thoughts on math@MIT! by Fatima A. '25

math month post

I asked a bunch of people, professors, current and former students why they do math and their thoughts on the MIT math department. Here is what they had to say!

Short introductions!

Jonathan: My name is Jonathan. I joined the department as a postdoc last year. I’m interested in topology, which is the study of flexible shapes like knots or rubber surfaces. We can’t use the traditional geometrical measurements like length or area to characterize these shapes since they aren’t rigid. So we develop more robust measurements like counting the number of holes in the shape. Fatima: I took Seminar in Topology last semester with Jonathan!

(let’s call them) Z: I’m a recent MIT alum who majored in 18, and now in a math PhD program.

Carl S. ’23: I’m Carl, and I’m a first-year math PhD student at Stanford. My main interest is extremal combinatorics, but I’m also interested in analytic number theory (and maybe some other analytic things).

Tristan S. ’23: I’m a 2023 graduate of the math and music majors, currently a master’s student at the University of Cambridge.

Matthew H. ’25: Hi! I’m Matthew, a ‘25 who lives in Next House. I’m a former Californian who enjoys playing tennis, chess-related activities, and watching local sports teams.

Misheel O. ’25: I’m a junior at MIT studying math and CS, and I watch anime and stuff.02 know an xkcd for every situation

Zimi Z. ’25: I am a third-year student in course 18C (Math with Computer Science). As the name of my major suggests, I do a lot of things involving math and computer science and both.

Why do you do math?

Professor Cohn03 Fatima: I took Algebra I with Professor Cohn last fall! : One of the things I love about math is that we’re all on the same team. Of course I’m not the world’s greatest mathematician: I’ve done some interesting mathematics, but there are others who are more accomplished. If math were a competition to be the best, I might as well give up now, because I have no hope of “winning math” (whatever that might mean).

Fortunately, that’s just not how it works. If someone else proves an amazing theorem, their achievement doesn’t make my work any less interesting or significant. Instead, the more we learn, the more fascinating and valuable mathematics as a whole becomes. We’re all working together to figure out the answers to questions we share an interest in. As in all human activities, there’s occasionally competition for jobs or prizes, but competition is not the point of mathematics. Other mathematicians are my friends and colleagues, not my rivals.

The reason I want to emphasize this is because of how incredibly successful MIT students have been in the Putnam exam (see here). There are many amazingly quick and insightful students at MIT, and I think people sometimes get the impression that success as a math major must mean being able to come up with ingenious solutions to carefully posed problems under time pressure. That’s an impressive skill, but it’s not the most important skill a mathematician can have, or necessary for success. It’s important to celebrate achievements in math competitions without letting them overshadow other forms of promise or achievement. Fatima: This answers a bunch of the questions that follow, but I didn't want to break it up because so true!!

Jonathan: I enjoy the mystery of it. You’ll often see some patterns that you can’t explain, but after working hard to understand the objects better, the patterns become natural and inevitable.

Z: I’ve been interested in math for basically my whole life, initially through math competitions but now for the math itself, which I find both interesting and beautiful.

Carl: I’ve pretty much always been more interested in math than anything else. This has maintained since I was about four years old, despite the fact that what “math” means has changed a lot — from arithmetic to short-answer questions to olympiads to college classes and now to research. I’ve essentially followed the “path of most math” for most of my life up to this point: I considered studying CS or physics in college, but liked math more; I explored quantitative finance and software development, but preferred research math. I’m pretty happy with where this path has brought me, and there’s nothing else I’d rather be doing.

Matthew: I like the combination of intuition and rigor–I believe the former is more important to me, but I think being able to understand concepts from the ground up is satisfying. I also think that understanding the explanations of various phenomena through math is something that I value, which is why I take a lot of classes in subjects that aren’t math, but involve math to a reasonably high degree (for example: I took Systems Biology, where a large focus of the class was understanding how systems of molecules that regulate each other behave, which can be modeled using dynamical systems.)

Misheel: I like doing math because I like the idea of figuring things out and/or understanding things. It’s really hard to say really general stuff, but if you assume a small number of reasonable things (axioms), then suddenly those concepts connect together and you get an explosion of connections and new concepts that you can learn and explore. I also play a lot of video games (or used to before college lol) so I like making video game analogies. In this case, the axioms are sort of like a seed for a world in minecraft and you explore what grows out of that seed and do cool stuff.

Another analogy is that research in general feels like exploration as well, like exploring every corner of a room before moving on, and I enjoy that a lot. Running a dfs05 depth first search in some sense.06 Misheel: it is important to contrast the need for dfs in research with the danger of dfs in learning because if you dfs in learning, you really can't stop and you'll have very specific knowledge about a few fields but not a lot of foundation. so for studying bfs (breadth first search) is better or something.

Zimi: Good question, I don’t know if I would call myself someone who “does math” as I have not engaged too much in math besides taking classes. I take these classes mostly because they are fun! I have not participated in a math UROP or DRP,07 directed reading program. during IAP, math majors can sign up to read some math topic of their choice with a mentor. which is honestly something I regret a little.

What do you think is one of the hardest things to get right in math pedagogy? Fatima: For some reason, I did not send this question to any of the students. I regret that! People probably had thoughts from the other side and a lot of students also TA classes anyway.

Jonathan: Math is very interconnected, and the best way to learn a subject is to get acquainted with examples and intuition from neighbouring areas. So it’s difficult to teach math in a linear way! I wish I could continue my classes with students after a couple semesters, and circle back to old material with new context from other subjects they’ve learned.

What are your thoughts on the math department at MIT?

Jonathan: It’s really nice being surrounded by people (students and faculty alike) excited about math and working on such interesting problems!

Z: The MIT math department (including both pure and applied) is one of the best in the world, and in particular the undergrad math population has been insanely strong the past few years (see Putnam results…). It’s also worth mentioning that the standard course offerings are pretty comprehensive compared to other schools (I think), and usually well-taught.

Zimi: Math is one of the bigger departments at MIT so you get to meet a lot of people with many different interests!

What are your favorite things about the math department at MIT?

Jonathan: cookies at tea time

Z: I really appreciated the variety of course offerings, which covered everything I could dream of wanting to learn (and much, much more). Also, the other math majors: I learned a ton from the friends I made.

Carl: The MIT math department is rather large, and there’s a great breadth of interests represented at every level (undergrad students, grad students, faculty, etc.). This makes MIT a great place to figure out what (within math) one is interested in. I also like how, by and large, the faculty genuinely care about the quality of the teaching.

Tristan: My favorite part of the math department is the active community of students willing to talk about anything math. It’s always great to have peers who are excited about the same things as I am!

What are your least favorite things about the math department at MIT?

Jonathan: cookies at tea time – I think I’ve eaten too many.

Z: The fact that there are so many strong math majors makes it very easy to fall through the cracks, which can make coursework and life difficult.

Carl: There are so many undergraduate students who are interested in math, which is fantastic! Unfortunately, many opportunities that the math department offers (UROPs, DRP, SPUR)09 summer program in undergraduate research are resource-constrained, and the upshot of this is that they end up being available mostly to the students who came in with a lot of background, or who have very visible and measurable accomplishments to distinguish them from the crowd. This means that, while some students get fantastic opportunities and support, some get very little. I believe this dynamic also pushes some students out of considering mathematics, which is a shame.

Misheel: MIT doesn’t have good infrastructure for grad students and undergrads to intermingle. For example, from what I have heard, Harvard students share their lounge with the grad students and they learn a ton from them. But on the other hand, the benefit at MIT is that the undergrads intermingle better. Still I think if MIT opened the grad lounge to undergrads it would be good XD

Zimi: The administration of the department seems to be a bit out of touch with the undergrads and there seems to be some divide between people who did a lot of math before coming to MIT versus people who maybe did not have as much prior exposure to say, competition math, but are just here to learn math.

What is your favorite math class you have taken at MIT?

Z: I liked a lot of them; two of them worth singling out are 18.78310 Elliptic Curves and 18.785,11 Number Theory I which I found to be organized and taught very well. Plus, I thought the material was fascinating.

Carl: Certainly the most influential class for me was 18.225 (Graph Theory and Additive Combinatorics), taught by Yufei Zhao. I loved the course, and, a few years on, it’s more or less defined my research interests.

Tristan: My favorite math class at MIT is 18.225 (Graph Theory and Additive Combinatorics), taught by Professor Yufei Zhao. I knew little about the subject before taking the class, but the course content helped me develop intuition and skills that are quite useful for both the topics covered and also math in general.

Matthew: It has to be 18.675, which is “Theory of Probability.” This class focuses on building a rigorous framework to understand what it actually means to define a random variable, and what it means to take the expected value of something, and finishes by talking about interesting topics such as martingales and Brownian motion. I’ve always been more interested in learning about more discrete forms of math (e.g. probability, theoretical computer science) and abstract algebra (group theory, field theory, etc.), but I think this class really made me appreciate why the fields of analysis and measure theory are important. Before I took this class, I had some vague operational notion of what probability meant (“if I flip this coin a bunch of times, I expect it to land on heads around half the time”), and this class taught me how to define probability rigorously. It also doesn’t hurt that a lot of concepts from this class showed up in my research this semester.

Misheel: I loved 18.786,12 Number Theory II which was taught by my advisor, and it was even heavily13 Fatima: Misheel loves algebra! algebra-oriented than other iterations of the class.

Another class I really liked is 18.102,14 Functional Analysis which was taught in a really fun way by Professor Melrose. I loved the fact that it was the intersection of algebra and analysis (or just the fact that it has really interesting algebra lol).

What is your least favorite math class you have taken at MIT?

Z: I hate to say it but 18.404,15 Fatima: Theory of Computation SOBBBB but ONLY because I discovered I was personally not really interested in the content after all; the class itself is very popular (for good reason) and organized very carefully. I’m still glad I took it, though. Fatima: I would probably say Real Analysis for me, but it was a little my fault. I took it too early without enough background with proofs. My thoughts on analysis have since changed (yay functional analysis!)

Any thoughts on how to best work with people with many differing backgrounds?

Jonathan: People with different backgrounds often have different names for the same thing – try to learn as many of these names as possible.

Z: (Unfortunately I don’t have too much experience with this.) My only experience with something like this at MIT came from taking 18.821.17 Project Laboratory. one of the communication intensive offerings in the math department where you choose some different problems over the semester and work on them with your teammates. at least that is what my former advisor's edition was supposed to look like. My teammates were extremely friendly, accommodating, and hardworking, and we were naturally interested in different aspects of our projects so there were no conflicts :P.

Tristan: When working with people from different backgrounds, it’s useful to get an idea of what type of things different people work better at. For example, mathematicians in different areas might bring different techniques to the table, and some might be able to be applied to various parts of the problem.

Did you ever feel at any point that you didn’t belong in the department? Did that ever go away? Why/why not?

Z: I experienced this freshman fall and was very intimidated by many of my classmates. For me the feeling did disappear eventually, because somehow I gradually became friends with some of those people, and found they were very friendly and willing to help me with math (and of course, they were normal people outside of math). I think I was pretty lucky, though.

Matthew: To some extent I have always felt this a little–I think that I really enjoy studying (some areas of) math, but at the same time I don’t feel like I really committed to taking many math classes–I enjoy taking classes from a wide range of courses (CS, biology, physics, math). I think the math department also tends to consist of a lot of students who are double-majoring in other fields because the requirements to be a Course 18 are relatively not as strict to satisfy, and I do have many friends (hi Fatima!!) who are Course 18s, so in that sense I do feel like I belong.

Zimi: Fun fact: when I first declared my major I thought I was going to be in the EECS18 electrical engineering and computer science department/more closely associated with EECS for some reason. So I was very surprised when I received an email addressed to “new math majors.” Before starting college I didn’t really think about studying math, so it took me a while to get used to calling myself a math major.

Sometimes I do feel like I don’t quite belong here. This feeling was stronger in the first two years in college—I didn’t do a lot of math in high school, so seeing people who were taking advanced classes and knew each other from high school math competitions was a bit fear-inducing. But there are a lot of other people in the department without too much background prior to coming to college. As I said before, I still wouldn’t consider myself someone who “does math,” but over time, I found myself feeling that I belong here more than I did in the beginning.

What are some skills you have found to be most useful as a mathematician?

Jonathan: This is not exactly a skill, but I’ve found it helpful to approach problems from a computational or a geometric/visualization perspective. But probably the most useful skill is figuring out how to interest other people in solving your problems!

Z: Passion and interest in the material is probably the most important for me, as it makes it much easier to find motivation to work. It’s also important to be willing/able to think very carefully and thoroughly, especially when first working with proofs. I also found it helpful to be unashamed to say things like “I can’t because I have to pset.”

Carl: Being able to look at a problem from different perspectives. So much of research mathematics is staring at a very hard problem and having no idea how to attack it. Solutions, when they come at all, often come from unexpected sources. (I think this can be explained by conditional probability — if a problem is interesting, conditioning on it not having been solved, progress is more likely to come from an unexpected idea than a standard one.)

Tristan: I think that being able to communicate effectively is extremely important to doing math. It’s easier to collaborate that way, to convince others of the correctness of your argument, and to teach math.

Matthew: I think being able to struggle on problems for long periods of time without giving up even when becoming frustrated is important–often problems can take a while, and it can feel like you’re getting nowhere. This seems to be true of many difficult things though, so I think this is just generally useful at MIT.

Misheel: Talking to people! Maybe this is specific to me, but I’m kind of bad at actually doing things,19 Fatima: completely incorrect! but a lot better at coming up with random ideas (most of them useless ones). So having someone to bounce and filter ideas and essentially “storing ideas externally” makes working on problems and stuff much better and way more fun. Also talking to people is a useful skill in general and later on in life if you want to collaborate with people. I think being able to write well is a really powerful skill as well.

What is the most rewarding part of your job/doing math?

Z: Kind of a lame answer, but mostly just the math itself. The friends are cool too but occasionally mid.

Carl: Going from being utterly mystified by something to understanding it, even just a little.

Tristan: The most rewarding part about doing math is getting to talk to other people about it. When I’m done with a cool problem or just finished learning something new, I always look forward to talking with peers about it, understanding the subject better and sharing our excitement over math.

Misheel: The feeling of learning cool things and making cool concepts connect in cool ways. The exploring aspect. Understanding someone explain super high level math and recognizing a few words you learned recently and then using those words in a sentence where it kinda makes sense is very cool.

Anything else you would like to add!

Z: Do math! math is good

Thanks to Paige B. ’24 for helping with the questions and to everyone else for answering the questions :))!

1. Fatima: I took Seminar in Topology last semester with Jonathan!
2. know an xkcd for every situation
3. Fatima: I took Algebra I with Professor Cohn last fall!
4. Fatima: This answers a bunch of the questions that follow, but I didn't want to break it up because so true!!
5. depth first search
6. Misheel: it is important to contrast the need for dfs in research with the danger of dfs in learning because if you dfs in learning, you really can't stop and you'll have very specific knowledge about a few fields but not a lot of foundation. so for studying bfs (breadth first search) is better or something.
7. directed reading program. during IAP, math majors can sign up to read some math topic of their choice with a mentor.
8. Fatima: For some reason, I did not send this question to any of the students. I regret that! People probably had thoughts from the other side and a lot of students also TA classes anyway.
9. summer program in undergraduate research
10. Elliptic Curves
11. Number Theory I
12. Number Theory II
13. Fatima: Misheel loves algebra!
14. Functional Analysis
15. Fatima: Theory of Computation SOBBBB
16. Fatima: I would probably say Real Analysis for me, but it was a little my fault. I took it too early without enough background with proofs. My thoughts on analysis have since changed (yay functional analysis!)
17. Project Laboratory. one of the communication intensive offerings in the math department where you choose some different problems over the semester and work on them with your teammates. at least that is what my former advisor's edition was supposed to look like.
18. electrical engineering and computer science
19. Fatima: completely incorrect!