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REU's:
The National Science Foundation makes possible a number of opportunities for undergraduates to join research projects each summer. This allows students to experience first-hand how basic research is carried out, and to contribute consequentially. The principal support by NSF of such activities is through the Research Experiences for Undergraduates Program.
REU "Sites" are established in all fields of science, mathematics, and engineering. Each Site consists of a group of ten or so undergraduates, who work in the research programs of the host institution. Students are in general accepted from throughout the country. Each student is assigned to a specific research project, where he/she works closely with the faculty, post-docs, and graduate students. In addition, seminars, lunch meetings, and social functions are organized to facilitate interaction between the undergraduates.
Summer 2002 REU information:
http://www.math.unl.edu/~womenws/opps/
http://www.lasalle.edu/~marano/ProjectNExT/mathMajors.html
http://www.ams.org/employment/reu.html
http://www.math.indiana.edu/reu.
The Indiana program generally assumes a background that includes at least one traditional "senior sequence" such as abstract algebra or advanced calculus. It's aimed at advanced students between their junior and senior years (relatively high level), although there are often some younger, but still advanced, students. There tends to be a variety of projects in quite a few different areas of pure and applied mathematics (wide scope). And there also tends to be individual research projects supervised by a number of individual faculty. (There are occasional group projects and various group activities, of course.)
You might want to check out a program called SUMSRI (Summer Undergraduate Mathematical Science Research Institute), at Miami University. This isn't (strictly speaking) an REU, but it has similar objectives. The program is targeted at, but definitely not restricted to women and minorities; in past year, there's been a good mix of students from across the nation. Typically, the students in the program are entering their junior or senior year of college, so perhaps younger students might want to wait a year before applying. Even though the program is young, it has had considerable success already; for example, SUMSRI students have won more than their fair share of awards at the Undergraduate Research Poster Sessions at the Joint Meetings.
-You can find more information at: http://www.muohio.edu/sumsri
Younger students shouldn't give up or be discouraged with REUs-- They should check the eligibility requirements. Some REUs are specifically for people with no prior REU-type experience (such as Hope College's). There are summer enhancement opportunities too.
Follow the links for "students" or "careers" or "opportunities" at any of the professional organizations' websites. Here's the MAA:
http://www.maa.org/students/reustuff/pages/smrpage.html
SIAM's site leads to:
http://www.math.okstate.edu/~wrightd/useful/resources.html#Summer%20programs
Here's where the internship page is on the AMS page:
http://www.ams.org/employment/internships.html
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