What is an REU, anyway?
The acronym REU stands for "Research Experiences for Undergraduates." Many REUs are sponsored by the National Science Foundation. REUs take place in the summer and most are 6--10 weeks long. There are REUs for pretty much every field of science. This page is about mathematics REUs. Laboratory science REUs are very different.
So... what happens at an REU?
The structure varies from REU to REU, but the common theme is for students to attempt research in some sense. Some REUs are more focused on the idea of developing mathematics that is new to the participants but perhaps already known to the world, and many have the creation of new mathematical knowledge as a goal. Most mathematics REUs have a group of students working with a research mentor, and the students may work on separate (but related) problems, or they may collaborate. Some REUs have several small research groups, and some have fewer larger research groups.
In terms of the day-to-day activities, participants have meetings, sit around and think, write computer code, read papers or books, sit around and think, take notes, go to talks given by visitors, work examples, and sit around and think. (Mathematics research involves a lot of thinking. Yes, it's challenging.)
Look, I have to get a job for the summer. I don't have the time or money to do an REU.
An REU is a summer job. Every NSF-sponsored REU provides a stipend to every U.S. citizen or permanent resident participant. Some REUs have funding for international students as well. Most REUs provide housing or a stipend for housing, and most REUs provide a stipend for food expenses. Many REUs also provide some travel funding.
Okay, fine. Apart from the money, what are the benefits of an REU?
There are tons!
A piece of advice: the structure of REUs varies widely, and not every REU matches the style of every student. Often this is not easily discernable from REU webpages, so if you apply for REUs and get into one, be prepared to ask questions, ranging from "how do participants deal with housing and meals?" to "how often do participants meet with their research mentors?" to "what support is there for continuing the work after the summer ends?"
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