How does actual academic research in mathematics work?

[Frion]

I know how homework works. I get a problem set, I have roughly a week to do it and I know that all of the problems in it have solutions. Furthermore, I know that the theorems and techniques I've been learning in the last 1-2 chapters would be very applicable to the homework problems.

But how does research in pure mathematics work? How do you know the problem you are working on has a solution? What if you accidentally chose to work on something like Fermat's Last Theorem? Where do mathematicians even find interesting problems to work on? What if you run into what seems to be a dead end? Is it all just pondering how to prove it or do computer simulations come into play as well? How do you even make progress on a problem? Suppose you wake up in the morning and it's still not solved... do you just think about it all day? Solve random problems that seem related to it? Talk it out with colleagues?

 

[chiro]

There will always be problems in mathematics and any kind of science.

One thing about research is that there is no guarantee you will find out the answer. Also another thing is that research is always filled with brick walls. If we knew the answer, we wouldn't call it research.

As with progress, its like any problem. You get a basic understanding of the background of the problem and that background will contain things like the type of thinking used to analyze the problem, the techniques used in trying to work with the problem and the history of labor that is put into solving the problem.

Then you take all that in and add your own tricks that you will try to solve the problem. If the problem has been around long enough then you will have a lot of ideas that don't work which can perhaps help you look into something not related to the trials of past researchers.

Also a lot of research is done by teams of people and mathematicians are people too ;). They throw ideas at one another and use the collective intelligence to get ideas about the problem from different perspectives.

Also I think its rare for anyone to get the luxury to at least work on their own problems all day. That kind of thing is done when all the "other necessary stuff" is done.

 

[deRham]

What if you accidentally chose to work on something like Fermat's Last Theorem?

As a PhD student, you have an advisor who can help you set reasonable goals. As an academic afterwards, you either have people more experienced than you or are so high up that you do not, in which case you would be in good shape.

 

[A. Neumaier]

But how does research in pure mathematics work?

It is lifelong learning - trying to understand what the others do, and see what you can do better.

How do you know the problem you are working on has a solution?

Because you see already a way that looks successful. Getting this sort of intuition takes a lot of practice and experience, and requires that you have seen so many dead ends before (because of lack of experience...) that you can smell in advance what looks like a dead end. Avoid it. there are enough open doors not yet tried. Go through one of them, even if it means having to learn new things. Learn how apparently closed doors can be open, and how to discriminate between those that are locked but have a key and those that are no doors at all.

What if you accidentally chose to work on something like Fermat's Last Theorem?

Only a fool chooses problems by accident. One choses problems that are expected to be rewarding in the sense that you learn a lot while working on them - even if you fail to reach the ultimate goal.

Where do mathematicians even find interesting problems to work on?

In the work of others. I their own attempts to do something. In the dissatisfaction with what you did or read.

What if you run into what seems to be a dead end?

Check out in which way it is dead (maybe there is an open end), and why you were running into it (maybe you can avoid it next time). Learn from dead ends as from promising avenues. Perhaps turn the dead end into a theorem that something is impossible. Read how others managed to get around this in a similar situation. Improve your toolkit by learning techniques that might bear on the problem. Or work on one of the many other problems on your list of ''it would be interesting to do this''.

Suppose you wake up in the morning and it's still not solved.

This is the usual thing. Get to know the structure of the problem; look at related problems that were already solved. Step back on what you did already and try to organize it more coherently so that you can see batter what's going on and why you are stuck.

Love never ends... Thousands of possibilities; each mathematician handles things differently (though there are many similarities).

 

[mathwonk]

Even if you choose to work on something like Fermat's last problem, you may well find partial answers. Indeed such partial answers were found for centuries before the full solution was provided. Any of those partial answers would have made a nice PhD thesis. The still mysterious Shafarevich Tate group of an elliptic curve is currently and continually providing useful work to generations of researchers as they add slightly to our understanding of it.

Indeed tackling portions of very difficult unsolved problems is one way to provide research problems, answering another question you asked. It is to me however more fun to find more accessible problems. One source of these is old papers that have fallen into obscurity, in which the author made good progress on an interesting problem but did not quite solve it. Or the author may have made a remark about some interesting he could not do, which might be accessible to someone with todays tools. You try to develop the knack of reading between the lines of research papers, to notice what they do not say,a s well as what they do claim to accomplish. Or sometimes you just solve the same problem someone else has already solved but you do so in a new way, which is easier or more general, or yields new connections of interest. If lucky your new approach applies to more situations and adds new cases of understanding, so you really do have something new.

The important thing is to keep working and if you do get stuck on a certain direction without making progress, after a while you change direction. You also talk and listen to other people to learn what they are doing. Sometimes what they say may be applicable to your problem. Once I was giving a talk and at the end, I saw a listener's jaw drop at my last comment. He rushed right out and the next day showed me that what I had said, suggested to him how to solve an important problem in a new way, and to prove something new as well about the object I was speaking on. The same thing happened to me in his talk, when he presented a fact that I had not known, which filled in an approach I had to an important result. It took me longer than one day to use it successfully, but I later sent it to him as well. So that one meeting where people were presenting their work, produced the solution of several new problems.

Every now and then it helps greatly to take off a semester or more and go somewhere to be around the top people in your area, where you can work and learn full time without the time taking obligation of teaching elementary courses and grading and administration and so on. This is the purpose of sabbaticals, but unfortunately some universities do not realize the value of these and do not provide them or are very stingy with them. they can greatly extend the creative life of a researcher. Places like the Institute for Advanced Study in Princeton, or the Math Sciences Research Institute in Berkeley provide a location for full time research to occur. Top universities like Harvard and Stanford are also intensely helpful places to spend time.

Here is some advice by someone much better qualified than me to advise:

http://terrytao.wordpress.com/career-advice/

 

[Frion]

Thank you all for the answers and wow thanks for that link. I never knew much about the guy except that he was a child prodigy. All the stuff I read on his blog is very pertinent and very enlightening. It's answering many questions I thought I wouldn't get answers to for years.