In a land-mark paper in 1956, J. Milnor showed that there are non standard differential structures on the 7-dimensional sphere. Six years later along with Kervaire, he introduced an abelian group structure on the set of equivalence classes of smooth structures on spheres of all dimension and determined these groups in several cases. We shall present some of the salient features of this work.

In this talk I will prove that the i-th homotopy groups of a sphere S^n are finite when i is greater than n, except in one particular case, using the Serre spectral sequence. In the first half of the talk I will give the background material needed to understand the proof.

This will be a continuation of the overview from the last week. Some details will be briefly recalled from the last time, for continuity and the benefit of new audience if any.

This will be an overview of the results on values of quadratic forms at integer points.

In this talk I will introduce generalities of the interaction between problems in Diophantine arithmetic and dynamics of flows on homogeneous spaces, and set the tone for subsequent lectures.