Martin Raussen and Christian Skau interview Prof. S R S Varadhan on his winning the 2007 Abel Prize; the interview is available in the February 2008 issue Notices of AMS (pdf). Prof. Varadhan has several interesting things to tell us about himself and his work: here is for example a couple of his answers on how and why he does mathematics:
I usually look at Mathematics in the following way: There is a specific problem that needs to be solved. The problem is a mathematical problem, but the origin of the problem could be physics, statistics, or could be another application, an economic application perhaps. But the model is there, and it is clear what mathematical problem you have to solve. But of course, if the problem came from physics or some application, there is an intuition that helps you to reason what the possible answer could be. The challenge is how to translate this intuition into rigorous mathematics. That requires tools, and sometimes the tools may not be around and you may have to invent these tools, and that is where the challenge and excitement of doing mathematics is, as far as I am concerned. That is the reason why I have been doing it.
What happens is, when you have a problem to solve, you have some idea of how to approach it. You try to work it out, and if you can solve it the way you thought you could, it is done, it is not interesting. You have done it, but it does not give you the thrill at all. On the other hand, if it is a problem in which everything falls into place, except for one thing you cannot do; if only you could do that one thing, then you would have the whole building, but this foundation is missing. So you struggle and struggle with it, sometimes for months, sometimes for years, and sometimes for a lifetime! And eventually, suddenly one day you see how to fix that piece. And then the whole structure is complete. That is the missing piece. Then that is a real revelation, and you enjoy a satisfaction which you cannot describe.
How long does the euphoria last when you have this experience?
It lasts until you write it up and submit it for publication.
A nice interview; take a look!