WiSe 20/21 Seminar: de Rham cohomology
Time and place: Monday 08-10 online
The goal of this seminar is to introduce and study de Rham cohomology, which is an invariant of smooth manifolds defined using differential forms. It is related both to analysis (via the fundamental theorem of calculus and Stokes’ theorem) and algebraic topology (as de Rham cohomology turns out to be isomorphic to singular cohomology). Moreover, the ideas underlying the definition of de Rham cohomology are quite versatile and can be applied in many other geometric contexts, for example in algebraic geometry.
The seminar will closely follow Chapter I of the book Differential Forms in Algebraic Topology by R. Bott and L. W. Tu and cover most of the results therein. The main theorem is Poincaré duality, which is a surprising symmetry in the cohomology of smooth compact manifolds. We will discuss it for both oriented and nonorientable manifolds.
| Date | Speaker | Topic |
|---|---|---|
| 02.11 | Yaxi Zhu | The de Rham complex of Euclidean space |
| 09.11 | Michael Friesen | The de Rham complex of a smooth manifold |
| 16.11 | Michael Friesen | The Mayer–Vietoris sequence |
| 23.11 | Arshay Sheth | Orientation and integration |
| 30.11 | Arshay Sheth | Stokes’ theorem |
| 07.12 | Matthias Uschold | The Poincaré lemma |
| 14.12 | Hua Jing | The Poincaré lemma with compact supports |
| 21.12 | Lukas Schweiger | Finiteness of de Rham cohomology |
| 11.01 | Zhelun Chen | Poincaré duality for oriented manifolds |
| 18.01 | Hua Jing | The degree of a proper map |
| 25.01 | Hua Jing | The Künneth formula and the Leray–Hirsch theorem |
| 01.02 | Marc Hoyois | The orientation bundle and the twisted de Rham complex |
| 08.02 | Zhelun Chen | Densities and Poincaré duality for nonorientable manifolds |