# Stanford Undergraduate Research Institute in Mathematics

## 2014

**Random Walks On Finite Groups - Card Shuffling** (Kevin Garbe, Randy Jia, Joesph Shayani, mentor: Evita Nestoridi)
Report
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?
** (Zhiming Wang, Robin Zhang, mentor: Niccolo' Ronchetti)
Report

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?** (James Allen, Amr Mohamed, 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.

** Errors in Prime k-tuples Sieves** (Grant Sanderson, mentor: Brian Conrad) Report

** Bounding the Prime Gaps ** (Vishal Arul, mentor: Kannan Soundararajan) Report

** The Bijective Correspondence Between Standard Young Tableaux Pairs and Irreducible Representation of S_n ** (Peng Hui How, mentor: Daniel Bump)

** Random Walks on Random Graphs ** (Dante Vela, mentor: Georg Menz)

** Wavelets and Applications** (Ahmed Bou-Rabee, mentor: Lexing Ying)