Project title:               Number Theory and Probability

Advisor:                     Steven J. Miller: Click here for my schedule.

Pictures for different projects: Click here for pictures page.        LINKS TO TALKS ON SUBJECTS

Greetings. I have general ideas of the topics I want to pursue, but I often don't finalize the projects until I know who is in the group and what their interests are. Thus, you should view these as starting points of a conversation. You can view all my papers from my homepage. You'll find many papers from previous years of SMALL. The articles below are meant to give you a rough sense of my interests, and once you tell me what you find fascinating I'll provide more specific reading.

Project Description:   We will explore many of the interplays between number theory and probability, with projects drawn from L-functions, Random Matrix Theory, Additive Number Theory (such as the 3x+1 Problem and Zeckendorf expansions) and Benfordís law. A common theme in many of these systems is either a probabilistic model or heuristic. For example, Random Matrix Theory was developed to study the energy levels of heavy nuclei. While it is hard to analyze the behavior of a specific configuration, often it is easy to calculate an average over all configurations, and then appeal to a Central Limit Theorem type result to say that a generic systemís behavior is close to this average. These techniques have been applied to many problems, ranging from the behavior of L-functions to the structure of networks to city transportation. For more on the connection between number theory and random matrix theory, see the survey article by Firk-Miller.

Below is a reading list for the 2014 summer SMALL program in Number Theory and Probability. It is important that we do some background reading and have some ideas about what we are going to study before the program begins, as we only have 9 weeks or so. It is thus essential to hit the ground running. Obviously, I'm not expecting you to be able to read and thoroughly master everything on the first pass. A lot of this reading is to give you the flavor of the problems and the methods; many of these results we'll build on and thus it is fine to accept them without fully understanding all the details of the proofs. (Sadly, accepting some things on faith is necessary for short programs in order to make progress; my hope is that you'll be interested enough to pursue the material and learn the additional details.) There are more projects listed than we can study; this is deliberate, so that we can choose projects based on your interests and skill sets and not just on mine. Additionally there may be an advanced L-function project with Professor Tian An Wong; click here for a short description and links to some background material.

For additional projects, see http://www.williams.edu/Mathematics/sjmiller/public_html/projects/index.htm (as well as the project summary sheet at http://www.williams.edu/Mathematics/sjmiller/public_html/projects/projects.pdf)