Jochen Brüning

A “Dirac system” is a model for Dirac operators which is a family of symmetric first order elliptic differential operators tied to the geometry of a Riemannian manifold. However, the model is much simpler than the Dirac operators themselves : it is merely a first order ordi- nary differential equation with operator coefficients. A point of the talk is to explain in what sense this operator valued differential equations behave like a matrix valued, true ordinary differential equation; or one might say, to explain in what sense the geometric situation modeled is “almost one dimensional”.

This is certainly not true in general since not all phenomena in higher dimensions can be reduced to one dimension but in certain situations this approach leads to very valuable in- formation. One example we will explain in detail is the case of complete manifolds with thin ends, like cylinders or cusps. Another case of interest arises in situations with symmetries (“scaling”) which applies locally in many situations.

The talk will proceed from simple notions to geometric situations, explaining the necessary definitions along the way. It will be acceptable to graduate students and non specialists of geometric analysis in general.