Stanford Undergraduate Research Institute in Mathematics

June 23-August 29, 2014

The Stanford Undergraduate Research Institute in Mathematics is a ten-week program that provides Stanford undergraduates the opportunity to work on mathematical problems in an extra-curricular context. Most students will work on interesting mathematical problems in a collaborative environment. A number will work one-on-one with faculty member. Summer funding will be available for some students, thanks to VPUE; others can obtain course credit in the fall quarter for participating. You can find the SURIM 2012 website here and SURIM 2013 website here .

Individual Research with a Faculty Member

Students working individually with a faculty member will decide on a project and the dates in consultation with their faculty mentor. Note that it is the duty of the student to find a faculty member interested and willing to work with them. A short project proposal will be requested with the application.

Collaborative Research

The remaining students will take part, full-time, in the ten-week program that will run from Monday, June 23 through Friday, August 29.

Goals of the program

At SURIM, students will be exposed to questions that are of interest in current mathematics, as well as the research and exploration aspects that accompany such questions. With their mentor's assistance, students will study the prerequisite materials to understand their program's topic and will then participate in exploration of their questions about the subject. The emphasis will be on self-discovery of examples and properties. In addition to knowledge of their subject and an understanding of what it means to explore a research question, participants will practice the ability to present mathematics in a formal seminar setting, use software such as LaTeX to typeset mathematics, use other programming languages to study mathematical questions, and interact with peers, graduate students and faculty.

Eligibility

All Stanford students who will be enrolled full/part-time during the Fall of 2014 are eligible to apply.

Format for the ten-week program

Students will be divided into groups depending on their mathematical interest and background. Each group will work closely with graduate students.

A typical week

There will be a couple of formal meetings with mentors each week. At the start, the mentors will lay out the beginning of the project, and the groups will decide how best to begin. Each group will prepare presentations to the entire institute each week, giving a status report to those working on other problems. (Practice with getting across ideas is essential to doing mathematics!)

Much of the week will be spent working individually and in groups, and in informal discussions with mentors.

There will be roughly two additional events per week. Some will be introductions to research tools (from writing with LaTex to the use of various software packages). Others will be lectures from researchers in academia and industry on what research is actually about --- how it is done, how to do it, and what it is like. Please check back for the schedule. The SURIM group will also have access to various classrooms during the summer, which will be listed later in the year.

Stanford-Berkeley Joint Conference

We expect to be holding a joint conference with the Berkeley REU program. Details to come later.

Projects

We are still working on choosing the projects for the summer. Below are two of the projects students will be working in 2014.

Random Walks On Finite Groups - Card Shuffling (mentor: Evita Nestoridi)

One of the main questions concerning a random walk on a finite group is finding the order of the mixing time of the walk. In particular, in card shuffling we are really interested in finding out how many shuffles are required to get the deck "perfectly" shuffled. In this project, we are going to learn techniques of bounding the mixing time and play with a lot of examples. We will try to actually solve particular problems-examples either from card shuffling or from a group of matrices over a finite field (or perhaps another finite group that we might find interesting) and ideally come up with new techniques for bounding the mixing time.

What happens when we iterate polynomials on the projective line? (mentor: Niccolo' Ronchetti)

Suppose you take a polynomial f(z), say with rational coefficients. You can try to iterate the polynomial: z -> f(z) -> f(f(z)) -> ? What happens to the orbit of some z? Is it periodic, or maybe dense? Does it contain infinitely many primes? More generally what arithmetic properties do the set of periodic points have? There are plenty of interesting questions that one can ask and try to figure out. We will try to explore them and learn plenty of exciting math in the process (for example we'll learn about Galois groups, Diophantine problems, local fields?)

What is the shape of molecule space? Can we develop topological OCR? (mentor: Ryan Lewis) Topological data analysis attempts to extract a topological understanding of scientific data from finite sets of samples. Usually data analysis assumes that the input is a point cloud and comes from some underlying geometric space. Topological data analysis focuses on the recovery of the lost topology of this underlying space. For this project we are looking for students to do topological data analysis. We will use a new computational topology library to analyze data sets. For those less interested in using and writing software, more mathematical problems can be solved.

Application

Please submit the following information, by email, to Nancy Rodriguez at nrodriguez@math.stanford.edu, by March 1, 2014 with subject line ''SURIM application''. Please include in the email the following information: