**NOTE: This article is from 1998. While some of the details for REUs have changed over time and the number of sites has increased, these successful and popular programs exist in mostly the same form today. The article below is meant to give you an idea of what REUs are like. Please look into specific REUs (both AMS and the NSF keep current lists) for current information.**

Did you ever wonder what it means to do research in mathematics? How do you start? Where do you get a problem? What do you do when you get stuck? What, exactly, are you doing when you're not stuck? Most people have to wait until graduate school to find answers to these questions. Increasing numbers of undergraduates, however, are getting a sneak peek at research mathematics by attending one of the many Research Experiences for Undergraduates (REUs) offered around the country. (There are approximately twenty mathematics REUs each summer. Check out the list of programs at the National Science Foundation Math REU Web page [ 1 ]).

We wondered what happens at an REU, so we contacted about two dozen alumni and asked them to describe their experiences. One thing we learned was that all REUs are not the same. Dan Isaksen, now a graduate student at the University of Chicago, participated in REUs at the Universities of Dayton and Minnesota-Duluth as an undergraduate and has several times returned to Duluth as an advisor. He comments, "I think it is very important for a student to find the right kind of program. Some programs are more motivational, and others are designed for self-motivated students. Some are aimed at original research, and others are aimed at thoroughly learning a body of advanced mathematical material. Some are computational, some aren't. Some are applied, others are pure." A little bit of research can help you here--most REUs have Web pages. (You can get the URLs from the NSF page cited earlier.) The undergraduate math club at Harvard maintains a Web page [2] that contains reviews by Harvard students of REUs in which they have participated.

Some REUs are very structured, with organized lecture courses, computer instruction and homework. More are unstructured--you are free to schedule your own work sessions and you meet perhaps once or twice a week with your faculty advisor to discuss progress. At some, people work in groups; at others, each person has his or her own problem. Whatever the structure, everyone we talked to had a positive experience. Lenny Ng of Harvard says, "Duluth, besides being rewarding mathematically, is a lot of fun!" Brian Wecht of Williams College says of the SMALL REU at Williams, "...you'll find it interesting, no matter what [you do]."Jade Vinson of Washington University describes the Cornell REU as "a rare opportunity" with "the ideal atmosphere."

Many of the people we talked to ended up with published research papers, lots gave talks at national or regional MAA or AMS (American Mathematical Society) mathematics conferences. Some did not get that kind of closure, but even they were enthusiastic about what they had learned. Suzanne Lynch of the University of Missouri-Rolla says of her experience at the Washington University REU, "As for me personally, I obtained no results about either problem, but rather a deeper understanding of them. Maybe someday I will stumble across something that will remind me of the still unsolved problem, and I will have a breakthrough. I don't feel I wasted my time just because I didn't obtain the main result I was looking for."

We have picked out four of our correspondents to tell their stories in more detail below. We thank all the students who wrote and shared their experiences with us: In addition to those already mentioned we also thank Aleksey Zinger of MIT, Kiran Kedlaya of Harvard, and Jessica Sidman of Scripps College. We wish that there was room to tell all their stories.

After **Ian Taylor's **junior year at Carleton College, he participated in the REU at Michigan Technological University. Work began around 9 a.m. with the six participants reading papers or bouncing ideas ("or bouncing chalk pieces") off each other. Their advisor, Anant Godbole, called the groups into his office individually where he would hear about recent progress and perhaps suggest new ideas or directions. After a leisurely lunch at the MTU cafeteria, work would continue until about 4:30 when the beautiful outdoors of the Upper Peninsula of Michigan called them to go walking or to play Frisbee or tennis.

"Dinner was usually cooked by a spontaneously created team, but such events as the great spinach quiche bakeoff would bring more structure. Some people would actually pass up the Olympics or a movie to go think about math, but most of us resisted such behavior. Weekends brought such events as concerts by the local teenage grunge bands, or the Strawberry Festival."

After some interesting excursions/false starts into a graph-theoretic application to food webs and extensions of the hat-shuffling problem (see box), Ian settled down into a group investigating the Wiener Index of a random graph. The Wiener Index is defined as the sum of all pairwise distances in a graph with n vertices and an edge between a given two with probability *p,* where distance is the minimum number of edges that must be crossed to get from one given point to another.

Simply dividing the Index by yields the average distance between two points in the graph. "Using cool probability stuff we actually found that this average converges to an easily describable function of *p.* I was the lucky one who got to talk about this at the MAA Mathfest in Seattle in August [1996]. "

The three are preparing a manuscript of their results for submission.

Location | Michigan Tech U |

Length | 8 weeks |

Stipend | $2500 |

Students:Advisors | 6:1 |

Housing | Share one big house |

A group of *n *people attend a party and all check their hats at the door. At the end of the evening the hats are randomly distributed to the partygoers. Everyone in possession of his own hat leaves. The others randomly redistribute the undeparted hats, when you get your hat you leave. What is the expected number of hat redistributions necessary before everyone gets his own?

Like Ian, **Sarah Spence** had just completed her junior year, at the University of Richmond, when she participated in the REU at the University of Minnesota-Duluth. Sarah's summer experience is better described as weekly-periodic: "Monday and Tuesday afternoons were reserved for group lunches and talks. After lunch, students gave talks summarizing their accomplishments over the previous week. These talks were generally around 30 minutes. We used this time to present new theorems and ideas, or show what had stumped us all week. Often one of the other students or advisors would see mistakes or offer advice on how to fix a proof. Preparing these talks was very helpful because it gave me a reason to organize my results each week. It also gave us practice presenting mathematics.

"We were then rewarded on Wednesdays with field trips. Our excursions included picnicking by a river, whitewater rafting, kayaking, visiting a paper mill, going to the zoo, strawberry picking, and sightseeing around Duluth. One week we took a trip to Michigan Tech to visit the REU there. We spent two days giving and hearing talks.

"For the rest of the week, each student decided how to spend his or her time. We all worked at different times, but we tried to reserve dinner time to cook and eat together as often as possible. I usually spent mornings doing work at the library or in the computer room. I did not use computers in my research, but I found the room to be a good place to work. I would break in the late afternoon to work out in the gym on campus and go running around Duluth/Lake Superior. After dinner I would continue my work. We were basically on our own Thursday through Sunday, and this is when I completed the majority of my work. Since we all worked on separate problems, most of our time was spent alone.

"Approximately halfway through the summer, we each started a preliminary paper summarizing what we had accomplished. We met with Joe Gallian to discuss writing techniques and progress. As the summer drew to a close, I spent most of my time writing up my results."

"Like many Duluth participants, I worked in graph theory. I studied stratified graphs, which are graphs whose vertices have been partitioned into color classes, and distance graphs, which are graphs whose vertices are graphs, and two vertices are adjacent if one of the corresponding graphs can be attained from the other by rotating one of its edges. I answered many open questions in these areas, and proved several theorems."

"I wrote a paper which has been accepted for publication in *Ars Combinatoria.* I presented a talk at a colloquium at the University of Richmond, and also presented my work at the Joint Mathematics Meetings in San Diego. I participated in the Undergraduate Poster Session and won an award for one of the top five projects. I also gave a talk at an AMS Contributed Paper Session."

Of her experiences in an REU, Sarah writes: "Participating in the Duluth REU cemented my goal of becoming a research mathematician. I feel that this experience helped prepare me for graduate school as well. It was wonderful to work in a stimulating environment, surrounded by extremely bright students who were equally excited about mathematics. By listening to other students' weekly talks and working through problems together, I believe that I increased my mathematical ability and creativity. This was a unique experience that certainly had an important effect on me as a mathematician."

Location | U of Minnesota-Duluth |

Length | 10 weeks |

Stipend | $2500 |

Students: Advisors | 7:3 |

Housing | 3-bedroom apartments |

**Kariane Calta** (Williams College) participated in the SMALL REU at Williams College after her sophomore year. "In an average day, I and the two other members of my group would typically read journal articles that were relevant to our research, then, working alone or together, we would try to develop conjectures and prove or disprove them, and, where possible, extend the results of the articles we read. Often, perhaps once a day, we would meet with our faculty advisor to discuss our ideas and to obtain further direction for our research. All of the groups of SMALL would meet once a week to discuss the program as a whole, including events of interest to all groups such as various math conferences, and also extracurricular activities. Twice during the summer, all six groups presented talks on their research. Weekly there would also be a mathematics colloquium, a talk on some area of science, and a lunch for all summer science research students on campus.

"The subject area in which I worked was number theory. Specifically, we investigated properties of integer sequences modulo m, in particular, the Fibonacci sequence. We produced many interesting results, a number of which were original, involving various properties of these sequences, such as their periodicity. Our research culminated in a paper which summarized the results we had obtained, both original and not. We are currently in the process of working on a paper which we hope to submit for publication in the near future.

"I thoroughly enjoyed this research program. It was great being able to delve deeply into a particular field of interest in a way that is not always possible in a classroom situation. In addition, our faculty advisor brought a level of enthusiasm to our research that was nothing less than contagious. I had a wonderful time learning new material, and meeting new people who also shared my interest in math."

Location | Williams College |

Length | 9 weeks |

Stipend | $2700 |

Students: Advisors | 18:6 |

Housing | dorm room |

**Cathy Isaac** (College of St. Benedict) participated in Carnegie Mellon University's Center for Nonlinear Analysis Summer Institute after her sophomore year. A typical day started with two classes: one on analysis and one on the software package *Maple;* afternoons were spent doing homework, meeting with project advisors, and working on projects. Sometimes afternoons were spent talking with the teaching assistants, who liked to give interesting and challenging problems for participants to work in their spare time. Tuesday afternoons were reserved for presentations by speakers on various areas of mathematics.

"Wednesday and Sunday nights we grabbed the comforters from our beds, made some microwave popcorn, and walked to a nearby park where movies were shown outside. We also organized informal nightly GRE study sessions in our lounge where anyone preparing to take the GRE could come to do practice problems. Most nights, the majority of us ended up congregating in the room with the pool table, where we liked to think that we were adding to our vast knowledge of vector calculus."

"We got to choose projects in the areas of mathematical finance, graph theory, and differential equations, and had the option of working alone or in groups. I worked with a partner on a project in finance that involved the pricing of barrier options. A barrier option is something you buy in place of buying stock. It comes with a set expiration date, barrier price, and strike price. It gives the holder the option of buying the stock at the strike price on the expiration date, provided that at no time did the stock price reach or exceed the barrier. If at any time the stock price reaches the barrier, the option becomes worthless. We modeled a system in which a transaction cost (i.e., broker's fee) must be paid at the point at which the barrier is reached. We derived the formulas needed in order to determine the value of the option and the amount of stock to buy/sell at each point of a discrete time model so that we would have enough money to pay the transaction cost as well as complete the hedge (a counterbalancing transaction). We also derived a partial differential equation which would give us the value of the option at the barrier as a function of time and stock price. We created a Maple program to compute the value for us so that we could create realistic examples and see what was happening. We gave a 45 minute presentation of our project at the conclusion of the program."

"My experience at Carnegie Mellon firmed up my decision to go to graduate school. It also had a negative effect: beforehand I thought I didn't like applied math, but by working on my project I discovered that I really enjoy applied math as well. So now it is much more difficult to decide where I want to focus my future studies! I found that I really love doing mathematical research, even with all the frustrations that come along with it, and can visualize myself with a career as a mathematician. The thrill that comes from doing original mathematical research, and actually making progress, is indescribable."

Location | Carnegie Mellon U |

Length | 7 weeks |

Stipend | $1800 |

Students: Advisors | 13:5 |

Housing | dorm room |

Mathematical maturity is more important than formal coursework, though, ideally, I like students to have had a serious class in probability and another in Rudin-style analysis, and to be familiar with combinatorial/graph theoretic concepts. At MTU, students should be prepared to make the transition from leisurely textbook-style learning to a rapid digestion of state of-the-art techniques in discrete probability.

However, I look at more than just a student's math background. I like the REU team to consist of eight interesting people. When composing my group I often look for the kind of multidisciplinary diversity that will make it more than the sum of its parts: travel abroad, piano, theatre, previous research experience, a strong position concerning vegetarianism. The guiding principle here is that these eight students will be living together for 2 months-an eclectic blend of interesting personalities is more likely to create a memorable summer.

My advice is to:

Spend time enriching your background (for these and other possibilities, check out MAA Online: http:/www.maa.org): conduct some research under the direction of your favorite professor; submit a solution to a Monthly problem; play an active role in the local chapter of Pi Mu Epsilon or MAA; spend time at another summer program such as the Carleton/St. Olaf, Berkeley, or George Washington programs for women; go to Budapest.

Submit a well-written application that directors are likely to remember. Talk about your relationship to mathematics; talk about why you want to participate in this particular program; let your personality show.

Spend a lot of time thinking about whom to ask to write letters of recommendation on your behalf. A well-throughout pageful of strengths and weaknesses from a caring college teacher is often more useful than a three-line letter from a research giant.

REU directors find an undeniable symbiotic relationship between their REU students' work and their own research. Often the ingenuity and simplicity of a 21-year old's attack on a problem teaches me far more than a scholarly article. Even though I have suggested the problem and suggested potentially successful lines of attack, I often find that students are able to lay down truly new paths for themselves. I ultimately learn as much from them as they learn from me. Over half of the papers that bear my name are now co-authored by REU students. Besides, it feeds my ego to have a posse of bright people around me, and it certainly makes my colleagues and co-workers envious.

**Professor Anant Godbole** is the director (or, as Ian Taylor says, guru) of the Michigan Technological University REU.

If you're looking for a research REU, keep in mind that the experience of working on open problems is more important than the specific topic in which you work. Even if, for example, you love algebra and hope to work in group theory, you can have a valuable REU experience working in topology or number theory. Most of the fundamental qualities of doing research transcend the field of inquiry: doing background reading and journal searches, trying simple cases, getting stuck and frustrated, discovering key connections and patterns, finding flaws in arguments and starting over, writing up results and presenting them to others, and feeling the exhilaration of mathematical discovery. Keep in mind also that mathematical subfields are very broad. If you choose an REU project in geometry because you love geometry, you may end up doing math that bears little resemblance to the geometry of your experience. But then, that's the point. An REU project of any format-research, classroom, open-ended, well-defined-is intended to broaden your mathematical skill and experience. If you're worried that you may not have the right background for a particular project, read program descriptions carefully and ask your own professors for guidance. If that doesn't help, write the REU organizers. Many programs don't expect participants to have training in specific areas, though mathematical maturity and familiarity with core undergraduate subjects are often important.

Be sure to consider factors other than subject matter. Are there talks or conferences during the summer? Do participants work individually or in groups? Do you want to live in a different part of the country? What are the housing arrangements? What kind of extracurricular activities are there? How many other students will be on campus during the summer? What kind of social life does the program foster? All these factors can affect your REU experience, though you probably don't want to weigh any one too heavily.

*-Deb Bergstrand*

The single most important factor in deciding on what program interests you is whether you want to participate in original mathematical research or whether you want an intensive classroom experience. Both have their advantages, and both mimic different aspects of the grad school experience.

The first gives you a realistic taste of research, something most students don't experience until their third year of graduate study in mathematics. Instead of working many years to reach the research level and then determining if you enjoy it, such programs give you the opportunity to try it now, and see if it is what you want to do. Many students find out that they love it, some find out it is not for them. In both cases, the experience is tremendously valuable.

The programs that provide an intensive classroom experience allow you to become totally immersed in learning mathematical material in a way you cannot during the regular school year. Without a multitude of courses in a variety of areas vying for your attention, you can concentrate on learning material to a much greater depth.

*-Colin Adams*

**Professors Bergstrand** and **Adams** run the SMALL REU at Williams College.

[1] http://www.nsf.gov/crssprgm/reu/list_result.jsp?unitid=5044

[2] www.math.harvard.edu/~mathclub/ (no longer working)

[3] Thanks also to the REU organizers who patiently answered all of our questions: Colin Adams and Deb Bergstrand, Anant Godbole, Joe Gallian, Bob Strichartz of Cornell, Morris Kalka of Tulane, and Cliff Reiter of Lafayette.

Deanna and Steve were Assistant Professors of Mathematics at Carleton College in Northfield, Minnesota at the time this was published.

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