Seminar: Exotic spheres and their curvatures (Winter Semester 2024/25)
- Lecturer: Prof. Dr. Wilderich Tuschmann
- Classes: Seminar (0126800 )
- Weekly hours: 2
Preliminary Meeting and Assignment of Talks Thursday, July 18 from 13:00-14:00 in SR 2.059
Here you find the seminar announcement.
The seminar is organized via Ilias. Here is the Ilias-Link.
| Schedule | |||
|---|---|---|---|
| Seminar: | Thursday 9:45-11:15 | 20.30 SR 3.069 | Begin: 31.10.2024 |
| Lecturers | ||
|---|---|---|
| Lecturer | Prof. Dr. Wilderich Tuschmann | |
| Office hours: by appointment | ||
| Room 1.002 Kollegiengebäude Mathematik (20.30) | ||
| Email: tuschmann@kit.edu | Lecturer | Dr. Artem Nepechiy |
| Office hours: by appointment | ||
| Room 1.004 Kollegiengebäude Mathematik (20.30) | ||
| Email: artem.nepechiy@kit.edu | ||
Contents
In the early 1950s, John Milnor startled the mathematical community by proving that in dimension 7
there exist so-called exotic spheres, i.e., smooth manifolds which are homeomorphic but not diffeo-
morphic to the standard (7-)sphere. His proof was based on the Signature Theorem that had been
discovered shortly before by Friedrich Hirzebruch, and the study of exotic spheres as well as their
Riemannian geometric properties has since then been an active and intriguing field of differential
topology and geometry alike.
The seminar will provide an overview about the basic results and open questions in this important
field of research and also prepare interested participants for writing a master thesis in this subject.
Along the way, we will encounter fundamental and important notions and tools like Chern, Euler and
Pontryagin classes, cobordism groups, the Thom isomorphism and the signature formula, as well
as gain a better understanding of positive, nonnegative, almost nonnegative sectional, positive Ricci
and scalar curvature, and basics of spin geometry.
Prerequisites
Sound knowledge of foundational results and concepts from differential geometry as
provided in the KIT course ’Differential Geometry’, as well as rudiments of algebraic topology