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.
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.
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.
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
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. You can see last summers
calendar of events (updated on to
be posted soon). The SURIM
group will also have access to various classrooms during the summer:
- 381-T available M-F from 8-5.
- 381-U available T,Th,F from 8-5 and M from 8-11 and W 8-3.
- 383-N available M from 11-5 and W from 3-5.
We will likely divide into three groups. Below are two of 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
(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
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
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.
Please submit the following information, by email, to Nancy Rodriguez at email@example.com, 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.
- (a) Name and year.
- (b) If you have a faculty member who has agreed to work one-on-one with you, please let us know. (This is not necessary to apply.) If this is the case, please include a short proposal, developed in consultation with your intended mentor.
- (c) Name of one or two professors who are familiar with you
(ideally in mathematics)
- (d) Mathematical background and interests.
- (e) For those not working individually with a faculty member, which of the
possible projects appeal to you?
- (f) Do you need funding in order to take part? Would you like course credit? (Note: it is not possible to get both funding and credit.)
- (g) Curriculum vitae and unofficial Stanford transcript.
- (h) Project proposal for those seeking to work one-on-one with a faculty member.
Directors: Gunnar Carlsson and
Rodriguez. Assistant Director: Ravi Vakil.
Mentors: Daniel Litt, Chris Henderson, and Otis Chodosh
Scheduled Speakers: Ravi Vakil, Rafe Mazzeo, Daniel Bump, Brian Conrad, Gunnar Carlsson
If you have any questions, or are even just curious about the program,
please contact Dr. Nancy Rodriguez (firstname.lastname@example.org).
She will also be available to chat during iDeclare week.