MATH-F310: Differential Geometry I

2021

Room: P.OF.2068-2070;     Time: Mondays, 10:00-12:30.
Room: P.A2.222;     Time: Thursdays, 10:00-12:30.

Course content: This course is the first part of the Differential Geometry course. In particular, basic notions and methods of differential geometry such as smooth manifolds, vector fields, vector bundles etc. appearing both in various branches of mathematics and physics will be introduced and developed.

Lecture notes (Last updated: 02 DEC 2021)

Evaluation: The final mark consists of 80% for a written exam and 20% for homeworks.
Exam: The written exam takes place on 18.01.2022 in P.FORUM.G (Auditoire DOLLO), 9:00 - 12:30 (approx.).
Mock exam: If you wish to test yourself before the real exam, download and print the mock exam and allow yourself 2h30min for solving. Notice that the mock exam covers only Sections 2 and 3 of the lecture notes. The real exam will cover all material of the course.

Literature

D.Barden, C.Thomas. An introduction to differential manifolds, Imperiall College Press.
J.Lee. Introduction to smooth manifolds, Springer Verlag.
L.Tu. An introduction to manifolds, Springer Verlag.
A.Shastri. Elements of differential topology, CRC Press.

Excercises

Room: P.2NO 707.
Time: Tuesdays, 16:00-18:00.

Excercise sheet 1 (Deadline: 05.10)
Excercise sheet 2 (Deadline: 12.10)
Excercise sheet 3 (Deadline: 19.10)
Excercise sheet 4 (Deadline: 26.10)
Excercise sheet 5 (Deadline: 09.11)
Excercise sheet 6 (Deadline: 16.11)
Excercise sheet 7 (Deadline: 30.11)
Excercise sheet 8 (Deadline: 07.12)

MATH-F419: Algebraic topology

2021

Room and Time: Wednesdays 10:00-11:00 in P.OF.2078 and Fridays 15:00-16:00 in P.1C3.203.

Course content: The idea is to associate to topological spaces algebraic objects (groups, rings etc). If this is done judiciously, one can hope for example to distinguish non-homeomorphic spaces or essentially different continuous maps (in a suitable sense). This in turn allows one to prove interesting results, for example that any continuous map from a closed ball in a finite-dimensional Euclidean space into itself has a fixed point (Brouwer’s theorem).

Lecture notes (Last updated: 25 NOV 2021)

Literature

J. Vick. Homology theory. An introduction to algebraic topology.
A. Hatcher. Algebraic Topology ( Chapters 1 and 2), available online: http://www.math.cornell.edu/~hatcher/AT/ATpage.html

Course "Introduction to Gauge Theory"

PCMI Summer School "Quantum Field Theory and Manifold Invariants", July 2019

Lecture notes (Last updated: 22 OCT 2019)