What Does Mathematics Research Really Involve?

[DarrenM]

Hello there,

Would someone be so kind as to elaborate on what research in the field of mathematics entails? Not necessarily what is being researched, as I've found quite a bit of that on my own; to be frank, most of it is currently over my head.

Rather, (this is going to sound a bit silly) "research" conjures up images of laboratories and humming machines and 'experiments.' I know that is a misconception, but I'm quite curious as to what the reality is.

 

[fantispug]

That's a deceptively hard question. I've studied as a pure mathematician, and I find it hard to tell you what I did.
The way I see it is when you start researching mathematics you've got a mathematical toolkit of all the ideas and techniques you've learned so far (you linear algebra, your calculus, etc.).
Then you look at a problem mathematics can be applied to - good problems often come from understanding mathematics itself. An example might be: I know a formula to solve linear equations and quadratics; how can I find one to solve cubics? Quartics? Quintics? (and this turns out to be a very rich question).
You then think about how to use your mathematical toolbox to solve these problems. Nearly always you'll get stuck somewhere, so you need new tools.

For new ideas we read books, articles and journals, we go to conferences, we correspond with other mathematicians. The tool we need might already be developed, or we might need to invent it ourself, or extend another tool.

This is the hardest bit - you have to guess what tool you need. Often you just try different things with varying amounts of success.

When you have an idea you work it through, on a computer or a whiteboard or in your head on the bus. If it works you pat yourself on the back and feel good. If it doesn't you try something else.

(You might design a new tool that doesn't solve your problem, but you still think it's pretty nifty. You might go and try to find problems that you can use your tool to solve.)

That's as close as I can think of a description without going into a specific example.

 

[sutupidmath]

That's a deceptively hard question. I've studied as a pure mathematician, and I find it hard to tell you what I did.
The way I see it is when you start researching mathematics you've got a mathematical toolkit of all the ideas and techniques you've learned so far (you linear algebra, your calculus, etc.).
Then you look at a problem mathematics can be applied to - good problems often come from understanding mathematics itself. An example might be: I know a formula to solve linear equations and quadratics; how can I find one to solve cubics? Quartics? Quintics? (and this turns out to be a very rich question).
You then think about how to use your mathematical toolbox to solve these problems. Nearly always you'll get stuck somewhere, so you need new tools.

For new ideas we read books, articles and journals, we go to conferences, we correspond with other mathematicians. The tool we need might already be developed, or we might need to invent it ourself, or extend another tool.

This is the hardest bit - you have to guess what tool you need. Often you just try different things with varying amounts of success.

When you have an idea you work it through, on a computer or a whiteboard or in your head on the bus. If it works you pat yourself on the back and feel good. If it doesn't you try something else.

(You might design a new tool that doesn't solve your problem, but you still think it's pretty nifty. You might go and try to find problems that you can use your tool to solve.)

That's as close as I can think of a description without going into a specific example.

This is one of the best descriptions I've ever heard so far, in this forum, as long as math research goes!

 

[Animastryfe]

I've been wondering about two things regarding mathematics research:

1. What do most mathematicians use when they're working? Blackboard, paper, computer?

2. How do grants work in regards to this? I would ask a more specific question, but I'm very fuzzy on grants and funding in general.

 

[tatiana_eggs]

Thanks, Darren, for asking this question. It is something that I have wondered about for a long time. And thanks, too, fantispug for a great answer.