V3D3/F4D1 - Grundzüge der Analysis und Geometrie auf Mannigfaltigkeiten / Analysis and Geometry on Manifolds

WiSe 2024-25

General information:

This will be a standard introduction to analysis and geometry on manifolds. We will aim to cover: basic theory of topological and smooth manifolds, transversality, Whitney embeddings theorems, vector fields, vector bundles, integration theory. Additional topics may include: Lie groups, Riemannian and symplectic manifolds and de Rham cohomology.

The primary source for the course will be the instructor's class notes, which will roughly follow the first sixteen chapters of Lee's classic textbook Introduction to Smooth Manifolds. Students are also responsible for material covered in the exercise sessions (which will be run by a team of tutors). Students are not required to purchase a textbook for this class.

The course will be taught in English.

Contacts:

Instructor: Laurent Côté
Email: lcote@math.uni-bonn.de

Course Assistant: Koen van den Dungen
Email: kdungen@uni-bonn.de

Course materials:

Appendix to Lee's textbook. This is a copy of the appendix to Lee's textbook, which is freely available on the Springer website. The rest of the book can be accessed through the University library.
On the classification of topological 1-manifolds. This is article, written by David Gale in 1987, explains the classification of topological 1-manifolds. The article is written in the style of a take-home exam, which means that the reader is asked to supply some missing arguments but the author also provides hints.

Final exam information:

Exam 1: Tuesday 11.02.2025, 9-11, Großer Hörsaal, Wegelerstraße 10
Exam 2: Friday 21.03.2025, 9-11, Großer Hörsaal, Wegelerstraße 10

In order to be admitted to the final exam, students must score at least 50% of the points in the exercise sheets and present at least one solution in the tutorials.

Exercise sheets:

Homework will be posted weekly on Fridays at the following link: Homework link.
Homework should be submitted electronically on Sciebo on the following Friday before noon. The homework will be returned in the tutorials. Each tutor will provide to his/her class the relevant information for uploading the homework.

Some relevant literature:

M. Hisch, Differential Topology. Springer, 1976. (This is a classic textbook on differential topology.)
J. Lee, Introduction to Smooth Manifolds. 2nd ed. 2012. (This is arguably the world's most popular textbook on differential topology. The proofs are extremely detailed and complete.)
J. Milnor, Topology from the Differentiable Viewpoint. Vol. 21. Princeton university press, 1997. (This remarkable book is only fifty pages but covers a lot of ground. Manifolds are considered as subsets of R^n.)
S. Ramanan, Global Calculus Vol. 65. American Mathematical Soc., 2005. (This is an advanced book; it takes a more modern (sheafy) approach on some of the advanced topics which we will cover.)
T. Wedhorn, Manifolds, sheaves, and cohomology. Springer Spektrum, 2016. (This book develops the theory of smooth manifolds from scratch via sheaf theory. Highly recommended for students also interested in algebraic geometry.)