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First of all, I don't know if this is the right place to ask about careers. If not, please direct me somewhere I can get more help.

So, I'm in my first year of university, majoring in applied mathematics. I went to IMO twice in high school (1 bronze and 1 silver), so I have a very strong foundation in mathematics. However I really have no idea what I'm gonna do in the future. There are a lot of options, but none seem to suit me well. I expect people here with actual experience on the related field could share some information.

There are two paths which I'm particularly interested in: financial stuffs and research. I've heard a lot about careers in financial maths: quants, actuaries, financial engineers, risk managers, etc. I don't exactly know what these guys do, but I heard it pays well. Can anyone explain briefly how these jobs differ one another?

Secondly, the research path. I was always studying math just for fun. But things started to get boring in university. Tonnes of new definitions, routine formal proving, and less problem solving. I don't know how things will go, though, because I'm still in first year. So, I want to know, is math gonna be even more boring in the future? I don't actually know anything about researchers in the first place. I know they prove theorems and publish papers. But what do they do on a daily basis?

I would really glad if someone actually has experience on both areas and share it with me. My future still looks dark for me, I hope someone here can help. Thanks in advance.

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There's a big transition from high school math to university math. The very first barrier is language, typically consists of "definitions, formal proving" etc, which could be boring but you need to get through them once. I think it's better for you to decide what math is like to you after you take more classes and learn some more math with content. –  Soarer Sep 16 '11 at 3:24
Thanks. But I also need to decide my career path now so I can take the appropriate classes. That's why I'm asking here in the first place. –  ben Sep 16 '11 at 3:31
If you enjoyed IMO-type mathematics, you may find the early part of university mathematics not to your taste. But have patience, and you may get to things you may find more enjoyable like, say, combinatorics or graph theory. Note there are also mathematicians who don't like problem-solving as much: read this paper of Gowers about the two cultures of mathematics. –  Zhen Lin Sep 16 '11 at 3:58
You might ask around as to whether there's anyone on the math faculty who would enjoy doing a little work on interesting topics with a student who has had some success in high school competitions. Someone who could point you to some books, set a few problems for you to solve, meet with you from time to time. This may not help you choose a career, but it may make your undergraduate days less boring. –  Gerry Myerson Sep 16 '11 at 5:40
Your choice of courses should not depend on what you think you might want to do after your degree. In fact, you should not even limit yourself to applied mathematics at this stage, since this is a very uninformed decision. Go to as many different courses as you possibly can, including pure and applied. As an undergraduate, I went to pretty much all the mathematics courses my department would offer for at least the first two and a half years, with varying degrees of time investment. See also my comment under Igor Pak's answer. –  Alex B. Oct 9 '11 at 3:41

3 Answers 3

What do researchers do on a daily basis? They teach classes (at all sorts of levels, in principle, and depending a bit on what kind of institution they are at or what country they are in, from beginning undergraduate to advanced graduate level), they advise thesis students (some mixture of undergraduate thesis students, masters thesis students, and Ph.D. students), attend research seminars (where speakers report on their latest research), and do their own research.

"Doing their own research" ranges from [here I am speaking about pure mathematical research, which is where my own experience lies]: studying other books and papers to get inspiration, learn new techniques, or look for a specific tool which could help them with a problem they are confronting; working out their own new ideas, perhaps just sitting by themselves with pen and paper, or perhaps in discussions with a collaborator; writing up papers explaining their new theorems; or many other possibilities (e.g. writing computer code to investigate certain situations or phenomena).

As for what the new results are that people prove, here one can say that in most fields of mathematics, there is a broadly understood and acknowledged (to the workers in that field) "frontier" of cutting edge knowledge, and some sense of what things that are not currently understood are important and should be investigated. People then try to push their way over that frontier into these unknown areas in some promising direction, with the hope that they will push the frontier a little way further. (Often there are very famous open problems in a field, such as the Clay Millenium Prize problems, but most people are not working directly on such difficult problems --- those problems are difficult and celebrated for a reason. Rather, people work typically work on more tractable, and incremental, problems, with the hope --- which seems borne out by the overall experience of many generations of mathematicians --- that consistent incremental progress in pushing the frontiers of understanding forward adds up over time to significant advancement of our knowledge.)

It is hard to be more specific about the kinds of theorems that people prove, because they are incredibly varied in subject area (the umbrella of "pure mathematics" is a very large one), and it is also often hard to explain cutting edge mathematical research to someone who doesn't have some background knowledge in the subject area. There are some famous results that you have probably heard of (e.g. Wiles's proof of Fermat's Last Theorem or Perelman's proof of the Poincare conjecture). As I noted above, most people's research results are typically on a smaller scale than these, but it's not unreasonable to take them as models, and just to imagine that people are proving somewhat similar things (albeit a little more specialized and less celebrated): perhaps proving properties of certain classes of Diophantine equations, or of certain kinds of manifolds.

If you go to the Arxiv, and click on the link for new mathematics papers, you will see lots of new mathematical research papers, on a huge array of subjects. The terminology will probably all look incomprehensible, but perhaps it won't look as boring as your current mathematics syllabus. (Certainly, people who go on to pursue a research career don't find mathematics boring; it is normally the opposite --- they do it because they find it too interesting to do anything else.)

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I will answer the second part, which is an important issue with a very clear answer. Essentially, there is no way you can learn what mathematicians do by taking undergraduate math classes or even by having some kind of undergraduate research experience (you might want to still do both, but for other reasons). The traditional answer is "start attending research seminars and colloquia". If you already know what kind of math you like in broad terms (like algebra vs. analysis vs. topology etc.) - choose that one. If not, try attending several and see where your interests lie. This is a precarious route, as it really depends on the speakers, topics of the day, accessibility of the material, etc. But in my opinion this is probably the best way to get excited about math research - see other people interested in the subjects, the types of problems that they work on, etc. Even if after an hour lecture you will learn a few new mathematical notions that you seem to like, the time is well spent.

If your university does not have research seminars, there is a less active way to do this, on the internet. Various universities and institutes (like MSRI, Banff) have large collections of videos of research talks, sometimes absolutely spectacular. Several blogs (e.g. Terry Tao and Gil Kalai) serve as interactive talks with discussions on important new math developments, among other things. Finally, plenty of excellent math books are free online (also my own), and some of them are great fun - try to browse through a bunch of them, see what catches your interest. In short, there is plenty of opportunities to learn if there is a will and an interest. Good luck!

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There really isn't much point for a first year undergraduate in going to research seminars. I think that the thing to do is be patient and try to find among the stuff you learn things that interest you. For what it's worth, I only found out that I wanted to become a mathematician when I attended a course on Galois theory in my 3rd year. That's when I knew that I wanted to do mathematics for the rest of my life and also, roughly, what kind of mathematics I wanted to do. –  Alex B. Oct 9 '11 at 3:40
Russian tradition is different. Like many others, I started attending seminars while a freshman. As Gindikin reports on the Gelfand Seminar, "no less half of the participants were undergraduates".link –  Igor Pak Oct 9 '11 at 5:23
Assuming ben is at an American university, I think this advice is unrealistic. My senior year in undergrad (Harvard) I decided to go to any seminar where I knew the words in the abstract. This was still only about a quarter of the talks available, and half the time I still got nothing out of them. Unfortunately, most seminar talks are aimed way above the level of an undergrad (often, way above the level of their audience!). –  David Speyer Oct 20 '11 at 13:30
Right, that matches my experience as well. When I attended Harvard seminars as a grad students, I also got very little (Basic/trivial notions don't count). However, the seminars I attended (or ran) at other places were markedly different. This has to do with the effort the organizers make to have talks at least partially understandable by general audience. –  Igor Pak Oct 25 '11 at 0:39

I think the best advice I can give to a prospective student is maximise your future entropy, that is take a wide selection of courses initially from all sections of math- pure,applied,statistics, computer science/programming, cryptography, that will leave your options open for future specialisation when your find your niche what you really want to do, also research shows that people who are motivated strictly by financial gain or the prestige of a degree often are unsatisfied with their degree and have higher rates of failure and dropping out. Also try, if you can, helping others to learn who are struggling, it is very rewarding and often the best way to really master a subject.

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