Project title:               Number Theory and Probability

Advisor:                     Steven J. Miller

SMALL 2011 Video: low quality click here          high quality version on youtube here

Pictures for different projects (if you want to see some general SMALL pics, such as the soccer game with physics, click here).

Also, click here for pictures from Bubbles with the Geometry Group from Sundays @ 6.

Click here for Joust Friends.

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 More Sum Than Difference sets) and Benford’s 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 system’s 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 2011 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.

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)