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I am currently helping my advisees (undergraduate math majors) plan their summer. As the obvious option, they've already applied to lots of REUs (some have gotten in and others not heard a positive response yet), but of course REUs can be very competitive to get and it would be nice to have other options I could suggest. Any recommendations? As background on the students, they are serious and high-achieving sophomores with an interest in pursuing graduate studies. Ours is an applied math program at a smaller polytechnic, and the students have some research experience. The program wouldn't necessarily have to be research-based, though - just something mathematical that they could learn and advance from and further their careers.

Thanks for any and all thoughts/suggestions!

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    $\begingroup$ What is an REU? $\endgroup$ – DavidButlerUofA Mar 22 '15 at 22:47
  • $\begingroup$ Research Experience for Undergraduates (e.g. these: ams.org/programs/students/emp-reu). In the US, the NSF funds a number of such programs at various institutions, mostly 8-10 weeks, in which students work in small teams with faculty mentors on research projects. Sorry, I forgot that NSF-funded REUs are specific to the US! $\endgroup$ – Idempotent Mar 22 '15 at 22:48
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    $\begingroup$ See also the the MO question "Research Experience for Undergraduates: Summer Programs," and the MSE question "Research Experience for Undergraduates: Summer Programs (that accept non-American applicants)." $\endgroup$ – Joseph O'Rourke Mar 22 '15 at 23:18
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    $\begingroup$ It doesn't have to be "NSF-sponsored", does it? If they don't win, they could work on their projects anyway. $\endgroup$ – vonbrand Aug 7 '15 at 17:41
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    $\begingroup$ @vonbrand That's true, it doesn't need to be NSF sponsored, just that the majority of programs I know about are. I'm just interested in finding ways for the students to take part in summer math programs with some funding, as a way to broaden their horizons (they work with me on their projects all the time - maybe they should collaborate/learn from others, too) and get a chance to build connections in the field. $\endgroup$ – Idempotent Aug 8 '15 at 16:12
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Some local (or not) businesses may have fairly mathematical internships. For instance, a biotech company may have nontrivial graph theory or other discrete computational problems to solve that will require something analogous to math research, though without proofs. (I'm not making this example up.)

Certain computer science internships would be heavily mathematical as well. For instance, the Google Summer of Code often partners with open-source mathematical software, such as lmona.de and SageMath - I think Sympy and others have also gotten these. Some of these projects can be quite mathematically technical.

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The tags say "undergraduate-research", but the question seems broader than that. I'd like to point out that summer teaching is an excellent alternative to research, which can give the students a much better understanding of the material they think they know.

At the advanced end, summer programs like PROMYS, Ross, Mathcamp and Hampshire hire undergraduate counselors, provide a great environment for learning mathematics, and are recognized as prestigious in graduate admissions. (Disclosure/endorsement I worked for PROMYS for four summers, and had a great experience.) Down a step, many local colleges run summer programs (my own university has MMSS) and hire undergraduate assistants.

And of course, more basically, tutoring jobs exist and usually the tutor winds up learning more than the students.

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