Seminar "Elliptic Boundary Problems for Dirac operators"


Consider a (linear) partial differential equation P (u) = g on a subset X of euclidean space with smooth boundary Y or more generally, on a smooth manifold with boundary (X, Y). It is natural to impose boundary conditions (also in view of applications). Furthermore, in contrast to linear equations between finite dimensional vector spaces (of the same dimension), uniqueness of a solution does not imply its existence (and vice versa). This is obstructed by the index of the operator P , which makes the index an object of fundamental interest. Another motivation to study indices of differential operators comes from geometry: Often the differential operator is motivated from the geometry and the index will carry geometrical information within itself. For example, the non-vanishing of the Atiyah-Singer-Dirac Operator on a closed manifold give an obstruction for a manifold to admit positive scalar curvature. In this seminar we set out to understand what good boundary are and how to get a grip on the index problem, primary for (generalised) Dirac operators. In a preparatory first talk we introduce (review) the necessary material on unbounded operators and pseudodifferential operators. After a short discussion of Clifford and Spinor bundles, we will define generalised Dirac operators and prove several analytical properties of them, for example, solutions of Dirac operators satisfy an identity theorem, like holomorphic functions. After some analytical preparation, will introduce elliptic boundary condition and, quite ambitiously, strive for the proof of the Atiyah-Patodi-Singer index formula, which relates the index of the operator P to the underlying geometry of X and its restriction to the boundary. This seminar will be a joined seminar between the Universit^auml;t G^omol;ttingen and the University of Augsburg and therefore be held online only.

Distribution of Talks


possible Modules:
Moduls: M.Nat.4812 (Seminar on analysis of partial differential equations); M.Nat.4912 (Advanced Seminar on analysis of partial differential equations); M.Nat.4813 (Seminar on differential geometry); M.Nat.4814 (Seminar on algebraic topology); M.Nat.4914 (Advanced Seminar on algebraic topology);