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This post was written on an airplane two and a half weeks ago, but I forgot to post it until now. Therefore, it is somewhat delayed.
I was at the Penn State REU for seven weeks this summer, so I thought that I’d comment on it here. I feel that I should probably say something about my problem at the REU, but I can’t do much better than what others have already said. Therefore, for an explanation of the problem, go climb Mount Bourbaki.
I’ll instead make some general comments about research.
The goal of the REU is to provide a Research Experience for Undergraduates, and that’s precisely what was attempted: We were divided into groups and each group was given a problem to solve. At the beginning, nobody had the background to approach their problems, so most of the first few weeks were devoted to learning the background material. Only then could we start to attack the problems.
There are a number of ways in which the idea of undergraduate research in mathematics is not ideal. In the sciences, anyone can participate in a research project; every lab needs someone to do labor-intensive and mind-numbing tasks such as washing bottles. Mathematics, however, is entirely different.
There are no bottle washers in mathematics. Instead, mathematical research is highly individual and requires a large amount of preparation and prior knowledge. As a result, many attempts at undergraduate mathematical research never get past the preparation phase. In this sense, many REUs are like summer reading courses. But there’s one important difference: Because the stated purpose of an REU is to do research and not to learn, the reading courses are very rushed and incomplete. Not only do students never get to doing research, they also do not properly learn.
Of course, the idea of mathematical research for undergraduates is not completely meaningless. In fact, it’s probably healthy for students to understand that research is distinct from problem sets. However, they shouldn’t expect to magically understand research from an REU.
It might be more sensible to redefine REUs as reading courses and stop advertising the research component; it would better match what the students actually need to do. There are several reasons why such a suggestion is impractical, however. Students would find REUs less attractive as they would no longer look as good on their resumes. Maybe more importantly, The NSF would find REUs less attractive and might stop funding them.
The lesson here is that in the real world, appearances matter more than reality. Despite whatever I would like to believe, mathematics does have a nontrivial intersection with the real world.