So you want to apply to an REU, and you are required to write a "personal statement," "statement of purpose," or something to that effect. Some students may be able to get good advice on how to write their statement from faculty mentors at their own institutions, but we know that access to such information is not always readily available. For those of you who are interested in this whole REU thing, here are some tips that I would tell my own students. Note that you will get differing opinions from different people, possibly opinions that disagree with what's written here. But hopefully, this will at least level the playing field a little bit. We receive many applications from students who meet the bar in terms of grades and academic performance, but we just can't tell whether they really are interested or have the motivation and desire to be successful at an REU.
1. Explain why you want to participate in an REU and why you think you will benefit from it.
Spots in REUs are very limited. We get maybe around 200 applicantions for 8-9 spots. We want to make sure to offer a spot to someone who is really going to appreciate and take advantage of the opportunity. What are your career goals? How will you use what you gain from participating to reach your goals? We also invest a lot of time and effort into our students, so we want to see that you really want to be in the REU and will stay engaged and motivated, even during the tough times.
Yes, that's right! Math research can be slow, and there will be times during the program where you are frustrated and feel like you're going nowhere. That's OK, as it's par for the course when it comes to doing original research. If you knew exactly how everything should go and can do it without hitting any roadblocks, someone else would have figured it out already! We want to make sure you'll be willing to push through those difficulties to reach your goals!
2. Explain what makes this specific REU exciting to you.
We want to know that if we offer you a position in the REU, that you will be excited to be here and will choose this REU over a different one. It's not necessarily that we just want the "best" students that will churn out lots of papers or that we are "competing" for the best students, but we can never have more offers out than we have spots remaining, and we usually like to give students enough time (usually a week or more) to make an informed decision about which REU among the offers they received is best for them. Going back over and over the pool of applicants to find the next person up if someone declines is time-consuming, and we don't want to drag it out for too long.
In addition, just as in the first point, the more excited someone is about being at our REU, the more likely they are to be successful. Do some research about each of the programs that you are applying to. Find out what the projects are about and information about the faculty member(s) that will be mentoring your project. Take a look at some papers that they have published. It's OK if you don't really understand them (and don't be intimidated if you don't! The intended audience for papers is experts who have studied the subject matter for years), but try to at least grasp a vague understanding of what they are about by reading the abstract and skimming the theorems that look important. Yes, this does take a lot of time if you are applying to a lot of REUs, which is almost a necessity given how competitive they are, but a bland, generic personal statement that has nothing specific about the REU you are applying to says (whether accurate or not), "I'm not that interested in this REU, but I'm applying to it because I might as well while applying to some other ones I'm really excited about."
3. Explain why you will have a successful REU experience.
As previously mentioned, math (and really any kind of) research takes resilience and determination to persist through the dry spells where nothing seems to work or make sense. It's part of the process of creating something that no one has thought about before. Excitement goes a long way (see points 1 and 2), but also give some concrete reasons as to why you're prepared for the challenge. Give an example of a time that you struggled in mathematics and what you did to get through. How will that experience help you at the REU?
Other things that are important are your ability to study and work on your own as well as work with your research team. What are some skills that you have that will make you successful in those aspects? How have you demonstrated them in the past?.
4. A final note.
It's OK if you don't have answers to all of the things mentioned above. We are not looking for perfect people (we aren't perfect either!). But highlight your strengths and the ways in which you can be a positive contribution to the REU. One of the things that we look for when choosing participants for the REU is how the individuals will fit together and complement each other. It's important that everyone gets along, but it's also important to have a range of different skills represented so that the sum is better than the individual parts. Giving us more information so that we can see how you will fit in will improve your chances of being selected.