Incoming freshman confused about how research works here
hey, I'm an incoming freshman and I've been trying to figure out how undergrad research actually works at MIT, specifically for math and theoretical CS. I'd love any honest advice from people who've navigated this
for context on where I'm coming from mathematically: I've worked through Axler's Linear Algebra Done Right, Rudin's Principles of Mathematical Analysis, Munkres' Topology, Lee's Introduction to Smooth Manifolds, Folland's Real Analysis, Conway's A Course in Functional Analysis, Aluffi's Algebra: Chapter 0, as well as some other texts, lecture notes, etc. i read most of these books front to back. am currently working through evans's pdes and a book on operator algebras. the areas that excite me so far are functional analysis, operator algebra, functional calculus, C*-algebras, pdes, and stuff about manifolds (i prefer working with manifolds analytically over algebraically), though I'm still figuring out what I actually want to spend years on. on the CS side, I've built a compiler, a networking library, and a game engine, mostly in Rust and C, but I don't find typical course 6 material all that interesting.
but i have a few things I'm generally confused about.
How do you actually find people to work with? ik the nominal answer is "email professors," but that feels impossibly vague. do you cold-email? do you need to take their class first? are there informal channels, reading groups, or seminars where you can just show up as a freshman? i have no idea how the hell this works, and I don't want to embarrass myself by approaching this wrong.
How do you connect with professors? one thing i'm also really lost on is how to actually form a relationship with a professor outside of just emailing them cold. like, is it normal to go to office hours just to talk about the material even if you don't have a specific question? i feel like showing up and saying "hey i just think this is cool" would be weird, but i also don't know how else you build any kind of rapport before asking someone to work with you. do people just attend seminars and hope someone talks to them, or is there a more natural way this happens? i guess what i'm asking is: what does the path from "i have no relationship with this person" to "they're willing to advise me on something" actually look like in practice, because from the outside it feels totally opaque. on a side note, i also really dont understand the idea of office hours. i know some classes you have to fill out appointments to have office hours with profs, and that just feels really odd.. what am i even meant to talk about? why would a prof ever want to talk to me? lmao
How do you figure out what you're actually interested in? i've read a lot, but reading textbooks isn't really the same as knowing what kind of problems you want to spend a year stuck on. is there a way to sample research areas without committing? do people do short rotations, or is the expectation that you pick a direction and go deep immediately?
Math vs CS? my background straddles both, but I'd probably be course 18, pure. I've heard that theoretical CS (things like complexity, information theory, or TCS adjacent to operator algebras) can have a different culture and a different relationship to "publications" and "results" than pure math. i'm not planning to take course 6 classes just to be in CSAIL, but I wonder if there's some version of theory research that would fit my interests and still be housed on the math side of things. and, very expressly, i loathe things like combinatorics and elementary number theory, and tbh this is basically all i see in cs research.
Why would a professor want to work with me? this is honestly what holds me back most. I can read at a decent level and I can implement things, but I have no research output, no publications, and no contest credentials. there are people here who have done all of that. what actually makes a professor say yes to working with an undergrad who is just enthusiastic and reasonably prepared?
any honest takes would mean a lot. ty all