Stanford Undergraduate Research Institute in Mathematics

June 24-August 30, 2013

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.

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 adviser. 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 24 through Friday, August 30.

Goal 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 2013 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. See the tentative calendar of events. The SURIM group will also have access to various classrooms during the summer:

Stanford-Berkeley Joint Conference

This summer we will also be holding a joint conference with the Berkeley REU program. The conference will be held Wednesday, July 31. The schedule can be found here.

Projects

Below are the three projects students will be working on this summer!

"Listening to Polygonal Drums: Theory and Numerics"

The goal of this project will be to study the spectrum of polygons (and potentially other domains) in the plane. The spectrum of a region are the possible frequencies that a drum of that shape would produce. Translated into mathematical language, this involves the study of the eigenvalues of the Laplacian, a simple partial differential operator with an incredibly rich history. The project will consist of two parts: (1) We will study/develop the necessary mathematical background/theory in order to understand the spectral problem. The exact topics will depend on the participant's background, but knowledge of multivariable calculus and ODEs at the level of the MATH 50 series should be sufficient. The topics will include basic partial differential equation theory and basic functional analysis in order to understand the eigenvalue problem. More advanced topics can include classical first (and higher) eigenvalue estimates, such as the Rayleigh--Faber--Krahn inequality, which states that the fundamental tone (the lowest tone/eigenvalue) of a domain is no less than the tone corresponding to a round ball of equal size, and/or Weyl's formula for the asymptotics of the eigenvalues. (2) We will model the eigenvalue problem on a computer. This will likely involve learning the so called "Finite Element Method," and implementing it to study a problem related to those discussed in (1), of the group's choice. No prior programming experience will be expected.

Rational Functions with Integer Coefficients

Let f(x)=p(x)/q(x) be a rational function, where p, q are polynomials with integer coefficients.  The coefficients of the power series expansion of f(x) about zero have a simple description via a linear recurrence.  For this project we will study the coefficients of such power series--for example, is there a good bound on the greatest integer n so that the coefficient of x^n is equal to zero?  (This question is open in general, and is likely quite difficult!)  What does a "random" such power series look like?  What about analogues over e.g. finite fields or other rings?  This project can include input from p-adic analysis, the representation theory of the symmetric group, Galois theory, or complex analysis, depending on the interests of the students involved--there are also many interesting numerical questions that can be answered by computer experimentation.

Stochastic Processes: Theory and Applications

The goal of this project will be to study the properties of random processes and their connections to partial differential equations and real world systems.  We will focus on discrete time and space martingales and Markov chains (with a view towards continuous time and space).  The project will consist of first developing the necessary mathematical background.  From there, it can go in many directions including, but not limited to, exploring questions and conjectures made by the participants and applications to partial differential equations, finance, or algorithms.  The program should be self-contained but the participants should have a background in the Math 50's series and a solid foundation in proof-based math.  Some simple programming could be useful for modeling but is not required.

Application

Please submit the following information, by email, to Nancy Rodriguez at nrodriguez@math.stanford.edu, by March 1, 2013 with subject line ''SURIM application''. Please include in the email the following information: Notification Deadline: Students will be notified of their acceptance by March 15, 2013.

People

Directors: Gunnar Carlsson and Nancy Rodriguez. Assistant Director: Ravi Vakil.

Mentors: Daniel Litt, Chris Henderson, and Otis Chodosh

Questions?

Scheduled Speakers: Ravi Vakil, Rafe Mazzeo, Daniel Bump, Brian Conrad, Gunnar Carlsson

Questions?

If you have any questions, or are even just curious about the program, please contact Dr. Nancy Rodriguez (nrodriguez@math.stanford.edu). She will also be available to chat during iDeclare week.