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Topology/Geometry Seminar
Thursday, November 3, 2005
5:00pm,
106 McAllister
Mathieu Stienon
Penn State University
Non abelian differential gerbes, I
Abstract: Gerbes and stacks are objects of algebraic geometry that were introduced by Grothendieck in the 60's. Recently, physicists have been increasingly interested in gerbes. Indeed, gerbes were originally developed by Giraud as geometric realizations of non abelian cohomology classes the same way a line bundle is as a realization of his Chern class. The differentiable version of those algebraic objects can be studied using Lie groupoid theory. More precisely, as observed by Behrend and Xu, (locally representable) stacks are presented by Morita equivalence classes of Lie groupoids. This is somehow similar to the presentation of manifolds by compatible differentiable atlases. In this first talk, we will recall the main definitions of Lie groupoid theory. In particular, we will discuss Morita equivalence and torsors. Then we will show how a gerbe and its band can be seen as a Lie groupoid extension and one of its torsors. Finally, we will discuss the particular case of gerbes over manifolds and how this relates to non abelian cohomology.
We meet weekly on Thursday afternoons, with the third meeting of each month in Altoona. If you are interested in giving a talk in our seminar please contact one of the coordinators for the Fall 2005 semester: Aissa Wade and Wojciech Dorabiala.
