Matrix ensembles and L-functions

Workshop theme

Supported by the European Commission, Sixth Framework Programme - Marie Curie Conferences and Training Courses - MSCF-CT-2003-503674

This workshop will explore recent interest in utilising random matrix theory to probe the Riemann zeta function and other L-functions. Montgomery and Dyson realised in the 1970s that the zeros of the Riemann zeta function show the same statistical behaviour as the eigenvalues of random matrices, and more recently Katz and Sarnak proposed that the same is true of statistical properties of zeros of L-functions averaged over families. Over the last few years, random matrix theory has provided the first viable conjectures for mean values of zeta and L-functions on the critical line where their complex zeros are believed to lie. To find more such problems where the subjects can be combined to great effect, this gathering of number theorists and random matrix theorists will measure what can be done in random matrix theory against the current interesting problems in number theory.

So far, progress in number theory via random matrix theory has followed the pattern that a rigorous random matrix calculation is made, then the structure of this result leads to a conjecture for the equivalent number theoretic quantity. These various predictions that random matrix theory provides, such as those for averages of L-functions over families, have been established unconditionally . These in turn have lead, for example, to subconvex estimates for special values of L-functions from which the resolution of some long standing problems in number theory and mathematical physics have been achieved. Much more is expected to be achieved along such lines and this is one of the themes of this meeting.

Thus the number theoretical conjectures based on work in random matrix theory have great merit in themselves, but another of the focuses of the conference will be the search for a way to make rigorous the jump from random matrix theory to number theory.

Applications for this workshop are now closed

Timetable

Monday 12th July 2004
11:00 to 11:45	Christopher Hughes Mollified \& amplified moments: Some new theorems \& conjectures	Room 1
12:00 to 12:30	Hugh Montgomery University of Michigan Primes \& pair correlation of zeros	Room 1
14:30 to 15:00	Aleksandar Ivic University of Belgrade On the moments of Hecke series at central points	Room 1
15:00 to 15:30	Matti Jutila University of Turku The twelth moment of central values of Hecke series	Room 1
16:00 to 16:45	Testing random matrix theory vs the zeta zeros	Room 1
17:00 to 17:30	Daniel Bump Stanford University Automorphic summation formulae and moments of zeta	Room 1
17:30 to 18:00	Parametric RMT, discrete symmetries, \& cross-correlations between zeros of L-functions	Room 1
18:45 to 19:30	Dinner at Wolfson Court (Residents Only)	No Room Required

Tuesday 13th July 2004
09:30 to 10:20	Emmanuel Kowalski A survey of elliptic curves	Room 1
11:00 to 11:45	Kannan Soundararajan University of Michigan Extreme values \& moments of L-functions	Room 1
12:00 to 12:30	Andrew Booker Poles of L-functions \& the converse theorem	Room 1
14:30 to 15:20	Peter Sarnak Princeton University Perspectives on L functions and spectral theory	Room 1
16:00 to 16:45	Jonathan Keating University of Bristol Negative moments	Room 1
20:30 to 18:00	Hendrik Lenstra Jr. Universiteit Leiden Escher and the Droste effect	Room 1

Wednesday 14th July 2004
09:30 to 10:20	Douglas Ulmer University of Arizona Introduction to function fields	Room 1
11:00 to 11:45	Nick Katz Princeton University Random matrix theory \& life over finite fields	Room 1
12:00 to 12:30	Eduardo Duenez Symmetry beyond root numbers: a GL(6) example (joint with S Miller)	Room 1

Thursday 15th July 2004
09:30 to 10:20	Fernando Rodriguez-Villegas Japan Atomic Energy Agency & University of Texas at Austin Computing twisted central values of L-functions	Room 1
11:00 to 11:45	David Farmer The geometry of zeros	Room 1
12:00 to 12:30	Ashkan Nikeghbali Zeros of random polynomials \& linear combinations of random characteristic polynomials	Room 1
14:30 to 15:20	Michael Rubinstein University of Waterloo Experiments in number theory \& random matrix theory	Room 1
16:00 to 16:45	Steve Gonek University of Rochester A new statistical model of the Riemann zeta function	Room 1

Friday 16th July 2004
09:30 to 10:20	Alex Gamburd Stanford University Applications of symmetric functions theory to random matrices	Room 1
11:00 to 11:45	Martin Zirnbauer Universität zu Köln Ratios of random characteristic polynomials from supersymmetry	Room 1
12:00 to 12:30	Chantal David Concordia University Vanishing of L-functions of elliptic curves over number fields	Room 1
14:30 to 15:20	Dorian Goldfeld Columbia University Multiple Dirichlet series, an historical survey	Room 1
16:00 to 16:45	Ratios of zeta functions \& characteristic polynomials	Room 1
17:00 to 17:30	Alberto Perelli The Selberg class of L-functions: non-linear twists	Room 1
17:30 to 17:50	Giuseppe Molteni Bounds at s=1 for an axiomatic class of L-functions	Room 1

Speakers

Andrew Booker
Daniel Bump Stanford University
Chantal David Concordia University
Eduardo Duenez
David Farmer
Alex Gamburd Stanford University
Dorian Goldfeld Columbia University
Steve Gonek University of Rochester
Christopher Hughes
Aleksandar Ivic University of Belgrade
Matti Jutila University of Turku
Nick Katz Princeton University
Jonathan Keating University of Bristol
Emmanuel Kowalski
Hendrik Lenstra Jr. Universiteit Leiden
Giuseppe Molteni
Hugh Montgomery University of Michigan
Ashkan Nikeghbali
Alberto Perelli
Fernando Rodriguez-Villegas Japan Atomic Energy Agency & University of Texas at Austin
Michael Rubinstein University of Waterloo
Peter Sarnak Princeton University
Kannan Soundararajan University of Michigan
Douglas Ulmer University of Arizona
Martin Zirnbauer Universität zu Köln

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Workshop theme

Applications for this workshop are now closed

Timetable

Speakers

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