Vertically Integrated Summer Program in Computational Mathematics

The Ohio State University, Columbus, OH - Summer 2004







Undergraduates, graduate students, post-docs and faculty will work for 10 to 12 weeks investigating interesting unsolved conjectures theoretically and experimentally. The goal is to help the undergraduates, graduate students and post-docs in making the transition from one level of their mathematical careers to another.

The vertically integrated nature of the program, in which faculty work with post-docs and graduate students, and all three work with undergraduates, will provide valuable training and exposure which is not seen in a typical REU program.


This program is based on guided undergraduate research classes taught at Princeton, NYU and Ohio State over the last four years, as well as a pilot summer program run last year at the American Institute of Mathematics. Previous year's investigations are available online at


After the summer program at AIM, one of the biggest requests by students was for background material. As the students are working on the forefront of mathematics (with mixed backgrounds), the background material is scattered through a variety of sources. A variety of introductory notes (many assuming just calculus) will be made available for students interested in getting a head start.

For the summer program, in addition to introductory lectures, students will immediately begin working on projects on arrival. The projects we plan all have an element which can be investigated on a computer with essentially no preparation. This allows the student to accomplish something tangible on the first day. Throughout the summer we will maintain a mix of computer programming, analyzing data, background reading, and theoretical work.

We expect students to have wildly different backgrounds. Both at the summer program at AIM and the Princeton/NYU/Ohio State classes, we have never had a problem finding problems appropriate to student background. While the techniques which (we hope) will someday prove many modern conjectures are quite sophisticated, often the statement, simple cases and numerical investigations require little more than simple algebra and calculus. Graduate students and post-docs will run series of lectures on the needed background material (linear algebra, probability theory, basic number theory, and so on); the faculty (and probably the post-docs and graduate students) will lecture on more advanced topics and the problems to be investigated.

In addition to mentoring students, graduate students and post-docs will conduct original research, under the supervision of the faculty; these projects will be related to the undergraduate investigations.

It is important that the participants stay focused. Much to our surprise, we found last summer that the students appreciated mandatory weekly presentations. For 5 to 10 minutes, students would lecture to the group on what they accomplished the previous week. The students said this kept them focused and working; we will continue this tradition this year.

The problems will be chosen from Number Theory and related disciplines. We have found it is very useful to have large numbers of students working on related problems, as they are then able to discuss and work together.

The undergraduates have told us that they love doing original computations -- it makes them feel like they own a problem. Thus, while we will suggest projects and avenues (and make sure the students are making good progress), we feel it is important that  the students are co-investigators, helping to choose what to investigate.

At the end of the summer, participants will LaTeX their investigations, which will be published on the web; in addition to announcing their results, these reports will also help future investigators. Also, all students will give a 40 minute talk, as well as introducing another student. Giving talks at conferences / schools is a useful skill, but often undergraduates and graduate students have no preparation before their first presentation. Several days will be spent giving advice; undergraduates and graduate students will also give several practice talks to the post-docs and faculty before giving their final presentation. Students from Princeton, NYU and AIM, while often worrying about their presentations beforehand, have uniformly said it was a very useful experience.

The program is similar in spirit to regular research classes; however, as participants' time will not be split with other classes, we envision significantly greater productivity and results. The goal is to prepare participants for the next level of their mathematical careers; we feel such a vertical environment in the summer is a natural way to do so.

Tentatively, the program will run for a little over two months, starting in late June.


Research problems will be drawn primarily from Number Theory and Random Matrix Theory; depending on participants' interest, we may also investigate some problems in Graph Theory and Probability Theory. For example, below are three types of problems which give a flavor for the types of topics to be investigated. By no means are these complete descriptions, or the only problems. More problems will be listed (with more details) later. If you are not familiar with some of the terms below (such as eigenvalues, for example), don't worry -- background lecture series will be run for the program.

Again, these problems are meant to give a feel for the types of problems we will study. These problems quickly lead to modern theory and techniques, but their statements are straightforward, and they have numerical components that can be investigated quickly. As an exercise, try and prove that it takes  log(a0) terms before you expect to hit the cycle 4, 2, 1, or try and figure out how to deal with such large numbers.



Undergraduates, Graduate Students, Postdocs and Faculty interested in participating in this program should contact Steven Miller at, who will provide more information and application materials. Financial support will be available for some participants.