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Random matrix theory has been a hot topic in number theory, particularly since the Rudnick and Sarnak landmark work on the spacing of consecutive zeros of L-functions. This highly accessible talk has a far more elementary flavour, focusing on eigenvalues of random integer matrices instead of the Gaussian Unitary Ensemble. For a fixed n, consider a random n×n integer matrix with entries bounded by the parameter k. I'll give a simple proof that such a matrix almost certainly has no rational eigenvalues (as k increases). Then we'll delve into more detail on the exact eigenvalue distribution of the 2×2 case. Along the way we'll rediscover a forgotten determinant identity and tackle some quadruple sums. This is joint work with Greg Martin.

In recent years the application of random matrix techniques to analytic number theory has been responsible for major advances in this area of mathematics. As a consequence it has created a new and rapidly developing area of research. The aim of this book is to provide the necessary grounding both in relevant aspects of number theory and techniques of random matrix theory, as well as to inform the reader of what progress has been made when these two apparently disparate subjects meet. This volume of proceedings is addressed to graduate students and other researchers in both pure mathematics and theoretical physics. The contributing authors, who are among the world leading experts in this area, have taken care to write self-contained lectures on subjects chosen to produce a coherent volume.• Self-contained lectures by world-leading experts in the field • The volume is integrated, indexed and cross-referenced • This title covers the most important and recent advances in the subjectContents1. Introduction; 2. Prime number theory and the Riemann zeta-function; 3. Notes on pair correlation of zeros and prime numbers; 4. Notes on eigenvalue distributions for the classical compact groups; 5. Compound nucleus resonances, random matrices and quantum chaos; 6. Families of L-functions and 1-level densities; 7. Basic analytic number theory; 8. Applications of mean value theorems to the theory of the Riemann zeta function; 9. L-functions and the characteristic polynomials of random matrices; 10. Mock gaussian behaviour; 11. Some specimens of L-functions; 12. Computational methods and experiments in analytic number theory.

References: Cipra, B. 1998. A prime case of chaos. In What's Happening in the Mathematical Sciences, Vol. 4. Providence, R.I.: American Mathematical Society. (Available at http://www.ams.org/new-in-math/happening.html.) ______. 1996. Prime formula weds number theory and quantum physics. Science 274(Dec. 20):2014. Davis, P.J., and R. Hersch. 1981. The Mathematical Experience. New York: Viking Penguin. Katz, N.M., and P. Sarnak. 1999. Zeroes of zeta functions and symmetry. Bulletin of the American Mathematical Society 36(January):1. Peterson, I. 1995. Cavities of chaos. Science News 147(April 29):264. Richards, I. 1978. Number theory. In Mathematics Today: Twelve Informal Essays. L.A. Steen, ed. New York: Springer-Verlag. Peter Sarnak's lecture on random matrix models in number theory and quantum mechanics is available at http://www.msri.org/publications/video/fall98/mandm.html. Andrew Odlyzko's Web page at http://www.research.att.com/~amo/ features computations of the zeros of the zeta function.

The Riemann hypothesis was first proposed in 1859 by the German mathematician Georg Friedrich Bernhard Riemann (1826-1866). It concerns the so-called zeta function, which encodes a great deal of information about the seemingly haphazard distribution of prime numbers among the integers (see The Mark of Zeta, June 19, 1999).

The connections between the zeros of the zeta-function and random matrix theory have become the most active and exciting threads of research in the hunt for the Riemann hypothesis. Rockmore devotes four chapters at the end of his book to various aspects of this research. He discusses the work of Sarnak and Katz on analogous results for function fields. He also discusses work of Tracy, Widom, and Deift that connects the distribution of eigenvalues of random matrices to properties of permutations. This chapter has the engaging title "God May Not Play Dice, but What About Cards?"