Parametric geometry of numbers and diophantine approximation

The first goal of the talk will be to explain the general principles of parametric geometry of numbers, which is a way to describe combinatorially, up to a bounded error, any one-parameter diagonal orbit in a space of lattices. Then we shall discuss applications of these ideas to diophantine approximation in flag varieties, using a general version of Dani's celebrated correspondence. (Part of the talk will be based on an on-going project with Emmanuel Breuillard, Omri Solan and Nattalie Tamam.)