Special Year 2009-10: Analytic Number Theory

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...

The Schmidt subspace theorem is a deep generalization of Roth's theorem in Diophantine approximation to the setting of hyperplanes in projective space. Another well-known generalization of Roth's theorem is the theorem of Wirsing, which extends Roth...

I will discuss several results on mean-values of multiplicative functions, notably Halasz's method for estimating such averages. In the second lecture I will show how this circle of ideas can be used to derive "weak subconvexity" bounds for L...

I will discuss several results on mean-values of multiplicative functions, notably Halasz's method for estimating such averages. In the second lecture I will show how this circle of ideas can be used to derive "weak subconvexity" bounds for L...