C5

A finite set of elements in a connected real Lie group is "Diophantine" if non-identity short words in the set all lie far away from the identity. It has long been understood that in abelian groups, such sets are abundant. In this talk I will discuss recent work of Aka; Breuillard; Rosenzweig and de Saxce concerning this phenomenon (and its limitations) in the more general setting of nilpotent groups. 

In the first part of the talk we are going to build up some intuition about limit-periodic functions and I will explain why they are the 'simplest' class of arithmetic functions appearing in analytic number theory. In the second part, I will give an equivalent description of 'limit-periodicity' by using exponential sums and explain how this property allows us to solve 'twin-prime'-like problems by the circle method.

This talk will discuss the discovery of Heegner points from a historic perspective. They are a beautiful application of analytic techniques to the study of rational points on elliptic curves, which is now a ubiquitous theme in number theory. We will start with a historical account of elliptic curves in the 60's and 70's, and a correspondence between Birch and Gross, culminating in the Gross-Zagier formula in the 80's. Time permitting, we will discuss certain applications and ramifications of these ideas in modern number theory. 

We define the notion of orbit decidability in a general context, and descend to the case of groups to recognise it into several classical algorithmic problems. Then we shall go into the realm of free groups and shall analise this notion there, where it is related to the Whitehead problem (with many variations). After this, we shall enter the negative side finding interesting subgroups which are orbit undecidable. Finally, we shall prove a theorem connecting orbit decidability with the conjugacy problem for extensions of groups, and will derive several (positive and negative) applications to the conjugacy problem for groups.