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.
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.
"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. 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."
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.
Take a look at http://www.ams.org/programs/students/undergrad/emp-reu 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.
Found this description of mathematical research at https://www.awm-math.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: "Monday--tried to prove theorem, Tuesday--tried to prove theorem, Wednesday--tried to prove theorem, Thursday--tried to prove theorem; Friday--theorem false."
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.
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.
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.
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..