Talk

The irrationality measure of a real number characterizes its Diophantine approximation property. The problem of counting the number of approximates to a given real number with respect to certain approximation order has been considered by a number of authors such as S. Lang, W. Adams,... We propose a local version of this problem and show that in certain cases the approximates are uniformly distributed, by using standard tools from analytic number theory.

A question going back to Serre asks which groups arise as fundamental groups of smooth complex projective varieties, or more generally, compact Kaehler manifolds.

By the slope method in Arakelov geometry, we can construct a
family of hypersurfaces which cover the rational points of bounded height on
an arithmetic variety, but do not contain the generic point of this variety.
By estimating some invariants of Arakelov geometry, we can control the
number and the maximal degree of this family of auxiliary hypersurfaces
explicitly. In this talk, I will explain the method of studying the problem
of counting rational points by using methods from Arakelov geometry.

I will talk about the classification of tensor ideals in the Deligne categories attached to the symmetric, orthogonal and general linear group and give some applications.