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.

Announcements:

22.10.2024: Correction: in Step 1 of the proof of the existence of partitions of unity, one should additionally observe that the collection of all regular coordinate balls *whose closure is contained in some U_alpha* forms a basis for M. Then apply Lemma 1.7 to this collection to extract the V_i. This is needed in Step 2 to ensure that you can find \tilde{V}_i contained in some U_\alpha and containing the closure of V_i.
15.10.2024: Correction: I stated incorrectly in class that a function is smooth if it admits partial derivatives of all order. Rather, the correct definition is that a function is smooth if it admits *continuous* partial derivatives of all orders. This is equivalent in dimension 1 but not higher dimension. Thank you to the student who asked me about this after class. A reference for a counterexample is N. Kimura and J. Burr, Discontinuous Function with Partial Derivatives Everywhere, The American Mathematical Monthly, Vol. 67, No. 8 (Oct 1960), pp. 813-814.
09.10.2024: Exercise 1.1(a) in homework 1 is now an optional bonus problem (because I haven't yet defined the term "manifold")

Contacts:

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

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

Course materials:

Unofficial class notes. These are unofficial class notes which a student in the course has generously offered to share. They are not monitored by the instructor.
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.
Category theory terminology. A two page summary of the basic definitions in category theory, taken from Lee's book. Note that his notation for some categories differs from the one in class. Another good source for this material is the wikipedia article on category theory.

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.)