Course: From Calculus to Cohomology.
Lecture + Exercise session (3+1): Tuesdays and Thursdays 10:00 - 11:30, Seminarraum 1 des Mathematischen Instituts (Raum 005).
EXAM: Tuesday, February 6th, 2018, 10 am, Stefan-Cohn-Vossen room (3rd floor)
Homeworks:
Course description:
The goal of this course is to introduce cohomology from differentiable viewpoint. We start with introducing de Rham cohomology for open subsets of R^n and use this differential perspective to prove some results usually discussed in an algebraic topology course (Brouwer fixed point theorem, Jordan Brouwer separation lemma). Then we define differential forms and de Rham cohomology for abstract smooth manifolds, and study very important cohomology classes: characteristic classes of vector bundles. Moreover we introduce the notions of the degree and index of a vector field, and prove the Poincare-Hopf theorem. If time allows, we also discuss the Poincare duality and the splitting principle.Students who wish to take this course as a 6 credits course will need to do some extra assignements given out during the semester (preparing a topic from the course book and giving a ~ one hour lecture, presenting solutions to some homework problems in class, possibly taking and organizing notes during other students talks and presentations).
Prerequisites: Multivariable Calculus (i.e. Analysis 2 and 3), Linear Algebra.
The lecture will follow closely the book:
"From Calculus to Cohomology: de Rham Cohomology and Characteristic Classes"
by Madsen and Tornehave.
Suggested additional reading:
"Differential Forms in Algebraic Topology" by Bott and Tu,
"Algebraic Topology" by Fulton.