# Thoughts on Theses: Will DeGroot ’23

Will DeGroot ’23 is a mathematics and computer science double major. His thesis works toward answering a “big open question” within the field of abstract algebra called the Galois embedding problem. In collaboration with the Amherst STEM Network.

**Nathan Lee: **To start off, tell me a little about yourself.

**Will DeGroot: **My name is Will, I’m a computer science [and] math major. Ever since I arrived in Amherst, I knew that I wanted to become a math major. I just started taking some slightly more advanced classes in high school, and I became really interested in it. I also programmed a little in high school, so the computer science major was a little bit of a passion project mixed with a practical decision, employability-wise.

**NL: **Why did you decide to pursue a thesis project?

**WD: **I’ve been involved in math research for two summers now, so I was always interested in potentially pursuing graduate school in math. During the Covid summers things were a little weird, so I didn’t really end up doing any math research and did other job-slash-internship opportunities. And then during the summer after my sophomore year, I did an REU [Research Experience for Undergraduates] program that was online during Covid, and then last summer — the summer after my junior year — I worked with Professor Benedetto and a few other students on campus. I just have [always] been interested in doing math research … the thesis is just a product of that.

**NL: **Why are you interested in math research?

**WD: **I always really enjoyed the problem-solving aspects of pure math. I’ve tried physics and some other slightly more applied math disciplines, and they just never quite had the same appeal to me. [There’s a] satisfaction in solving mathematical problems and doing mathematical proof. I discovered a type of math that I really enjoyed during my sophomore year, which is abstract algebra. [I was introduced to it] in ‘Groups, Rings, and Fields,’ a class that I took with Professor [of Mathematics Harris B.] Daniels. Ever since that course, I’ve been very interested in doing research in algebra. My thesis that I’m currently pursuing is about abstract algebra.

**NL: **Normally, we’d dive straight in and ask what your thesis is about. But we’re dealing with abstract math here, very high-level pure math. I think it would be helpful if you could give a little bit of background. What interests you specifically about abstract algebra?

**WD: **Typically, when we think about doing math, we consider doing arithmetic on real numbers — decimal numbers, any normal numbers that we’d think of. Abstract algebra asks the question ‘What if we wanted to come up with a system of doing math with an operation that’s something like multiplication, but instead of using all the numbers, we only wanted to use the numbers 0, 1, 2, 3, 4, 5?’ If you take 3 + 3, that equals 6, and 6 is [not contained] in the set, 0, 1, 2, 3, 4, 5. When I first took ‘Groups, Rings, and Fields,’ you’d generalize: The first part of the course is about generalizing the notions of addition, subtraction, and multiplication into operations that can act on sets that are just [abstract] symbols.

And I don’t know — it’s tough to isolate exactly what appealed to me about that. But I think, in general, I’m just really interested in how form appears in math. Oftentimes, when you’re thinking about a mathematical structure, or an algebraic structure, there are many different ways to describe that structure. In certain ways in which you describe it, it might lend itself to certain proofs or certain ways of thinking about something. And other descriptions of the same thing might be useful in a different context. So I think it’s exciting to try to look at certain very basic ideas from a variety of different perspectives, and try to understand the advantages of each of those perspectives, if that makes sense. It’s a little abstract.

**NL: **What is your thesis?

**WD: **My thesis deals with a branch of math called Galois theory. Galois theory has to do with something called a field, and field extensions, which are fields that contain other, smaller fields. A field is like the set I was talking about earlier, with operations like multiplication and addition that act on it and some more rigorous properties that make a field a field. For example, a field might be the set of all rational numbers. And you can consider that field as being a sub-field of all the real numbers. And Galois theory provides some tools to understand how many fields there are between a field which contains another field. If you have a field A which is contained in a field B, are there any fields that contain A but are also contained in B?

My thesis, in particular, deals with a big question in Galois theory. If I had to give a quick explanation, when you look at one field which contains another field, there’s a particular mathematical structure called a Galois group that you can assign that field extension. And there are many techniques in Galois theory that you can use to calculate Galois groups of field extensions. But it’s unknown if, for every Galois group, you can find a field which has that Galois group. It’s called the inverse Galois problem. (Not sure if any of this is actually making sense — this is usually just a conversation between me [and] myself or me [and] my advisor.) The inverse Galois problem, you don’t have to understand what it is. It’s just a big open question in Galois theory. There’s a really similar unknown problem that is actually a generalization of the inverse Galois theorem. It’s called the Galois embedding problem. And my thesis deals with a couple of small cases of different Galois embedding problems.

A lot of the content of my thesis is about navigating these several interlinking correspondences where you plug one thing into one form, it spits out another different form, then you plug it into something else, and then you trace it all the way back through. My thesis very much is about looking with different perspectives at the same sort of object. It’s kind of weird how there are so many different ways that you can look at the same structure, and it’s amazing how all these different ways have developed throughout the history of that.

**NL: **What was the process like working with your advisor?

**WD: **It’s the best. I love my advisor. He’s the best professor ever — my advisor is Professor Daniels in the math department. At the beginning, I was meeting with him once a week. Now that it's getting closer to the deadline, I’ve been meeting with him twice a week. Pretty much, I usually go in with something I don’t understand — rarely do I go in just totally knowing what I’m doing. I’ll go in with a question or two, or there’s something I’m confused about, or a possible new direction. And I’ll just go into his office and scrawl everything out on the chalkboard, and we go over it together and try to clear up misunderstandings. But most of my project has been self-directed. At the beginning of the thesis, Professor Daniels gave me a lot of different roads that I could explore, and I figured out which topics were most interesting to me.

All of the most rewarding experiences of my thesis have been in meetings with Professor Daniels, because I’ve been grinding hard trying to figure out something, and ending up having an idea that I’m really excited about, and I’ll go see him the next day. And sometimes — usually — it’ll be pretty wrong. But sometimes, it’ll be on the right track, and we’ll work it out together. I feel like we’ve been working together as a cohesive team, and I come out of his office feeling that I’ve usually accomplished something.

**NL: **If you imagine a researcher in chemistry, you have a very clear image of what this person is doing, yes? It’s not as clear for math. What is the research process like for you?

**WD: **Like I said before, my topic has been to try to solve a new Galois embedding problem or a very small case of one embedding problem. At the very beginning of my thesis, I found a paper which had a technique of solving Galois embedding problems, and it looked like a technique that could be applied to some new circumstances and derive some new results that aren’t necessarily super original, but still might be worth publishing a paper about. So really, the majority of my thesis work was looking at the paper. At the beginning, it was just complete gibberish to me — I didn’t understand anything. And I’m just slowly dissecting the terminology bit by bit scouring through different textbooks. And when I would encounter a theorem that seemed like integral to what I was reading, then I would try to understand that theorem. And if it was feasible, I’ll prove the theorem and understand the proof, and notate that all down for my thesis. Eventually, I got to the point where I understood all the mechanics of how the paper worked.

Some of these theorems, along the way, it’ll be like, ‘Here’s the theorem.’ And then I was stuck probably for two weeks on a small section of a proof that said, ‘From this, you do a couple easy calculations, and the rest follows.’ And I was in Professor Daniels’ office for two weeks trying to figure out those ‘easy calculations,’ and they were not so easy. I ended up having to use a variety of different sources to construct a [sorta original] proof of my own.

That was a bit of a sidetrack. But yeah, most of my work was figuring out this paper and this technique. After I figured out the paper and the technique, I applied it to a couple new areas, and based on the results of applying that technique, I tried to do a little bit of mathematical description of my results by what I saw. Then, I’ve just been furiously typing.

**NL: **You mentioned earlier that you are interested in going to grad school for math. Could you tell me a little bit about your plans for the future?

**WD: **Well, my future is very much up in the air because I applied to a bunch of grad schools, and I’m waitlisted at three. So I may not be going to grad school next year, and maybe hanging around for a bit reapplying, pursuing other job opportunities.

**NL: **Do you have any advice for people entering the thesis process, or students in general?

**WD: **Well, I’ll give the typical advice to just write as much as you can early on, so you’re not feeling stressed. I think I did a reasonable job of that, but I’ve [still] been struggling in the last couple of weeks, like really getting everything on paper. I would also say, just don’t worry too much about spinning your wheels for a week or two, not accomplishing much. Sometimes I feel like that’s actually a good thing, in the sense that you can let yourself be a little creative. If you’re early on in the thesis process, don’t stress that much about time and read what seems interesting to you and try to do something that seems fun and interesting, even if it might lead nowhere. I think some of the proudest things I included in my thesis was work I did after or just at the conclusion of a long period of accomplishing nothing or just doing random calculations.

**NL: **Is there anything you might want to say to the Amherst Student?

**WD: **The math department is great. It’s my favorite thing about Amherst, probably. Everyone in the math department is fantastic. Also, have fun trying to be creative, and don’t think you can’t do anything new.

**NL: **Will, it was nice talking to you. Best of luck on the rest of your thesis.

**WD: **Nice talking to you.

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