What is mathematical research like?

Solution 1:

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.

Theory building is a somewhat different activity that involves the creation of new structures, or the extension of existing structures. Usually, the motivation behind the study of these new structures is coming from a desire to generalize (in order to gain better insight or be able to apply particular techniques of one area to a broader class of problems) or there might be a need to these new structures to exist, due to some application in mind. Typical activities would include a lot of reading on relevant structures, understanding their global role, figuring out what generalizations or new structures would make sense, what the aim of the new theory will be, and then a long process of proving basic structure theorems for the new structures that will necessitate tweaking the axioms. A striking example of this kind of research is Grothendieck's reformalization of modern algebraic geometry. Cantor's initial work on set theory can also be said to fall into this kind of research, and there are many other examples.

Of course, quite often a combination of the two approaches is required.

Today, research can be assisted by a computer (experimentally, computationally, and exploratory). Any mathematics research will require extensive amount of learning (both of results and of techniques) and will certainly include long hours of thinking. I find the entire process extremely creative.

I hope this helps. As should be clear, this is a rather subjective answer and I don't intend any of what I said to be taken to be said with any kind of mathematical rigor.

Solution 2:

"A mathematician is a blind man in a dark room looking for a black cat which isn't there." Attributed to Darwin (but I'm not convinced).

EDIT: A friend of mine found a discussion of this quote at wikiquote. It says (among other things),

1. The attribution to Darwin is incorrect,

2. In a publication of 1911 it was attributed to Lord Bowen (who died in 1894), but it was about "equity", not about mathematicians, and it was a hat instead of a cat,

3. It was published in 1898 as being about metaphysicians and hats,

4. William James, 1911, had it about philosophers,

5. The first reference to mathematicians seems to be in a 1948 collection of essays edited by William Schaaf.

EDIT 30 August 2016: Expanding on the last point. The collection is Mathematics, Our Great Heritage, edited by William Leonard Schaaf, published by Harper in 1948. An essay by Tomlinson Fort, Mathematics and the Sciences, appears on pages 161 to 172. A footnote states, "Address delivered at the dinner of the Southeastern Section of the Mathematical Association of America at Athens, Ga., March 29, 1940. Reprinted, by permission, from the American Mathematical Monthly, November, 1940, vol. 47, pp. 605-612." On page 163, Fort writes,

I have heard it said that Charles Darwin gave the following. (He probably never did.) "A mathematician is a blind man in a dark room looking for a black hat which isn't there."

Solution 3:

I guess its something like what you said, but not so much euphemic :) Mostly, researchers are dealing with problems in which there are several people working at it at the same time, so there's some kind of communication as they often work in groups. They also have to attend to conferences to get to know what's new on research world. Sometimes they are trying to "mix" different branches of mathematics in order to develop some new techniques to solve the problems.

I used to have an advisor who once explained that its contributions as a researcher involved solving problems that appeared in engineering & physics literatures but that the authors didn't had the tools and/or time and/or interest to work them out.