Abel Lectures at the University of Oslo

Pierre Deligne:

Hidden symmetries of algebraic varieties

Abstract:
If a complex algebraic variety is defined by equations with rational coefficients, the set of its points whose coordinates are in that is are algebraic numbers, carries an action of the Galois group Gal ( / ), the symmetry group of  .  This, and related symmetries, have a strong influence on the topology of complex algebraic varieties, and a mysterious connection with Hodge theory.

Nicholas Katz

Princeton University:

Life Over Finite Fields

Abstract:
We will discuss some of Deligne's work and its diophantine applications.

Professor Claire Voisin

École Polytechnique and CNRS

Mixed Hodge structures and the topology of algebraic varieties

Abstract:
Deligne proved that the cohomology of a complex algebraic variety carries a functorial "mixed Hodge structure",  which generalizes the previously known results for smooth projective complex varieties. Formal properties of morphisms of Mixed Hodge structures then allowed him to deduce a number of important consequences, including the "theorem of the fixed part".  I will describe these fundamental results, and some applications to the study of Hodge classes.

Science Lecture:

Ravi Vakil

Stanford University

Algebraic geometry and the ongoing unification of mathematics

Abstract:
When we look carefully at nature, we can discover surprising coincidences, which suggest deeper underlying structure.   The centrality of mathematics comes in part from the fact that seemingly unrelated ideas are often unified by some grand theory, which is far more powerful than the sum of its parts.  Mathematics is most exciting when different ideas come together unexpectedly to give a new point of view.  This is typified in algebraic geometry, and in the work of Deligne in particular, which brings together many themes in mathematics, including geometry, number, shape (topology), algebra, and more.  This magic is the reason I became an algebraic geometer.  For example, the theory of Pythagorean triples (such as 3² + 4² = 5² ) connects geometry to the theory of numbers by way of algebra.  This ancient example grows up to be the Weil conjectures, a wondrous prediction whose proof was finally completed by Deligne. 

I will try to share a glimpse of this strange unification of many different ideas.  This talk is aimed at a general audience, and no particular background will be assumed.