Previous Special Year Seminar

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...

In this mini-course we give a quick tour of sieve methods. The first lecture will deal with the rudiments of the basic theory and mention a few simple examples. The second talk will, for the most part, feature some more recent applications. The...

In this mini-course we give a quick tour of sieve methods. The first lecture will deal with the rudiments of the basic theory and mention a few simple examples. The second talk will, for the most part, feature some more recent applications. The...

A sequence of primes $p_1, \dotsc , p_k$ is a prime chain if $p_{j+1} \equiv 1 \pmod{p_j}$ for each $j$. For example: 3, 7, 29, 59. We describe new estimates for counts of prime chains satisfying various properties, e.g. the number of chains with $p...

We improve Kolyvagin's upper bound on the order of the p-primary part of the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely the one predicted by the Birch and...