What are some good ways to figure out one's interests in mathematics, before applying to PhD programs?

Question

Are there mathematics seminars to attend in the summer, or do they typically only occur during the academic year? I am referring to American universities.

An idea I had last night was to YouTube some math seminars to see what talks I can watch quickly - talks are about an hour long. I found a good PDEs talk from a Princeton seminar. I hope to find more talks that are available online.

Is this not advisable? These are talks given by experts, and as a lowly master's student, I naturally do not understand about 90% of the mathematics being talked about and written on the chalkboard. I am mainly just trying to get some sense of purpose of, say, studying PDEs, its difficulties, etc.

I am also making progress on a master's thesis, but my gut feeling is that this problem is too narrowly focused and in the end may not help me figure out better my research interests in a PhD program.

What else could I do?

Answer 1

Are there mathematics seminars to attend in the summer, or do they typically only occur during the academic year?

I think none of the answers thus far actually addressed this. At most US universities, seminars don't happen, at least not regularly, during the summer. One reason is many faculty will be traveling. However, sometimes some people, possibly grad students, will organize an informal seminar to read through some books or papers. If other people aren't doing it, you can ask around to see if people are interested and organize it yourself.

What else could I do?

In place of seminars in the summer, there are usually lots of conferences and workshops as well as summer schools specifically geared towards young people. Look around for what's available (you can try asking some professors if they have suggestions) and go to what you can. Many of these have funding for students, but unfortunately most of the funding may have already been given away by now.

Edit: By the way, it's not expected that you know what you want to research when you enter PhD programs (in math in the US, say). Many people who start aren't even sure if research is for them. Most programs are designed so that you can use the first couple of years figuring out what you want to do. (And even later, research interests may change---I may be doing something completely different in a year or two from what I'm doing now.)

Answer 2

Try reading the Princeton Companion to Mathematics. The editor insisted that the articles were actually comprehensible!

The fundamental problem in selecting research areas is that how an area looks from the inside once you actually know it, is very different to how looks from the outside. This is particularly the case when people tell you about the applications or nice pictures -- these really tell you nothing about what doing research in the area is like.

What sort of problems do you like to solve? What sort of techniques do you like to choose? can you see yourself fiddling with an integral all day? do you like abstract nonsense or prefer things more concrete?

Answer 3

In pure math, research seminar talks are notoriously specialized, and not always well done. To give you an idea, I'm a postdoc doing research in PDE, and I've gone to many a PDE seminar that left me in the dark after the first slide, because the subject matter was too far from my own corner of PDE, or because the speaker did a bad job of introducing his work to non-specialists, or a combination of the two. At the very least, as someone who's "looking to get into PDE", you're very far from being in the target audience of these seminars.

You're better off looking at graduate-level textbooks in various subjects. This isn't really a window into the research world, but it's likely the best thing you can do right now to get a sense of what you like. For PDE, Evans's book is a standard choice.

Answer 4

I think it's entirely reasonable for you to be at the point you are "but still wondering"... Indeed, of necessity, most master's theses in math are pretty narrow or specialized or not-entirely-"research", and that's fine, but that experience is different from the "more genuine" version that should happen in a PhD.

Not clear that you'll find YouTube videos of everything you'd want to see, but it can't hurt anything, unless you simply get discouraged.

Also, I'd bet that you'll find more instances of advanced graduate course notes by major contributors/experts than you'll find videos of their lectures. This would allow you to examine the stuff more calmly, and at your own pace. Maybe more coherent than lectures, too.

It is certainly difficult to make choices with insufficient information, but/and there's sorta the "meta" question of "which information to try to acquire?", which makes it worse. Thus, the perhaps-better coherence of advanced grad course lecture notes might be both grounded in more careful (less time-constrained) discussion, but still try be aiming forward and touching contemporary issues.

Answer 5

If you want to figure out if a PhD in a subject is right for you, get involved in research. PhDs are about doing research, not taking classes or attending lectures. A PhD program is an apprenticeship for becoming a researcher and the only way you'll know if you like it is by gaining experience with research.

Answer 6

I have a Ph. D. in electrical engineering. Most people would think my dissertation was full of math -- applied mathematics (linear algebra). Most math professors [at least my U.S. university] were mostly interested in infinite dimensional work and the like, so a distinction should be made between PhD mathematics and mathematics in general.