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 main theorem of the seminar 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.

• Organisational meeting/distribution of topics: Wednesday February 7, 16:15 in M104, or by email
  at marc.hoyois@ur.de
• Registration for course work/examination/ECTS: FlexNow

• Presentation: Giving a seminar talk of roughly 90 minutes

• Detailed written report of the seminar talk