Mathematics is such a huge field, and I am just wondering how does a mathematician keep focused? For example to learn Stochastic Calculus you need some Analysis, Measure Theory, Probability, Set Theory, and all of those topics usually need some other branch of mathematics. So, how do you know when to say enough with Measure Theory? or enough with .....? And stay on topic, say learning Stochastic Calculus?

Also, I find that there is usually one course that just captivates me (Real Analysis) more than another and I often find myself neglecting the others (DFQ, Probability).

Since, many here have already finished graduate school, I figured perhaps they might have some good advice for maintaining a healthy and productive balance.

  • $\begingroup$ I think that sudden run of (+1)s means "that's a good question" $\endgroup$ – J. W. Perry Oct 10 '13 at 2:39
  • $\begingroup$ @J.W.Perry Yeah I am not sure what happened? I all the sudden got a bunch of points on MathOverflow $\endgroup$ – JimmyJackson Oct 10 '13 at 2:40
  • $\begingroup$ @J.W.Perry So do you have any advice? $\endgroup$ – JimmyJackson Oct 10 '13 at 2:46

How terrific that at least one course bowls you over.

The reason you keep discovering you need tools from one "branch" of mathematics to proceed in another "branch" is that there really are no branches. There is just mathematics and it is all thoroughly intertwined. We split things up so that we can teach them, or focus on them, but any split we make is at least partly artificial. So just about the time we think we've learned geometry, someone comes along and solves our hardest problem with algebra. Or we're working away at number theory, and someone brings our attention to elliptic functions.

The mathematicians who accomplish the most are the ones who know the most, and the amount, depth and breadth of what they know is amazing. So if you want to be really good, the answer is you just keep on learning, as much as you can in as many areas as you can.

However, you must also survive graduate school.

To deal with the courses you are less interested in I suggest (after years of experience doing things I don't like) that you attend to what needs to be done for those courses first. Do what you have to do, but not more.

Then plunge into what you like, and do more than you have to do. Discuss the interesting stuff with others who are also interested. That is a huge motivator.

Now -- do you want more of a life than mathematics, or not? The best mathematicians spend huge amounts of time on the subject. They work all day at it, then at night they go home and for recreation they -- do more mathematics. Is that the lifestyle you want? (I'm not knocking it, it's either for you or it isn't.)

If the answer is yes, you will have plenty of time do learn tons of math. If the answer is no, you have to budget your math time so it doesn't take over your life. And that definitely means concentrating on what fascinates you.

The good news is that what fascinates you is entrenched in mathematics, and as you concentrate on it a lot of peripheral knowledge will just naturally come to you.

Best wishes.

  • $\begingroup$ Thanks for your advice! I am going to try to discipline myself to not touch my favorite subject until I have finished all my other work. (I really hadn't thought about this before). $\endgroup$ – JimmyJackson Oct 10 '13 at 15:15

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