort of limit as $n$ goes to infinity?

One very interesting thing that I would like to learn more about posets is Möbius functions. Perhaps you might have learned about the Möbius function in a number theory class. There is a nice generalization to posets. Say $\mu_{k,n}$ is the Möbius function for $k$ consecutive integers starting at $n$. Can we find the limit of $\mu_{k,n}(n+j)$ as $n$ goes to infinity?

Projects on these topics will include some reading and learning about combinatorics and posets (and maybe number theory); some exploration of working out different examples, which might be a combination of both working out some examples by pencil and paper, and also computing examples on a computer; looking for patterns, and maybe even coming up with a conjecture. And maybe even proving it!
Billiard trajectories

Imagine a ball rolling around inside a polygon and bouncing off the sides. What kinds of trajectories are possible? Can we set up a polygon to make the trajectory be a perfect five-pointed star? How about a capital letter L? How about a capital letter K or X, or if we can’t exactly get those, then how close can we get?

Taking it a step further, what if the billiard ball is flying around in three dimensions, inside a 3-D polyhedron? Can we make the path be a trefoil knot? Can we set it up so that two billiard balls will bounce around along linked loops (a Hopf link)?

There are some papers about these problems (actually, there are a lot of papers about the 2-D version) so this project will involve picking one or two papers to read and try to explain; and if possible, setting up some computer models and producing some animations.
Percolation on unusual tilings

Imagine pouring syrup on a waffle. When each little waffle square gets filled, the syrup pours out into the neighboring squares; after a while they fill up too, and the syrup spreads further. The “limit” of the syrup shape is roughly a circle.

Now what if the waffle squares get replaced with a different shape, such as the Penrose tiling or the brand new discovery of the hat tile, or pentagonal tilings, or lots of other interesting things?

Besides syrup, this topic ties in with how fluids, contaminants, or other things percolate through a porous medium such as soil. Or, it could be energy spreading out from a point. Mathematically, this topic usually falls under labels like abelian sandpile model or chip firing.

This project would mostly involve setting up a computer model to run examples. It should produce some really neat images and animations! This is a topic where I don’t really know what to expect mathematically: what will be the “limit” shape? Will we discover any interesting patterns or trends? Well, I don’t know what they will be, but I’m sure that lots of interesting things will pop out.
Algebraic properties of graphs

I would like to learn more about algebraic properties of graphs such as the eigenvalues of the graph’s adjacency matrix and the graph Laplacian (basically the adjacency matrix, with the vertex degrees along the diagonal). This combines graph theory, combinatorics, and linear algebra. It can be approached both theoretically (reading, pencil and paper, proofs) as well as computationally (computing examples).

It would be interesting to learn about applications of graph eigenvalues, as well as to work out examples for various types of graphs.

Because the Laplacian shows up in the heat equation and in the wave equation, eigenvalues of the graph Laplacian are related to flow and vibrations along the graph. (Among other things, this relates to the abelian sandpile model and chip firing.) I don’t know a lot about this but I would love to learn more.

It’s also related to Markov chain models on the graph. So, this topic could be taken in a statistics/probability direction.
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. What is its homology? How does the homology change as $k$ and $n$ change? Is there a “limit” as $k$ or $n$ goes to infinity?

To work on this you would need to know some topology. A computer could be used to try examples. If it’s even possible to make a conjecture, then trying to answer it would involve number theory.
Poset of nonvanishing minors of a matrix

If you can answer this question on MathOverflow or even come up with a single good idea, it will almost certainly be a publishable research paper, not to mention a guaranteed accepted answer on MathOverflow (karma points!).

Let $M$ be a matrix. A minor of $M$ is a square submatrix, e.g., the $2 \times 2$ submatrix given by rows $1$ and $3$, columns $5$ and $8$ — the rows and columns don’t have to be consecutive. Let $P$ be the set of minors of $M$ with nonzero determinant. This gives a poset (partially ordered set). What are the properties of this poset? For example, can the width of the poset be related to any linear algebraic property of $M$?
Explorations in Number Theory

There are many theorems in number theory that would be interesting to explore. For example, building from Lagrange’s theorem that every positive integer can be written as a sum of four or fewer squares, we could investigate: how many triangular numbers does it take to write every positive integer? How about pentagonal numbers?

There’s a theorem by Euler relating pentagonal numbers, partitions, and generating functions. I would like to learn more about this.

We could study statistical properties: for example, what’s the average number of triangular numbers needed to write a positive integer; what’s the average number of ways to write a positive integer as a sum of triangular numbers?

The normal definition of squares can be changed. Normally we go from one square number to the next one by increasing both of the factors that get multiplied, going from $xx$ to $(x+1)(x+1)$. We can randomize this: when our current number is $xy$, we can pick the next number randomly as $(x+2)y$ or $(x+1)(y+1)$ or $x(y+2)$. How would Lagrange’s four-squares theorem work for one of these “pseudo-square” sequences? This project would give opportunities for computational exploration, by simulating a lot of different random samples.

There’s a famous result that the infinite sum of the $1/n^2$ gives $\pi^2/6$. What if we did this with a “pseudo-square” sequence?