Geometry working seminar

Wednesday, July 9, 2014 1:00 pm - 1:00 pm EDT (GMT -04:00)

Talk 1: Mateusz Olechnowicz - 1:00 pm

Pure Mathematics Department, University of Waterloo

"Group schemes"

The Mordell-Weil theorem says that if $E$ is an elliptic curve over $\mathbf{Q}$, then the abelian group $E(\mathbf{Q})$ is finitely generated. Mazur's theorem then pinpoints the torsion part of $E(\mathbf{Q})$ as one of fifteen finite groups. Central to Mazur's proof is the notion of a group scheme, which is a group object in a category of schemes. In this talk we will define group objects and give examples in various categories. We will also discuss affine groups and see how they can be realized as group schemes."

Talk 2: Ritvik Ramkumar - 2:30 pm

Pure Mathematics Department, University of Waterloo

"The Hodge Theorem: Part III"

In this final talk we will discuss some applications of the Hodge Theorem. The main application will be the Hodge decomposition for compact Kahler manifolds. A hermitian manifold is said to be Kahler if the associated (1,1) form is d-closed. It turns out that this condition is enough to relate the various operators corresponding to d,\partial, \bar(\partial) together. This will then enable us to simplify a lot of relations obtained by the Hodge Theorem. We will give examples of Kahler manifolds, prove the Hodge decomposition and use it to discuss the structure of Kahler manifolds.