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.

**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.

- (a) Name, Stanford ID number, 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? (we are still working on project 3…if geometry and topology intestes you state that on your application)
- (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.
Notification Deadline: Students will be notified of their acceptance
by March 15, 2014.
Directors: Gunnar Carlsson and Nancy Rodriguez. Assistant Director: Ravi Vakil.## People

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.## Questions?