043. Doing Math

tldr: I did a lot of math this summer

I truly believe that being able to cold-email [emailing someone who knows little/nothing about you] is one of the best skills to have, especially at MIT. In general, being able to email is an important skill to have at MIT. I don’t simply just mean be able to compose a message and send it– I mean be able to compose numerous important feeling/actually important messages and be able to handle the influx of emails from clubs, advisors, and tetazoo-glounge [in my head, pronounced “glounge”– not “g-lounge”]. Off the top of my head, I can name the top 5 cold emails I have ever sent:

  1. One to Sal Khan. I didn’t get a response, but still it’s Sal Khan.
  2. One to Lydia, a past MIT blogger who wrote this very interesting blog post about Hilbert’s Third Problem. This particular email had the subject line “I Really Like Your Cow(s)” [you can see why it was this in the blog post]. I talked to her about what it is like being a math major, and asked for any math book recommendations. Her blog post was actually what got me to apply for and get accepted into a UROP that studied tetrahedra last IAP [Independent Activities Period]. I even sent her a follow-up after that UROP to tell her about it.
  3. One to Haynes Miller, my freshman advisor and my current academic advisor. I told him my name [as it’s different than what was in my advising folder] and my pronouns. This actually was one of the reasons why I received a First-Year award for diversity, much to my surprise.
  4. One to OpenCourseWare, asking if they had any positions open for students. This ultimately led to an ELO [Experimental Learning Opportunity], and I still keep in contact with them. Working at OCW would be one of my dream jobs in life.
  5. Five to five different mathematics professors who study analysis, asking if they would be available for a summer reading program with me [Professors Guth, Staffilani, Melrose, Dyatlov, and Lawrie]. Professor Larry Guth happened to be available on such short notice, and thus began a very interesting summer.

What follows is what I did during my UROP this summer. [Truth be told, I wrote a bit about this in 039 but now I want to write about the math and the process.]

This spring, I had no idea what I would be doing this summer. I had applied for a few summer programs but those didn’t quite work out. Plus, I didn’t really know what I should do this summer. Everyone seems to know whether or not they should do an internship, and at the very least they know if they should go home. As someone who certainly wants to go into academia, finding a job over summer feels a bit more difficult; I know that I should do research at some point, but don’t even know what that entails. If I go to the UROP listings and type in mathematics, I get two opportunities for AI and 5 options for studying abroad.

Thus, somewhere around May 11th of this year during 18.102 [Functional Analysis] office hours, I asked my professor if he had any suggestions. I was, and am, finding analysis to be one of my favorite areas of math so far, and thought he might have some suggestions for books to read or things to do. He then suggested that I send out emails to other analysis professors asking if they would be free this summer to do some sort of guided reading, and thus the cold emailing began.

Three of the professors already had summer commitments, which is more than completely fair– I was asking pretty late all things considered. But, most of them said that if I wanted to look into a UROP with them in the future to reach out again, which was super exciting. One of them said that while they couldn’t do a program with me this year, they would be happy to talk to me about analysis classes. This, by far, was one of the most influential meetings I have had at MIT regarding academics. He suggested that I look into the 18.155-156 [Differential Analysis] sequence. I don’t know much about these two classes, but looking at the material I am very excited. The last professor, Larry Guth, said he would be available this summer, and we began planning.

To be clear, I specifically wasn’t looking for a UROP at first. In the first 3 emails I sent to professors I asked about a UROP and this wasn’t working. In the email to Larry, I simply wanted to be able to sit down and read a math book and put in the time. Furthermore, this program wasn’t exactly with Larry, at least not the entire time. I spent most of my time this summer with one of his graduate students Yuqiu Fu, who just so happened to be a TA for 18.102.

Fun fact about Yuqiu: during the first week of 18.102 I was feeling really insecure. The class felt at my level? Yet also so much more advanced than I could handle. I asked Yuqiu after an office hour working with him if he thought I should continue the class (i.e., if he thought I might “have what it takes” for lack of a better current phrase), and he said yes. And 18.102 is my favorite class I have taken so far.

We started by talking about possible books to read, but my functional analysis professor said that he would just go with anything Larry thinks I should read, which was great advice. I was considering something Big and Formal like Fourier Analysis by Stein and Shakarchi or Rudin’s Real and Complex Analysis. Larry suggested something out of left field: Alex Iosevich’s A View from the Top: a book that strives to introduce high school students and beyond into math research and techniques/topics in analysis, combinatorics, and number theory. 

At this point Larry suggested the program being a UROP for credit. Something they don’t tell you about UROPs is that you need to fill out an application. Not quite an application that’s fill-in-the-blank, but rather more like UC Personal Insight Questions.

“How do you plan to work remotely? What are your responsibilities and goals for this research? Why do you want to do this?”

Writing these answers down helps put things into perspective. One line from my applications states: “I hope that this will give me the opportunity to grow as a mathematician, as well as to become a better independent reader of mathematics materials. In the long term, I hope that both of these skills will make me a better graduate student.” Which is very true, I just hadn’t verbalized it until I did this application. I ordered the book, and one week later Yuqiu and I started reading.

I tried to figure out how I would hold myself accountable for doing work during the UROP. It is one thing to read a math book, and another thing to work through a math book, and I wanted to do the latter. Thus, I created a LaTeX document, and just put everything in there. Every question and finished exercise– anything to document my thought process through this material. We read through (thoroughly) 8 chapters of the book in 5 weeks. Technically I read chapters 9 and 10, but we didn’t work through all of it. We went through the exercises, asked (and answered) additional questions, and I presented proofs from the book. This helped me Really understand what was going on in every line. At this point I had the following notes (if you are interested in reading, but it is kinda dense and not particularly interesting in hindsight). 

If nothing else, I really like how these notes developed. There were a lot of places where I had to go back and revise my proofs and ask more, and it helped serve as a way to convey what I was working on with Larry. After five weeks, I met Larry on zoom for the first time, and we started discussing a question he wanted me to try and solve. Note at this time this blog is math math math so, I totally understand if you aren’t interested in reading. There is quite a bit of calculus as well.

I would really want to type all of my project out here, but WordPress isn’t LaTeX compatible. In fact, if I could I would create a however-long-it-takes-to-film YouTube video describing it, but I wasn’t quite certain how to do that/if I would have the time to do that. Thus I have decided to add the document in below, just like I did with my pre-project notes.

I found this project particularly interesting. I know my solution isn’t the most elegant, especially given that there is a Much More General theorem that one can prove using n-dimensional functions as opposed to a two variable function like the one we considered. But still- this was something I was able to derive. Furthermore, the material we covered in this project will be discussed in 18.155.

I don’t really know what math research is. I can picture what research may be like in many other fields–going through data and spreadsheets to create more accurate meteorological models, or writing computer programs to better understand the flow of foot-traffic in a city. Do I know exactly what is going on in these fields? Certainly not. But I can picture it to some degree, and at the very least I can talk to a friend about these interesting topics they are studying. Then, I spent this summer in my very own math UROP,  and I still don’t quite know.

I have learned that the biggest difference between reading a text and engaging with it is asking the next question– especially in math.

How do you generalize this theorem? What happens if f is n-dimensional? How else can I solve this problem? Why did the author do this?

Throughout my notes are little scatter-brained thoughts and comments, which I was fortunate enough to write down.

It was nice getting to spend this summer actually feeling like I did math. After 2033 lines of LaTeX code, and nearly 60 pages of typed notes, I made it to the end. I am very excited to do math again, and I am even more excited to do it in person.

Published by Paige Bright

Hi- my name is Paige Alexandria Bright. I am a rising junior at MIT interested in mathematics and philosophy. I have been writing this blog since the beginning of COVID. Lets see where this goes.

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