The origin of Hodge theory goes back to many works on elliptic, abelian and multiple integrals. In this lecture series I am going to explain how Lefschetz was puzzled with the computation of Picard rank and this led him to consider the homology classes of curves inside surfaces. This was ultimately formulated in Lefschetz (1,1) theorem and then the Hodge conjecture which is one of the millennium problems of Clay Mathematical Institute. The Hodge theory of hypersurfaces and in particular Fermat varieties is emphasized.

Lecture 1:

14 June Room E. Picard (1R2-129) 10:30.

Title: Lefschetz puzzle and Picard’s formula

Lecture 2:

16 June Room E. Picard (1R2-129) 13:30

Title: Lefschetz theorems on the topology of smooth projective varieties and Picard-Lefschetz theory

Lecture 3:

30 June Room J. Cavailles (1R2-132) 13:30.

Title: Toward a computational proof of Lefschetz (1,1) theorem.

Lecture 4:

05 July Room E. Picard (1R2-129) 13:30.

Title: Griffiths theorem on the cohomology of hypersurfaces.

Lecture 5:

07 July Room E. Picard (1R2-129) 13:30

Title: Hodge cycles for the Fermat variety.