John Baez
The Octonions
The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.
Pierre Py:
Bounded cohomology and Complex hyperbolic geometry
I will provide an introduction to bounded cohomology, for manifolds and groups. This theory, which received its impetus from the work of Gromov in the 80s, has many applications in differential geometry, especially in the context of negatively curved spaces. As an application, we will prove a classical theorem of Toledo on "maximal" representations of surface groups into the group PU(n,1) of isometries of complex hyperbolic space.
Pierre Bayard:
Introducción a la geometría espinorial
Presentaremos una breve introducción a la geometría espinorial: empezaremos con la descripción de las álgebras de Clifford, de los grupos Spin(n) y de sus representaciones, continuaremos con la noción de conexión en un haz principal y en sus haces asociados, y construiremos finalmente los haces de espinores y el operador de Dirac sobre una variedad espinorial; presentaremos algunas aplicaciones a la geometría de las subvariedades.