Organisation: Prof. Werner Ballmann, Stéphane Félix

Ort und Zeit: Freitag, um 14:15, LWK 008, ab 17. April 2009.
Nächsten Vortrag: 17.4.09, 14:15

Voraussetzungen: Elementare Kenntnisse über differenzierbare Mannigfaltigkeiten und in Riemannscher Geometrie (etwa im Umfang der Kapitel 1 und 2 aus [2] und des Kapitels II aus [1].)

Prerequisites: Basics on differentiable manifolds and riemannian geometry (approximately the content of Chapters 1 and 2 of [2] and Chapter II of [1].)

Inhalt: Grad differenzierbarer Abbildungen, Satz von Poincaré-Hopf, gerahmter Kobordismus und Homotopieklassen von Abbildungen, kritische Punkte differenzierbarer Abbildungen, Morse-Funktionen, Morse-Ungleichungen, Morse-Theorie des Energiefunktionals auf Wegeräumen, Anwendungen auf die Topologie Liescher Gruppen und symmetrischer Räume.

Content: Degree of a differentiable map, Poincaré-Hopf Theorem, framed cobordism and homotopy classes of maps, critical points of differentiable maps, Morse functions, Morse inequalities, Morse theory of the energy functional on spaces of paths, applications to the topology of Lie groups and symmetric spaces.

Program:

1. (Öznur Albayrak, 17.4.09) Chapter 4 and 5 of [2]: homotopy invariance of degree mod 2 and Brouwer degree of a differentiable map.
2. (Blanka Horvath, 24.4.09) Chapter 6 of [2]: Poincaré-Hopf theorem for vector fields.
3. (Stéphane Félix, 8.5.09) Chapter 7 of [2]: Framed cobordism and related properties, Hopf theorem.
4. (Shehryar Sikander, 15.5.09) Sections I.2, I.3 and I.4 of [1]: Morse functions on a manifold, homotopy types of sublevel sets of a Morse function (thm 3.5), Reeb thm (4.1)
5. (Jesko Huettenhain, 22.5.09) Sections I.5 and I.6 of [1]: weak and strong Morse inequalities, existence of Morse functions via embedding
6. (Nikolai Nowaczyk, 12.6.09) Sections III.11, III.12 and III.13 of [1]: path spaces, energy of a path, first variation formula, critical points of the energy (cor. 12.3), second variation formula
7. (Robert Nabiullin, 19.6.09) Sections III.14 and III.15 of [1]: Jacobi fields and relations to the second variation, index theorem
8. (Stéphane Félix, 26.6.09) Sections I.7 and III.19: Lefschetz theorem, panorama of some results on curvature in riemannian geometry
9. (Peter Kreyßig, 3.7.09) Sections III.16, III.17 and III.18 of [1]: finite dimensional approximation of path spaces, homotopy type of path spaces, example of the standard sphere
10. (Johannes Niediek, 10.7.09) Sections IV.20 and IV.21: Lie groups and symmetric spaces
11. (Lara Skuppin, 17.7.09) Sections IV.22 and IV.23: Bott periodicity theorems

References:

• [1] John Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, 1963.
• [2] John Milnor, Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, Based on notes by David W. Weaver, Princeton University Press, 1997.
• [3] Werner Ballmann, Lectures on Kähler Manifolds, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2006.