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:

- 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 3-5 and W from 11-3.

**"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.

- (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.

Mentors: Daniel Litt, Chris Henderson, and Otis Chodosh

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