For Canadian undergraduates (citizens and permanent residents) who do not hold a similar American status, the door to research is not entirely closed. Math departments at some Canadian universities offer undergraduate research assistantships (funded by the federal government), known as "summer NSERCs", to qualified applicants, regardless of where they are studying. To qualify, you must have at least a B average, you must NOT have undertaken any graduate studies, and at the time of your work term, you must have n semesters to complete your undergraduate degree, where 0<=n<=4. Because of funding changes, NSERCs can now only be held in industrial research firms or at "minor" (I don't know what that means; check the NSERC gopher for more info) Canadian universities.
The application deadlines are on average earlier than those for other summer jobs. Deadlines for NSERCs range from November to February, depending on the place to which you are applying. It's a good idea to pinpoint the universities you are interested in (you are allowed a maximum of two), and contact their math departments as early as possible--they should have application forms as early as September. You should also have some idea as to what kind of math you would like to do, so they can determine which projects are appropriate for you, given your background and interests. The actual application materials required are personal data forms, a college transcript, and a (very short) recommendation from a professor who knows you and your work.
Two summers ago, I spent three months working on a research project at Simon Fraser University, just outside Vancouver, B.C.. My supervisor was Dr. Alistair Lachlan, a prominent recursion and model theorist, and my project dealt with homogeneous 3-graphs--a 3-graph is simply a set of vertices together with a symmetric ternary relation. (to give some sort of geometric intuition to this concept, the "edges" of a 3-graph contain 3 instead of 2 vertices) "Homogeneous" means that every isomorphism between finite subgraphs extends to an automorphism of the whole graph. It has been proved by Allyson Tripp (a student of Dr. Lachlan) that there are only 4 finite non-trivial homogeneous 3-graphs up to isomorphism; the conjecture on which I was working was that under certain complicated conditions, there is only 1 countable non-trivial homogeneous 3-graph. Although the above sounds somewhat daunting, there are useful results which allow one to restate the problem in simpler terms.
The first two weeks of my project were spent reading theses which provided background material for the work I was to do later. Next, I wrote a computer program in an attempt to get results which might help with the proof of the abovementioned conjecture. While the computer was generating data, Dr. Lachlan and I worked (for about two months) on proving a non-trivial theorem which provided us with some insight as to the nature of the 3-graphs we were working with.
NSERCs differ from REUs in several respects. At least as I understand it, REUs generally involve team work; NSERC projects, on the other hand, are designed to give students the experience of working one-on-one (or sometimes two-on-one) with a professor, in preparation for graduate studies. Also, NSERCs typically last for three months or more--it's difficult but not impossible to get one for less than three months, given that most Canadian schools end their semester at the end of April. Finally, NSERCs are primarily academic experiences, unlike REUs, which are more structured in the way of residential and social life.
If you are interested in learning more about NSERCs or the work I did two summers ago (I did leave out a lot of details!), feel free to e-mail me. If you are interested in working in a non-math discipline, a lot of university science departments offer NSERCs for those areas, too. Industries, too, hire students interested in physics, engineering, and other sciences appropriate to their area of focus.
As an alternative to NSERCs, there is also something called the "Challenge" Program. I know very little about the mechanics of applying for it; last year, I applied for an NSERC and got a Challenge instead--it basically involves 7 weeks, (officially 30 hours per week) of whatever the professor you are working for wants you to do. I ended up doing a reading project on the geometry of elliptic curves. IT MAY NOT BE TOO LATE TO APPLY FOR A CHALLENGE POSITION THIS SUMMER. I'm not sure which universities offer them, but it's worth making enquiries at universities which interest you. (I did mine at the University of British Columbia.)
P.S. Any Canadian citizens or landed immigrants (permanent residents) who are planning to undertake graduate studies in any scientific field during 1996-97 (or later) should apply for NSERC postgraduate scholar- ships. They are valued between CAN$15,600 and CAN$17,500 per year for two years (possibly renewable for another two years) and are usually tenable in Canada, but can be held outside, too. Contact me (or NSERC) for more information. NSERC has a gopher site which will probably provide you with the necessary information. The deadline for applications is around Dec. 1, 1995 (for a scholarship for 1996-98).
--Reza Akhtar '95 ( reza@math)