The London-Paris Number Theory Seminar

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by a grant from the London Mathematical Society, and ANR Projet ArShiFo ANR-BLAN-0114.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev.

Paris organizers: Pierre Charollois, David Harari, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

Sixteenth meeting, London, 9/06/14.

The 16th meeting of the LPNTS took place in London, at Imperial College, on the 9th of June.

The schedule:

1200-1300: Farrell Brumley (U. Paris 13): "Sup norms of automorphic forms"
1300-1500: Lunch
1500-1600: Harald Helfgott (CNRA, Ecole Normale Superieure): "The ternary Goldbach problem"
1600-1630: Coffee
1630-1730: Andrew Granville (U. de Montreal, Cambridge U): "The pretentious Riemann Hypothesis and the short proof of Linnik's Theorem"


Brumley: There has been a lot of recent activity on the problem of estimating the sup norm of L^2 normalized automorphic forms. Most attention has been devoted to the compact setting, where there are firm conjectures to work with. In recent preprint, N. Templier and I have explored the question of what happens without the hypothesis of compacity. Cusp forms on non-compact finite volume locally symmetric spaces decay rapidly at infinity, but before doing so, they evince a sort of automorphic Gibbs phenomenon, attaining their largest value before dying. We are able to quantify this for SL_n(Z) and find bounds on a different scale in comparison with the compact setting.

Helfgott: The ternary Goldbach conjecture (1742) asserts that every odd number greater than 5 can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant C satisfies the conjecture. In the following decades, there was a succession of results reducing C, but only to levels much too high for a verification by computer up to C to be possible (C > 10^{1300}). (Works by Ramare and Tao solved the corresponding problems for six and five prime numbers instead of three.) My recent work proves the conjecture. We will go over the main ideas in the proof.

Granville: In this talk we will discuss several recent developments in our understanding of the basic questions of analytic number theory. Based on the work of many authors, Sound and I have been proposing, for the last few years, a coherent version of analytic number theory which makes no use of the zeros of L-functions. Until recently there were several lacunae in this theory, but now, based on ideas of Koukoulopoulos and Harper, we are able to achieve the same (or better) results than the "Riemann Theory". We shall present here a suitable reformulation of the Riemann Hypothesis, as well as indicate a new proof of Halasz's Theorem, leading to a surprisingly easy proof of Linnik's Theorem.

Previous few meetings:

Tenth meeting (London, 01/06/2011)
Eleventh meeting (Paris, 21/11/2011)
Twelfth meeting (London, 30/05/2012)
13th meeting (Paris, 22/10/2012)
14th meeting (London, 3,4/6/2013)
15th meeting (Paris, 18/11/13)

This page is maintained by Kevin Buzzard.