- Target audience: advanced bacheler student, master students, PhD students.
- Recommended background: Solid foundations on analysis on ma- nifolds and vector bundles; functional analysis, complex analysis in one variable and a bit topology is recommended but not a necessity
- Responsible: Thomas Schick, joint with Thorsten Hertl
- Date: to be decided by the participants
- Place: Virtually via BigBlueButton, details follow.
- Contact/Questions: thorsten-hertl@stud.uni-goettingen.de, thomas.schick@math.uni-goettingen.de, Tel. 39-7799
- Preliminary meeting: Friday Oct 9, 16:15 videoconference via stud.ip. Please register there to participate!
- credit points for presentation of a talk (with a handout)

Description:

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.

Literatur:

- Main: Bernheim Booss-Bavnbek and Krzysztof Wojciechowski, Elliptic Boundary Value Problems for Dirac Operators, Mathematics: Theory & Applications, BirkhĂ¤user, 1993.
- further: Bernheim Booss-Bavnbek and David Bleecker, Index Theory in Mathematics and Physics, International Press of Boston, 2013.

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);