9 de Marzo. Instituto de Matematicas de la PUCV, Blanco Viel 596, Cerro Barón, Valparaíso. La sesión consistira en dos charlas, dadas por Philippe Gille (CNRS/ENS) y Alvaro Liendo (U. de Basilea). En cada charla, la primera hora estará dirigida a estudiantes de postgrado y la segunda a un público de investigadores.
Expositor: Philippe Gille (CNRS/ENS)
Hora: 11:00-12:00 y 12:10-13:10
Titulo: Rational points of algebraic groups
For the linear group GL_n, the Bruhat decomposition provides a
parametrization of GL_n(k) for an arbitrary field k. Additionally,
we know that the quotient of the group SL_n(k) by its center is
simple when k is of cardinal larger than 4.
For an algebraic group G/k, a natural question is to ask whether nice
parametrizations of G(k) exist and also to find simplicity criterions
for the group G(k) modulo its center.
For a good class of algebraic groups, the Kneser-Tits problem links the
two questions. More precisely, the problem is to find a criterion
deciding whether the group G(k) is generated by additive algebraic
elementary matrices. After discussing the state of the Kneser-Tis problem,
the next issue is to investigate possible extensions
for all reductive groups.
Expositor: Alvaro Liendo (U. de Basilea)
Hora: 14:30-15:30, 15:40-16:40
Título On normal T-varieties.
A T-variety is a normal variety endowed with a faithful action of the
algebraic torus T=(k^*)^n. The complexity of a T-variety is the
codimension of a generic orbit. The best known examples of T-varieties
are toric varieties, i.e., T-varieties of complexity 0. Toric
varieties have a nice combinatorial description in terms of certain
collections of polyhedral cones, called fans.
In the first part of this talk, we present a combinatorial description
of T-varieties given by Almann, Hausen and Süss in 2008. This
description generalizes the usual description of toric varieties to
higher complexity in a natural way.
After the result by Almann, Hausen and Süss, the theory of T-varieties
has quickly developed by generalizing results known for toric
varieties to this more general setting. In the second part of this
talk, we give a survey about the theory of T-varieties. We restrict to
the case of rational T-varieties of complexity one for simplicity.