Over the past few years, several students at various levels of studies have emailed me with some questions after reading my "A day in the life..." piece. So for similarly-minded inquisitive readers (if there still are any), I decided to post some of the questions I've gotten, along with my answers, with some very mild editing. If you have other questions, feel free to ask, though I may take a while to respond depending on how busy I am.

Q: I read your article "a day in the life of a mathematician" and I thought it was very interesting. Have you developed any new ideas on how a movie star and mathematician are similar?

A: Thanks, I'm glad you like it.

Ha ha. That was mostly tongue-in-cheek, but there was something behind it. Roughly what I meant was there seems to be a sort of mystique about mathematicians for most people, like we are in a different world.

There are some other similarities also (a lot of traveling, it's not a 9-to-5 job, you get to choose what you want to work on and how).

Q: Was it easy to become a mathematician?

A: It was easy in the sense that it was natural---I just did what I enjoyed, which was studying mathematics. However, I needed to work a lot at it, and understanding mathematics is often a (hopefully rewarding) struggle.

Q: Were there any tests that you had to take? How many years of college did you have to go through?

A: 4 years of college, then 5 years of graduate school. You have the normal exams in college, and exams the first couple years of graduate school. (Different grad schools are different, but this is pretty common in the US.) Then it's just research, and maybe a few more courses. There's no specific exam to be a certified mathematician (unlike the bar exam in law). You just write up your research to get a Ph.D, and get it approved by some professors.

Q: Do you enjoy your job?

A: Yes, I do, though sometimes I need a break. Research takes a lot of devotion, and I don't think you can really do it if you don't enjoy it. Apart from the subject itself, one thing I enjoy about being a professor is the freedom I have in my job (schedule, choice of projects, etc.).

Q: I infer from your article that you specialize in number theory. You've already detailed what you do, but how did you get interested in it? Where did it start?

A: I initially got interested in math in middle school when some friends convinced me do to a math competition, MathCounts, with them. We spent lots of time doing practice problems, and I really liked both learning new ideas and problem solving.

While I always enjoyed math, I went through various phases of thinking I wanted to do computer animation, physics or computer science in high school and college.

As for number theory specifically, I can't pinpoint my interest, but is has a mystique about it with a rich history and long-standing conjectures, with the solution of Fermat's Last Theorem, announced with great fanfare when I was starting college.

My interests were probably shaped by my experience in problem solving contests (which tend to be heavy on number theory + combinatorics), an REU (a summer research program where I studied algebraic graph theory), as well as my interest in cryptography (which uses a fair bit of number theory) and spending a summer working at the NSA. My professors and the course offerings played a big role also.

I also really liked combinatorics and geometry/topology (and still do, and still use both). When I was applying to grad school, I thought I wanted to do either (algebraic) number theory or (algebraic) geometry. After learning more about both, I just felt that number theory was more my style. (Though now, I'm also thinking about projects in other areas, such as hyperbolic geometry and combintorial optimization.)

Q: You went to incredibly prestigious institutions, but for the average student, is it possible to obtain research/academia positions?

A: I actually got a degree from the University of Maryland Baltimore County for my undergrad, which is not at all a prestigious institution. (My advisor at Caltech told me later they had a long debate about whether to admit me or not, because they knew nothing of this school.) What's important is that you learn serious math, and then do good research. Of course, being at more prestigious schools often makes this easier by being surrounded by a lot more good mathematics.

Q: If I go to U***, how can I improve my chances of getting into grad school at Caltech, MIT, etc.?

A: If you go to U*** [not a top-ranked school] and want to go to a top grad school, it helps to have some grad coursework already. (Most people entering top Math PhD programs have some grad coursework.) Another thing that's helpful is to do summer programs like REUs. That way you can get a letter from someone from the REU who can compare you (hopefully favorably) with lots of other students from different schools. You should of course try to get A's in all your major courses and get to know your professors so they can write you good recommendations.

Q: How were your job prospects out of college?

A: I didn't apply, but several of my friends with a math degree placed into research/industry jobs with places like the NSA or Northup Grummond. (Those are the kinds of places around Baltimore.) Other places like financial institutions will hire you simply because they think you're smart, and just train you on the jobs (I know plenty of math Ph.D. who got jobs at banks, insurance companies, and investment firms---and these places will also hire people with just a BS or MS, as well as physics Ph.D.s---it's just most of the people I know have math Ph.D.s).

Q: What are you doing now, and do you feel satisfied with your work?

A: I'm still working on some similar projects in number theory as when I was a postdoc, though I'm also branching out some into geometry and combinatorial optimization. In some sense I feel satisfied with my work, but I don't want to just keep doing the same thing if you know what I mean. That's why I like to work on different problems and learn new things.

Q: Can you give me a brief overview of the several mathematical fields that are typically presented in grad school?

A: PURE MATH typically is divided into 3 main branches (though these are not so separate)

Analysis - This starts off as "calculus with proofs," and is an advanced study of functions, derivatives and integrals.

Algebra - This is the study of mathematical structures (e.g., number systems such as integers, rationals, reals and complexes, or "groups" of symmetries, e.g., symmetries of a cube or tetrahedron). As such, typically number theory falls under this branch, as does linear algebra.

Topology/Geometry - You can imagine what geometry is about--the shape of typically rigid objects; one can think of topology as a more general study of flexible objects.

Other main areas are

DISCRETE MATH - This includes things like combinatorics (counting problems, like given 5 men and 5 women, how many ways can you pair them up so each man is paired with a woman) and graph theory. This is most closely related to computer science and number theory.

APPLIED MATH - I know less about this, and different people have different definitions about what "applied" means, but in our department it means doing things like finding numerical (approximate) solutions to equations (computational aspects, rather than formal proofs).

PROBABILITY AND STATISTICS - Probability is the study of chance (e.g., poker odds and what not), and Statistics is about how to infer general information from some sample data (e.g., polling public opinions).

Q: As a mathematics grad student, what can I expect? In terms of required classes versus elective classes? I understand that in graduate school one specializes in a certain area of math, and I think that that would fall under an elective. But what else is there?

A: It depends on the school, but the following is pretty typical (for pure math). There are 3 standard first year classes: algebra, analysis and geometry/topology. (Or some places you can choose 2 of these 3 and do something like combinatorics/applied math or probability/statistics for the 3rd first year course.)

Then you will probably have qualifying exams on these topics (written and/or oral). The next year you have elective advanced topics depending on your interests (an advisor will typically help you choose them), and there may be exams on them afterwards or not. If you're just doing a master's you will stop here.

Otherwise from your 3rd year on, you should do individual reading/research with your (potential) PhD advisor. Though you can (and are sometimes so advised) still take other classes to either help you for your research area or just broaden your mathematical horizons (but probably not more than 1 course per term at this stage). Depending on the school, you should typically finish in 5 or 6 years. There are also typically weekly seminars to help broaden your perspectives and various opportunities like summer schools or 1 week workshops or special semesters at research institutes you can attend, as well as normal conferences. (There is generally funding, and I enjoyed several of these opportunities. It's also a good way to meet lots of people, which is helpful particularly when you're looking for a job.)

Kimball Martin [main] [math] [writings]
kimball.martin@ou.edu
Wed Aug 7 16:42:56 CDT 2013 [updated Fri Nov 13 15:29:39 CST 2015]