<p>I'm a second-year math undergrad at Northwestern. I have a few questions, thanks for helping!</p>
<p>1) I plan on taking the Math GRE subject test during my senior year. By that point I will have taken real analysis, algebra, topology, number theory, probability, principles of higher math (set theory, logic, cardinality, functions), and upper-level linear algebra. We don't really offer a standard complex analysis course for undergrads, so I'm planning on independent studying it with a professor during my senior year. Is complex analysis on the GRE and how much is necessary to know? What about DiffEq? My advisor told me not to take it...will this hurt me?</p>
<p>2) What's the best way to get involved with research? I plan on doing an honors thesis during my senior year, but I want to get involved sooner. I'm hoping to do something over the summer at my home in Los Angeles, but people at Caltech seem uninterested in working with non-Caltech students. I will be applying for a program at UCLA, but it definitely seems to have an applied edge to it. Is it possible to work on my own on pure math research? Where can I find unsolved problems that are somewhere near my level of abilities? I won't be taking analysis, algebra and topology until next year...but I do know how to write proofs and I have no problem with studying math on my own time.</p>
<p>Okay yeah I'm sounding like an over worried, nerdy undergrad loser.....but I hope you guys can help. Thanks so much!</p>
<p>There will be just one or two questions relating to complex analysis and diff eqs on the GRE. You can easily memorize the couple of formulas you would need to know about complex analysis to nail the complex analysis questions (Cauchy-Riemann equations, Cauchy's integral formula). I know someone who got something like the 95th percentile on the math subject GRE without having taken either of those classes, for what it's worth. You'll be a lot better off reviewing craptons of calculus and practicing hard calculus problems when the time comes than trying to fill in small knowledge gaps that would only apply to one or two questions.</p>
<p>No no no! As a techer, I can tell you that every year there are many non-Caltech students in our SURF program. What you need to be aware of is that the coursework you've described does not prepare you for pure math research here - in fact, very very few of our own pure mathematics students are able to find SURFs. (I'm talking Putnam fellows!) Instead, many math majors do SURFs in applied mathematics, or computer science, or even things like physics. You have much better odds of finding a project at your current level in those fields, whether at UCLA or at Caltech. Think of it this way: what you do this summer won't translate into your thesis project anyway, since you won't be working at Northwestern, so why not do research a little bit beyond your main interests?</p>
<p>Thanks snowcapk, your post made me go back and look at some of the opportunities Caltech lists in other fields. There are some JPL Computer Science projects that I think I could do if I take a programming class next quarter. You're right, at this point I need any type of research experience I can get because I'm so inexperienced.</p>
<p>Still, when the time comes and I'm ready to do some pure math research, is there a particular resource for finding unsolved problems? I mean we all know the twin prime conjecture and the Riemann hypothesis and such, but surely there are tons of other things to work on that haven't eluded mathematicians for centuries.</p>
<p>You need to know your calculus cold for the GRE math subject test. Linear and abstract algebra are much more important for the test than diffeq, but you should know the basics of diffeq.</p>