Special Year 2009-10: Analytic Number Theory

A result of Kim-Sarnak (2003) gives the best known bounds towards the Ramanujan conjecture for Maass forms. The technique employed has not, until now, been made to apply to general GL2 cusp forms over number fields whose unit group is infinite. In...

A bounded Apollonian circle packing (ACP) is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four...

For the last 5 years or so Terry Tao and I have been working on a programme to prove certain conjectures of Hardy and Littlewood concerning the number of primes vectors p = (p_1, . . . ,p_n) in some box which satisfy the equation Ap = b . The number...

In the first half of the talk I will give details of my joint work with Henryk Iwaniec. We use half-dimensional sieve to obtain a lower bound for the density of rational points on the cubic Chatelet surface. The cubic Chatelet surface can also be...

An overview of methods and results concerning small gaps between primes, between almost primes, and the consequences on values taken by certain arithmetical functions will be presented. The material is from the works with Goldston, Graham and Pintz...