Project title:               Number Theory and Probability

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 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.

• Hayes: The Spectrum of Riemannium: a light description of the connection between random matrix theory and number theory (there are a few minor errors in the presentation, basically to simplify the story). This is a quick read, and gives some of the history.
• Conrey: L-functions and Random Matrix Theory: This is a high level description of the similarities between number theory and random matrix theory.
• Katz-Sarnak: Zeros of Zeta Functions and Symmetry: Another high level article similar to the others.
• Diaconis: Patterns in Eigenvalues: this is a bit more readable than the others, and is based on a distinguished lecture he delivered.
• Miller and Takloo-Bighash: An Invitation to Modern Number Theory: This is the textbook I and a colleague wrote, based on years of supervising undergraduate research classes. I know several of you already have a copy -- it will be a good resource for the summer, as a lot of the background material we need is readily available here. Particularly important chapters for us are:
• chapter15 (which discusses the connections between random matrix theory and number theory, and is available online);
• chapter 18 (which does the explicit formula for Dirichlet characters);
• chapter 3 (which reviews L-functions and is also online).
• Firk and Miller: Nuclei, primes and the Random Matrix connection: a survey paper on the history of the subject, including both the nuclear physics experiments and the theoretical calculations.
• L-functions
• Iwaniec-Luo-Sarnak: Low lying zeros of families of L-functions: This is a must read. This is the first major paper calculating the 1-level density for families of L-functions. We will not need much of the paper, but you must read section 1, skim the beginning of section 2 (section 2 is devoted to developing good averaging formulas for specific families; as we will be looking at different families, we won't need this), skim section 3 (same comments apply), section 4 (very important: here they prove the explicit formula relating sums over zeros to sums over primes, and while we will use the explicit formula for different families, the calculation will follow similarly), section 5 (just skim, same caveats as before).
• Conrey-Snaith: Applications of the L-functions Ratios Conjecture: This is a very recent conjecture which is enjoying remarkable success in predicting answers. There is a lot I'll say about this during the summer. I somewhat jokingly call it the conjecture of the four lies, as there are five steps and four of the steps are provably wrong (ie, the assumptions in those steps fail); however, miraculously, all the errors seem to cancel to phenomenal level! I've become very interested in testing this conjecture as much as possible, and have written several papers in this (and have ideas for a few more which will be very accessible to us). Read through page 17.
• Rubinstein: Low lying zeros of L-functions and Random Matrix Theory: this is his dissertation, and in it he analyzes the 1-level density of the family of quadratic Dirichlet characters, and shows agreement with Random Matrix Theory. This is one of the easiest families to look at, and a great testing ground.
• Hughes - Miller: Low-lying zeros of L-functions with orthogonal symmetry: this paper finds a more tractable version of the Katz-Sarnak determinantal expansion for the n-level density, but for restricted support.
• Miller: A symplectic test of the L-functions Ratios Conjecture: This paper builds on those by Conrey-Snaith and Rubinstein and uses the Ratios Conjecture to predict the lower order terms up to square-root cancellation, and then shows (for suitable test functions) that this is the correct answer. An obvious project is to generalize this test for other families or to enlarge the support.
• Miller: Extending the support for families of Dirichlet characters (work in progress): This is a project I've been working on occasionally over the past few years. I have studied the 1-level density for Dirichlet characters, and subject to some reasonable assumptions I can extend the support and smash all world records. Is it possible to do this unconditionally? I don't know, but I hope so! I would love to work on this during the summer; we will almost surely need the results in the primes in arithmetic progression section below.
• Graph Theory
• McKay: Eigenvalues of Large Random Graphs: This readable paper calculates the density of eigenvalues for d-regular graphs; the answer is different than the semi-circle Wigner found for the family of all real symmetric matrices. Excellent projects are finding the density of states for special sets of matrices.
• Jackobson-(SD)Miller-Rivin-Rudnick: Random graphs: Numerics about the neighbor spacings of d-regular graphs.
• Womald: Models of Random Graphs: various ways to generate random graphs.
• Graph Theory (Virus Propagation)
• Email me for working notes from last summer.
• Random Matrix Theory
• Benford's law of digit bias
• Hill (general theory)
• Raimi (survey)
• Kontorovich-Miller (connection to L-functions, random matrix theory and 3x+1).
• Miller-Nigrini:
• Brown University Independent Study Group: Chains (applications to Bayesian theory, related to products) (very important read!)
• One of my summer goals is to understand this paper by Lemons. The general idea is quite interesting, but the paper has some leaps and needs to be put on a rigorous foundation. We made much progress in a continuous version last summer, and the goal is to do the discrete case this summer.
• Additive number theory (MSTD sets)
• Additive number theory (Generalized Zeckendorf Decompositions)
• Zeros of Random Polynomials

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)