Steven J Miller (email address: Steven.J.Miller AT williams.edu): I'm starting at Williams College in Fall '08; click here for some personal information.

Research interests: analytic number theory, random matrix theory, probability and statistics, graph theory. I'm also running the Green Chicken contest.

My main research interest is in the distribution of zeros of L-functions. The most studied of these is the Riemann zeta function, ζ(s) = Σ_{n=1 to ∞} n^-s. The importance of this function becomes apparent when we notice that it can also be written as Π_{p prime} (1 - p^-s)^-1; this function relates properties of the primes to those of the integers (and we know where the integers are!). It turns out that the properties of zeros of L-functions are extremely useful in attacking questions in number theory. Interestingly, a terrific model for these zeros is given by random matrix theory: choose a large matrix at random and study its eigenvalues. This model also does a terrific job describing behavior ranging from heavy nuclei like Uranium to bus routes in Mexico! I'm studying several problems in random matrix theory, which also have applications to graph theory (building efficient networks). I am also working on several problems in probability and statistics, especially (but not limited to) sabermetrics (applying mathematical statistics to baseball) and Benford's law of digit bias (which is often connected to fascinating questions about equidistribution). Many data sets have a preponderance of first digits equal to 1 (look at the first million Fibonacci numbers, and you'll see a leading digit of 1 about 30% of the time). In addition to being of theoretical interest, applications range from the IRS (which uses it to detect tax fraud) to computer science (building more efficient computers). I'm exploring the subject with several colleagues in fields ranging from accounting to engineering to the social sciences.

I have a webpage with additional details and papers related to research projects; I advise you to look at that and then chat with me. A brief summary of items is below.

Possible thesis topics:

• Theoretical models for zeros of elliptic curve L-functions (in the number field and function field cases).
• Studying lower order term behavior in zeros of L-functions.
• Studying the distribution of eigenvalues of sets of random matrices.
• Exploring Benford's law of digit bias (both its theory and applications, such as image, voter and tax fraud).
• Propagation of viruses in networks (a graph theory / dynamical systems problem).
• Additive number theory (questions on sum and difference sets).
• Sabermetrics.
• Statistics (especially concerning Williams data and education)

Possible colloquium topics: same as the above, plus anything you find interesting. I'm also interested in applications, and have worked on subjects ranging from accounting to computer science to geology to marketing....

Some references for thesis topics:

• Number theory and random matrix theory: Chapter 15 of my book and Miller-Dueñez (click here for applications to bus routes in Mexico).
• Sum and difference sets: papers by Nathanson, Martin and O'Bryant and Hegarty and Miller.
• Benford's law: papers by Hill, Raimi and Kontorovich-Miller.
• Sabermetrics: Pythagorean Won-Loss formula and some useful links.

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