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)