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.

7Gromov said when he was young he could work on problems from early morning to midnight. – alancalvitti Feb 13 '13 at 03:19

4This question has several good answers on Quora [here](http://www.quora.com/Mathematics/Whatdogradstudentsinmathdoallday). – David Robinson Feb 13 '13 at 08:02

2You should read *Théorème Vivant* (Alive Theorem) by Cédric Villani, as soon as it gets translated to English (or right now if you can read French). It tells the story of a research that led to the Fields Medal, and it's fascinating. – Laurent Couvidou Feb 13 '13 at 09:48

@LaurentCouvidou That sounds extremely fascinating, is there another book that tells a similar story? – TheHopefulActuary Feb 14 '13 at 04:51

@Kyle Not to my knowledge. Which doesn't cover lots of math ;) – Laurent Couvidou Feb 14 '13 at 08:42

Please avoid the tags [tag:undergraduateresearch] and [tag:research]. We are trying to remove them. – 6005 Jul 24 '16 at 19:49
9 Answers
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 openstanding questions include: The Riemann hypothesis and $P\ne NP$ (which are examples of theorem proving) and the solution of the NavierStokes 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.
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"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),
The attribution to Darwin is incorrect,
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,
It was published in 1898 as being about metaphysicians and hats,
William James, 1911, had it about philosophers,
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. 605612." 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."
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3And in the world of this quote, what is the blind man's goal? To find out (and prove) whether the cat is there? I'm having a hard time applying the quote to mathematics in a meaningful way, but maybe it's not meant to be very meaningful. – LarsH Feb 13 '13 at 16:03

2I think it's meant to be funny. I don't think it stands up to a rigorous analysis. – Gerry Myerson Feb 13 '13 at 23:10

@LarsH If anyone has a hope of finding the black cat in such a dark room, it is the Mathematician who isn't inhibited by such darkness. The main point is that the mathematician will often need to prove that something *isn't* the case, which is often as hopeless as trying to find said nonexistent cat. If the blind man walks to every spot in the room, the cat may have simply just moved. Even if absolute confidence is reached, it may not necessarily be the case. – Display Name May 27 '14 at 00:41

@DisplayName, can you explain why the mathematician has hope of finding the cat? As opposed to, say, a physicist, an engineer, or an animal trainer. I'm not seeing it. The mathematician is wellversed in logic and proofs, but this won't help find the cat in the complete absence of empirical data. I think we're straying from the original point, but since I don't understand the original point it's hard to get back there. Maybe the search for the meaning of this quote is like a search for a cat that doesn't exist. – LarsH May 27 '14 at 21:20

@LarsH Sorry the key at that part was the blindness aspect. Being blind he is more prepared for such darkness, which is irrelevant to him. That was explanation for the blindness aspect, the rest was in regards to being a Mathematician. – Display Name May 28 '14 at 04:23

1Of possible interest: http://quoteinvestigator.com/2015/02/15/hiddencat/ (which indicates the attribution to Darwin is first found in Tomlinson's essay). – Barry Cipra Aug 30 '16 at 01:53
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.
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Found this description of mathematical research at https://www.awmmath.org/noetherbrochure/Robinson82.html:
At one point, writes [Elizabeth] Scott, [Julia] Robinson was required to submit a description of what she did each day to Berkeley's personnel office. So she did: "Mondaytried to prove theorem, Tuesdaytried to prove theorem, Wednesdaytried to prove theorem, Thursdaytried to prove theorem; Fridaytheorem false."
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Take a look at http://www.ams.org/programs/students/undergrad/empreu for topics near your interests. I happen to know that these are not all the REU programs coming up. there is also http://mathcs.emory.edu/~ono/REUs/ and likely others not listed. Oh, I get it, that one is already full. Anyway, these are pretty well organized. Faculty give an overall picture, students do research projects and write up both group and individual reports. Programs in other countries may or may not be this well organized.
Right, the individual sites listed should give lots of information about past year summer programs, sometimes the reports by the students.
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I can recommend that you read Richard Hamming's book The Art of Doing Science and Engineering: Learning to Learn. It gives many examples about research work.
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For me as an independent mathematical researcher, it includes:
1) Trying to find new, more efficient algorithms.
2) Studying data sets as projected visually through different means to see if new patterns can be made visible, and how to describe them mathematically.
3) Developing new mathematical language and improving on existing language.
It is not so different from how you describe biological or economical research, only that you try to find patterns linked to mathematical laws rather than biological or economical laws.
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We know that mathematics is the subject which is not invented it's just found out from nature and as we all know that research means doing a work which is already been searched or done just to get the more accurate results and minimise some how the approximate errors .mathematics research means some how researching the nature so that we are able to do any work more logically and get the most accurate results related to anything with the help of our previous knowledge in mathematics..
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I think the ultimate goal of mathematical research is to discover all possibilities. The way to do so is to think all possible thoughts, discover all possible rules, and find out all possible objects which follow the rules.
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