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Research Group Differential Geometry

Kollegiengebäude Mathematik (20.30)
Room 1.003
Ute Peters

Institut für Algebra und Geometrie
Englerstr. 2
D-76131 Karlsruhe

Office hours:
Mo-Fr 09:00-15:00
Für Studierende:
Mo-Fr 09:15-11:15

Tel.: +49 721 608 43943

Fax.: +49 721 608 46909

AG Differentialgeometrie (Winter Semester 2019/20)

Lecturer: Prof. Dr. Wilderich Tuschmann
Classes: Seminar (0126600)
Weekly hours: 2

Seminar: Thursday 11:30-13:00 SR 2.058
Lecturer Prof. Dr. Wilderich Tuschmann
Office hours: by appointment; during lecture period: Thu. 10:15 - 11:15
Room 1.002 Kollegiengebäude Mathematik (20.30)
Email: tuschmann@kit.edu



Lashi Bandara (Universität Potsdam)

Boundary value problems for general first-order elliptic differential operators

The Bär-Ballmann framework is a comprehensive machine useful in studying elliptic boundary value problems (as well as their index theory) for first-order elliptic operators on manifolds with compact and smooth boundary. A fundamental assumption in their work is that an induced operator on the boundary can be chosen self-adjoint. Many operators, including all Dirac type operators, satisfy this requirement. In particular, this includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator. Recently, there has been a desire to study more general first-order elliptic operators, with the quintessential example being the Rarita-Schwinger operator on 3/2-spinors. In general dimensions, every induced boundary operator for the Rarita-Schwinger operator is non self-adjoint.

In this talk, I will present recent work with Bär where we and consider general first-order elliptic operators by dispensing with the self-adjointness requirement for induced boundary operators. The ellipticity of the operator allows us to understand the structure of the induced operator on the boundary, modulo a lower order additive perturbation, as bi-sectorial operator. We use a mixture of methods coming from pseudo-differential operator theory, bounded holomorphic functional calculus, semi-group theory as well as methods arising from the resolution of the Kato square root problem to extend the Bär-Ballman framework.

If time permits, I will also touch on the non-compact boundary case, and potential extensions of this to the L^p setting and Lipschitz boundary.