How to discover other fields of mathematics?

Question

I am currently an undergraduate and thinking about applying to graduate school for math. The problem is that I don't know what field I want to go. Taking graduate classes even more confuse me because the more I learn the less I know what specifically I want to do. My question is to where to find an information about different fields of mathematics? Maybe you can recommend me some good journals about math with overview of top areas of math or popular fields. I already spoke with my professors, asked graduate students about their history but I think that my knowledge about math in a broad sense grows slower than I want it.

Maybe there is some good website with people chatting about different fields of research? What about conferences: is there a conference available for undergrad about top-trands in mathematics?

All sources and all answers are welcome.

I am mostly interested in pure math, but I also like applied math.

Answer 1

If you are in the U.S., you could try attending AMS Sectional Meetings. These sorts of conferences happen often, and they feature talks in a huge number of different topics in current mathematics. Otherwise, I'm sure that in your locale, there are probably analogous conferences.

Also, you could sign up for email notifications from arXiv mathematics. You will get daily notifications of new papers in your choice of mathematical subfields.

Answer 2

It is hard to get a good overview of all of mathematics. The best really is to take a good broad variety of classes. This you usually do (to some extent) the first couple of years in graduate school. Here you will learn the basic language of the main areas of mathematics. Here you will be get to meet various professors.

For now, then, I wouldn't worry too much about what field you want to go into. If you don't already have an interest in an area, then it is hard to say much that can push you in a certain direction. You might be thinking that you want to find an area so that you can go to a school that specializes in that area. Again, don't worry too much about it.

One thing that you probably can decide on now is whether you want to do applied mathematics or pure mathematics. Some departments are very strong in one and not the other. It sounds like you are more into pure mathematics.

If you are wanting to do a Ph.D., then I would also point out the importance of having a good advisor. Some are highly motivated and work hard on their own, but others need good guidance. If you take an area like algebra, then you have to remember that this area is huge. There are many subareas of algebra and what you will end up doing research in will (usually) depend heavily on what your advisor does. I say this to point out that the choice of advisor for many is almost as important as picking a good area. (Again, others already know what they want to do and just need an environment where they can work in peace.)

About conferences. Ask your university/department about funding to go to MathFest or the Joint Meetings. Do you have a Mathclub at your school that might be able to help you? Next week happens to be MathFest and here many undergraduate students will present work they have done over the summer (or past semesters). These talks are a great introduction to various areas. The level is low enough that an advanced undergraduate student should be able to learn something. There are also student activities at the Join Meetings in January. Talk to your department about this now!

Another great suggestion is to attend an REU (Reseach Experience for Undergraduates). These are usually summer programs where you get to go into depth on a specific topic. Usually there isn't much of a requirement for background). Google this and start to talk to your department about how to apply.

Lastly, try to talk to as many people as possible about what they do. It sounds like you have already been doing this, but ask each professor in your department what they work on. Talk to graduate students about what they do.

Answer 3

As you finish your undergraduate studies, you should have had at least some introduction or passing acquaintance to the following subjects:

Algebra (the abstract kind)

Discrete mathematics (maybe some number theory)

Linear algebra

Real/complex analysis (post-calculus)

Topology

Differential equations

Statistics/probability

Of these, which did you enjoy the most? All of these areas will have "niches" in which investigations are still being made. Pick an area that speaks to you, and "go deep". Finding the right journals, or the right texts, is a lot easier when your search is focused. Savor this moment, you have a brief window of true freedom to choose. Two or three years in, and your choices will be a lot more constrained, because there is too much math out there to master it all.

I understand, I do, because with the freedom to choose, there is also bewilderment. What would be best? For my money, I say trust your gut, and I believe it's far better to be happy than trendy. If you are too driven on trying to find the most "fruitful" area, there's a real risk of it turning first to drudgery, then to feeling enslaved. You'll do far better, in the long run, having a passion for what you study.

As the old saying goes: "if you do what you love for a living, you never have to work a day in your life."