JHU Number Theory Seminar, Spring 2007

Small Gaps Between Primes: The Ideas of Goldston-Pintz-Yildirim

The study of small gaps between prime numbers is motivated by the twin-prime conjecture that there exist infinitely many primes p such that p+2 is also prime. Recently, D. Goldston, J. Pintz and C. Yildirim achieved a breakthrough in the field and proved results which had previously been considered beyond reach. In this talk, I will survey the history of the problem and some of the techniques used to study it. I will then go on to sketch the Goldston-Pintz-Yildirim method.

Maass forms and generalizations of Dyson's rank

George Andrews recently defined a generalization of partitions called Durfee symbols, and found that these objects possess some striking combinatorial properties. In particular, he proved that the Durfee symbols satisfy linear congruences for the primes 5 and 7 that are very similar to those that Ramanujan famously proved for partitions. Furthermore, he also found a generalization of Dyson's rank statistic that decomposes these congruences combinatorially, just as the original rank did for the partition congruences. Work of Bringmann and Ono used the theory of Maass forms to show that the rank has a much deeper relation to partition congruences than first suspected, and in fact satisfies nearly the same sort of congruences. Their work is a special case of the main result of this talk, which shows that there is a similar infinite framework of congruences for Durfee symbols and the full rank.

On mass equidistribution for the Siegel modular variety

If f(Z) is a holomorphic Siegel modular form of weight k which is a simultaneous eigenfunction for the Hecke operators, analytically normalized so that its L-2 norm is 1, then we can associate to it a probability measure on the Siegel modular variety. The mass equidistribution conjecture states that as the weight increases these measures approach the normalized volume form on the variety, that is, the mass of the measures become equidistributed. This conjecture grew out of questions in quantum chaos and quantum ergodicity (work of Zelditch, etc.) which were transferred to the arithmetic situation through the work of Sarnak and his collaborators and became of interest to number theorists, particularly with their connection to triple product L-functions. Recently, in joint work with Wenzhi Luo, we have found an approach to mass equidistribution ``on average'', so after one averages over all forms of a given weight, which relies only on elementary estimates on the Bergman kernels for the Siegel modular variety. This method also works for Hilbert modular varieties (work of S-C. Liu). In this talk I would like to explain a bit of background and then our proof of mass equidistribution on average in the Siegel modular case.

TBA

Large values of eigenfunctions on arithmetic hyperbolic manifolds

Extremal behavior of high-energy eigenfunctions on Riemannian manifolds is not well understood and depends heavily on their global geometry. For hyperbolic manifolds, this problem is part of understanding the so-called quantum chaos. In this talk, we will present recent omega results for $L^{\infty}$-norms of Hecke-Maass eigenforms on certain arithmetic manifolds. These are achieved by constructing two spectral averages involving twists of the pre-trace formula by various Hecke operators, which brings up the question of optimizing weights to maximize the quotient of certain two quadratic forms similarly to the recent method of resonators in the context of $L$-functions.