The language of geometry has changed drastically in the last decades. New and fundamental ideas such as the language of sheaves and cohomology are now indispensable in many incarnations of geometry, such as the theory of complex analytic spaces, algebraic geometry, or non-archimedean geometry. This seminar is intended as an introduction to these ideas illustrating them by example with the most ubiquitous branch of geometry, the theory of manifolds.

A sheaf (of abelian groups) is an assignment which associates to every open subset of a topological space an abelian group, satisfying some locality condition. Examples in the theory of manifolds include the sheaves of continuous or differentiable functions, the sheaves of vector fields and sheaves of differential forms.

Differential forms can be used to define De Rham cohomology groups which measure the failure of certain differential equations on a manifold to be solvable. One goal of the seminar will be to understand De Rham's theorem which tells us that the De Rham cohomology groups do not depend on the differentiable structure of the manifold, but only on the underlying topological structure. Using some sheaf-theoretic machinery, this result is reduced to the fact that on an open ball of R^n, de Rham's differential equations are always solvable, i.e. the Poincaré lemma.

Despite this, we will spend a lot of time to gain intuition for the abstract concepts of sheaves and cohomology.

If you are interested in the seminar, please join the GRIPS course (link below).

• Organisational meeting/distribution of topics: 27.07.2022 at 16:00 (4 p.m.). More details will
  be announced on GRIPS.
• Registration for course work/examination/ECTS: FlexNow

• Presentation: Giving a seminar talk of roughly 90 minutes

• Detailed written report of the seminar talk