Graduate seminar: Oberseminar on the Atiyah-Patodi-Singer index theorem (online) - Details

General information

Course name	Graduate seminar: Oberseminar on the Atiyah-Patodi-Singer index theorem (online)
Subtitle
Course number	503109
Semester	WiSe 2020/21
Current number of participants	11
expected number of participants	15
Home institute	Bereich Mathematik
Courses type	Graduate seminar in category Teaching
Preliminary discussion	Friday, 09.10.2020 16:15 - 17:00
First date	Friday, 09.10.2020 16:15 - 17:00
Type/Form
Learning organisation	This seminar will be a joined seminar between the University of Göttingen the University of Augsburg and therefore be held online only.

Lehrende

• UnivProf.Dr. Thomas Schick

Tutor/-in

• Thorsten Hertl

Rooms and times

No room preference

Tuesday: 14:15 - 15:45, weekly(11x)

Friday, 09.10.2020 16:15 - 17:00
Monday, 30.11.2020 (all-day)

Fields of study

• Vorlesungsverzeichnis WiSe 2020/2021 > Fakultät für Mathematik und Informatik > Lehrveranstaltungen der Lehreinheit Mathematik > Masterstudiengang Mathematik (M.Sc.) > Studienprofil F "Forschungsorientiert - allgemein" > SP 1 - Seminare
• Vorlesungsverzeichnis WiSe 2020/2021 > Fakultät für Mathematik und Informatik > Lehrveranstaltungen der Lehreinheit Mathematik > Masterstudiengang Mathematik (M.Sc.) > Studienprofil MDS "Mathematical Data Science" > SP 1 - Seminare
• Vorlesungsverzeichnis WiSe 2020/2021 > Fakultät für Mathematik und Informatik > Lehrveranstaltungen der Lehreinheit Mathematik > Masterstudiengang Mathematik (M.Sc.) > Studienprofil W "Wirtschaftsmathematik" > SP 1 - Seminare
• Vorlesungsverzeichnis WiSe 2020/2021 > Fakultät für Mathematik und Informatik > Lehrveranstaltungen der Lehreinheit Mathematik > Masterstudiengang Mathematik (M.Sc.) > Studienprofil Phy "Physik" > SP 1 - Seminare
• Vorlesungsverzeichnis WiSe 2020/2021 > Fakultät für Mathematik und Informatik > Lehrveranstaltungen der Lehreinheit Mathematik > Masterstudiengang Mathematik (M.Sc.) > Studienprofil Phy "Physik" > Profil Phy - spezielle Zyklen aus SP 1 - Seminare

Comment/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.