UChicago REU 2024

Hi everyone, it's been awhile. I've been too lazy to write a first year reflection post, so I probably just won't. Overall the REU was alright.

I had massive FOMO from seeing my PROMYS (summer math camp I did) friends return as counselors. The REU wasn't bad by any means—just harder to meet people and learn math. The REU was very independently driven: from finding a mentor to the few lectures we had, the mathematical activities were pretty self-driven. I still had a decently fun time—I got to spend a lot of time with my girlfriend and some UChicago math friends. I also met some of the more extroverted external students. We even had a road trip and got food! This was because they were old enough to rent a car. For the REU, internal students have to get their own housing, and I had some housing complications which were so so annoying but happened. I found that all that independence made self-control way harder. It was so tempting to just socialize or cook as excuses to not work. Because I made a calendar for the REU and was pretty active in the slack (made a notes channel), when I'd introduce myself to people, they'd ask if I was the Vincent Tran from slack 💀.

More on the mathematical side, the lectures are heavily dependent on professors being available and volunteering. This year we got unlucky and didn't have many algebra lectures, which sucked. For my mentor, I just yoloed and asked a famous faculty member at UChicago, and he agreed! I was his first REU student, which made things a bit awkward because both of us didn't know what we were supposed to do. In our first meeting before the summer, he suggested that I learn more general math that could be applied in other fields, saying that "you never know, you might become a biologist". He would proceed during the summer to fall back to this possibility a lot lol. The first week, I suggested that I spend some time thinking of a topic to write on for the REU, and my mentor dropped this hilarious line:

Sometimes you fall in love and have a baby and sometimes you just fall in love.

The first week I worked on Hartshorne as I initially planned but eventually we discussed my background more. He found it unacceptable that I have some holes in my math foundations, so I went back and filled some of those in. I didn't know a proof of Jordan decomposition and now I do. We continued going backwards in my background and he found it really terrible that I have such a weak linear algebra background. I self-studied an MIT open courseware class meant for engineers (the Strang one). So he gave me some linear algebra problems to work on while we find a text to learn more advanced linear algebra. I did most of the problems alright, the only issue was this one variation of a question where I needed to work basis-free. I couldn't figure out the right canonical mappings and this really frustrated him and me. Eventually he just explained the identification to me.

After asking another professor, we found a text to use—Linear Algebra and Geometry by Kostrikin and Manin. This is a great textbook, 9.5/10. It's an old USSR book, but it's well-written, covers interesting topics, and has lots of good exercises. Only thing is that the first chapter has a mixture of elementary linear algebra and computation though. The physics explanations are also a bit sketchy. I strongly recommend this to anyone that wants to strengthen up on their linear algebra. I read the entire book, and plan on doing more of the exercises.

My mentor often recommended I work on things that also had ongoing lecture series, and there was some lectures on representation theory. The lecturer was amazing, shoutout Justin Campbell. His notes on his website are also great. I started with chapter one of Serre's Linear Representations of Finite Groups, which was pretty good. It was written for chemists, so it wasn't terribly sophisticated. Then I worked on Etingof et al.'s text on representation theory, which is pretty good, but poorly organized. Chapter two has all the foundations/definitions for the rest of the book, and the problems can be so so difficult. I started with learning about some quivers, which are this combinatorial object that captures some interesting representation theory stuff. After doing this, I asked him to give me some motivation for learning representations outside of group theory, and he gave me a bunch of physics examples. He just really loves mathematical physics lol.

We then had a lecture sequence on quantum stuff and the Heinsenberg representation, which actually had two lecturers: a physicist at UChicago and a just graduated student. The math side was pretty good, we just didn't get a lot of it. For this, my mentor had me read a master's thesis by a student he once mentored. It was a good experience—learned an interesting representation, some subtleties of it, and surprisingly, distributions came up! Anyways this representation that is useful for quantum physics is very mathematical rich as well, having connections to modular forms and number theory! It can even be used to prove quadratic reciprocity! After this, I returned to Etingof and finished reading the representation theory parts of the book. I read on representations of finite dimensional algebras, finite groups, and more sophisticated techniques in representation theory of finite groups.

Finally, towards the end of the REU, Justin Campbell resumed his lecture series on representation theory (this time on representations of sl2(Fq)). For this, I worked through his notes on representations of GL(Fq), which involved the Heisenberg representation again! His final lecture was crazy: it started with the general idea of schemes, the important adjunctions, and then got into geometric representation theory! He got deep into it. We got to famous results in this field that were proven in the past 40 years (which for math is insanely advanced) by someone who is still living and is in our department! At the end we were allowed to give some presentations, and I gave one on the Riemann Roch theorem on graphs.

His recommendations for me during the summer is to learn distributions, ODEs, and hopefully the representation theory of GL2(R). I did an engineering class on ODEs at a community college in high school, so he wants me to fix this foundation. He also strongly recommends I learn complex analysis. In terms of course advice, his was pretty much learn the right things in the right order, be concrete, and learn everything 💀. For example, I thought that Riemannian geometry would be a low priority topic for me, but he suggested that I learn it. And you can also just tell from the amount of analysis he recommended I learn that he believes this. I suppose that's part of how he's so successful.

Some final remarks on my mentor: he's a surprisingly laid back person (he's realistic in the sense that he tries to limit how much math I try to learn, saying "there's only so much math a person can do a day"), and is quite caring. He told me how to avoid certain things that he regretted in his youth, and he even contacted other people for me. I really unfortunately had tech issues so often, which sucked so much. The tech issues are relevant because we met over zoom—the summer was too hot for him to come down to campus LOL.

Overall I'd rate the REU a 7/10. Advice for anyone wanting to do it is to be out there and plan events. I planned a board game event and a decent number of people came. Mathematical takeaways are that knowing how to write clean proofs is important, physics is kind of interesting, foundations and learning things in the right order are important (my mentor had bad experiences learning things out of order), and that representation theory is interesting. Also for anyone (even full program) doing the REU, I recommend going to Rudenko's apprentice talks. He covers a lot of interesting number theory and geometry that is not covered in classes. The number theory will be especially useful if you ever want to take algebraic number theory.

Really long post, so for making it to the end: here's one more quote from my mentor: "everyone needs ODEs, biologists, lawyers, dentists, you know, my first time in the dentist in the US, I told him that I was a mathematician, and he told me he took calculus in college. Anyways, he gave me a terrible bridge".