# Overview

Number theory enjoys a privileged position within Mathematics as a fertile source of fundamental questions. Among the seven Millenium problems listed by the Clay Institute, not less than three—the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, and the Riemann hypothesis—were handed down by the Queen of Mathematics. Even by the standards of a subject which has remained vibrant since the days of Fermat and Gauss, the last two decades have witnessed a real golden age, with landmarks too numerous to list completely, such as the striking progress on the Birch and Swinnerton-Dyer conjecture arising from the work of Gross-Zagier, Kolyvagin, and Kato; the proofs of the Shimura- Taniyama-Weil conjecture, Serre’s conjectures, the Fontaine-Mazur conjecture for two-dimensional Galois representations, and the Sato-Tate conjectures which grew out of Wiles’ epoch-making proof of Fermat’s Last Theorem; the revolutionary ideas of Bourgain and Gowers blending techniques in harmonic analysis and additive combinatorics, the Fields- medal winning breakthrough of Green and Tao on primes in arithmetic progressions, and the work of Goldston, Pintz, and Yildirim, and its spectacular recent strengthenings by Zhang, and Maynard and Tao, on bounded gaps between primes. Number theory has also spurred the growth of a wide array of new techniques, from p-adic Hodge Theory to additive combinatorics and probabilistic methods. The goal of the Thematic Year 2014-2015 will be to take stock of the most recent developments to emerge from this prolonged spate of activity.

One of the most dynamic and penetrating emerging themes, which will be the most important focus of the the special year, will be on the exciting topic of counting arithmetic objects. Led by Bhargava's pioneering work, this has lead to new and important results on counting elliptic curves with small rank, fields of certain galois types, counting points on families of higher genus curves, etc. The summer school will introduce many of the finest junior mathematicians to these ideas, lectured by many of the key players in the field. There will be an advanced workshop in November, and we hope to have some of the top people from this area in residence for part of the special year.