# What is mathematical research like?

I’m planning on applying for a math research program over the summer, but I’m slightly nervous about it just because the name math research sounds strange to me. What does math research entail exactly? For other research like in economics, or biology one collects data and analyzes it and draws conclusions. But what do you do in math? It seems like you would sit at a desk and then just think about things that have never been thought about before. I appologize if this isn’t the correct website for this question, but I think the best answers will come from here.

Largely (very largely, so please take everything here with a grain of salt), there are two types of mathematical research, commonly referred to as ‘theorem proving/problem solving’ vs. ‘theory building’. Typical characteristics of theorem proving/problem solving type research is to try and tackle a famous open problem, usually stated in the form of a conjecture as to the validity of a statement or the specification of a problem. Quite often this will entail spending a lot of time learning the relevant material, analyzing particular attempts at solutions, trying to figure out why they don’t work, and hopefully come up with some improvement to an existing attempt, or a whole new attempt, that has a good change of working. Very famous open-standing questions include: The Riemann hypothesis and $P\ne NP$ (which are examples of theorem proving) and the solution of the Navier-Stokes equations (an example of problem solving), all three are in the Clay’s Institute millennium problems list.