Oxford

(Note change in time and location)

The purpose of this talk is to give an introduction to the theory and

practice of integer factorization. More precisely, I plan to talk about the

p-1 method, the elliptic curve method, the quadratic sieve, and if time

permits the number field sieve.

Direct products of finitely generated free groups have a surprisingly rich subgroup structure. We will talk about how the finiteness properties of a subgroup of a direct product relate to the way it is embedded in the ambient product. Central to this connection is a conjecture on finiteness properties of fibre products, which we will present along with different approaches towards solving it.

Discrete problems have a habit of being beautiful but difficult. This can be true even of discrete problems whose continuous analogues are easy. For example: computing the surface area of a sphere of radius N^{1/2} in k-dimensional Euclidean space (easy). Counting the number of representations of an integer N as a sum of k squares (historically hard). In this talk we'll survey a menagerie of discrete analogues of operators arising in harmonic analysis, including singular integral operators (such as the Hilbert transform), maximal functions, and fractional integral operators. In certain cases we can learn everything we want to know about the discrete operator immediately, from its continuous analogue. In other cases the discrete operator requires a completely new approach. We'll see what makes a discrete operator easy/hard to treat, and outline some of the methods that are breaking new ground, key aspects of which come from number theory. In particular, we will highlight the roles played by theta functions, exponential sums, Waring's problem, and the circle method of Hardy and Littlewood. No previous knowledge of singular integral operators or the circle method will be assumed.

The class of fields with a given absolute Galois group is in general not an elementary class. Looking instead at abstract elementary classes we can show that this class, as well as the class of pairs (F,K), where F is a function field in one variable over a perfect base field K with a fixed absolute Galois group, is abstract elementary. The aim is to show categoricity for the latter class. In this talk, we will be discussing some consequences of basic properties of these two classes.

Graeme Segal shall describe some of Dan Quillen’s work, focusing on his amazingly productive period around 1970, when he not only invented algebraic K-theory in the form we know it today, but also opened up several other lines of research which are still in the front line of mathematical activity. The aim of the talk will be to give an idea of some of the mathematical influences which shaped him, of his mathematical perspective, and also of his style and his way of approaching mathematical problems.