# Fedor Bogomolov (Courant Institute)

**Vendredi 28 septembre 2012, 16h00 / Friday, September 28, 2012, 4:00 pm**

Salle / Room 6214

Centre de recherches mathématiques

Pavillon André-Aisenstadt

Université de Montréal

2920, chemin de la Tour

CETTE CONFÉRENCE S'ADRESSE À UN LARGE AUDITOIRE *SUITABLE FOR A GENERAL AUDIENCE*

**External and Internal Symmetries and their Role in Mathematics and Nature**

People were always fascinated by the presence of symmetry in objects of nature.

The study of different symmetries in cristals and other natural objects has a long history.

However the goal of studying symmetry has changed substantially over the last centuries.

Understanding internal symmetries became one of the main tools to uncover the properties of complicated mathematical and physical objects. This new approach was initiated in number theory with the works of Gauss and Galois and then expanded to algebra and geometry. Einstein's relativity theory demonstrated the power of this approach in physics. Since then it became one of the main tools in understanding the fabric of the Universe.

In my lecture I will discuss some of the discoveries which were obtained using symmetries.

**Une réception suivra la conférence au Salon Maurice-L'abbé, Pavillon André-Aisenstadt (Salle 6245).
A reception will follow at the Salon Maurice-L'abbé, Pavillon André-Aisenstadt (Room 6245).**

Conférences dans le cadre de l'atelier sur la topologie des variétés algébriques (21 au 28 septembre 2012)

*Lectures at the Workshop on the Topology of Algebraic Varieties (September 21-28, 2012)*

**Mardi 25 septembre 2012, 16h00 / Tuesday, September 25, 2012, 4:00 pm**

Salle / Room 6214

Centre de recherches mathématiques

Pavillon André-Aisenstadt

Université de Montréal

2920, chemin de la Tour

## Universal Spaces in Birational Algebraic Geometry

One of the most important ideas in topology is the idea of universal space and universal class for a contravariant functor. In my talk I will discuss our joint results with Yuri Tschinkel which show how this general scheme can be applied to the theory of cohomological birational invariants of projective varieties. Most of the results are currently established for projective varieties defined over finite fields but I am going to discuss also possible extension of the theory to projective varieties defined over fields in zero characteristic.

**Jeudi 27 septembre 2012, 16h00 / Thursday, September 27, 2012, 4:00 pm**

Salle / Room 6214

Centre de recherches mathématiques

Pavillon André-Aisenstadt

Université de Montréal

2920, chemin de la Tour

## Several Aspects of Infinite Transitivity

It was discovered about ten years ago that there are many algebraic varieties with infinitely transitive action of the group of algebraic automorphisms. This class consists of uniratonal varieties only and many unirational varieties are becoming birationally isomorphic to algebraic varieties with infinitely transitive action of the group of algebraic automorphisms after multiplication by an affine space. I am to discuss our joint results with K. Kuyumzhiyan and I. Karzhemanov about this class of varieties in the first part of my lecture. In the second part, I will consider the group of projective automorphisms as a subgroup of the group of permutations of the set of points in the projective space over an arbitrary field. My goal is to sketch a proof of our joint result with M. Rovinsky. We have proved that by adding to the projective group an arbitrary automorphism which maps some three points on a line in the projective space into a set of three non collinear points we generate a group with infinitely transitive action on the set of points of projective space.