Topology

''The Whitney-Trick''. Taken from Scorpan, Alexandru. The Wild World of 4-Manifolds. Providence: American Mathematical Society, 2005, Figure 1.26, S.46, with the consent of the AMS.
© AMS

The subject matter of topology are discrete invariants of topological spaces (for example smooth manifolds) and maps between them. The simplest such invariant is the winding number of a curve in the plane. One of the principal goals is to study whether a manifold is uniquely determined by its discrete invariants. Of particular interest in the focus subject are stable homotopy theory, K-theory, differential topology, index theory and geometric group theory.

Topology is not an isolated discipline, but has intense connections to other areas of theoretical mathematics, such as algebra, differential geometry, geometric group theory and the theory of operator algebras.

Prerequisites

Prerequisites for the specialization in topology are the lecture courses ''Foundations of analysis, topology and geometry'' and ''Topology I'' (or equivalent courses), with the following contents:

  • Topology I: singular homology theory, CW complexes and cellular homology, the cross product and the Künneth theorem.
  • Foundations of analysis, topology and geometry: notions of point set topology, fundamental group and covering spaces, smooth manifolds.

 

Courses for the specialisation in Topology

Summer semester 2021

Prof. Dr. Arthur Bartels: Topics in Topology
apl. Prof. Dr. Michael Joachim: Topology II
Prof. Dr. Michael Weiss: Vector Bundles, J-Homomorphism and Adams Conjecture

Winter semester 2021/2022

Prof. Dr. Johannes Ebert: Index theory I (prerequisites: introduction to functional analysis) (Type I, II)
Dr. Fabian Hebestreit: Higher category theory (Type II)
apl. Prof. Dr. Michael Joachim: Homotopy theory (Type I, II)
Prof. Dr. Linus Kramer: Geometric Group Theory I, possibly in German (Type I, II)
PD Dr. Jakob Scholbach: Category Theory (Type I, II)

Summer semester 2022

Prof. Dr. Johannes Ebert: Index theory II (Type I, II)
Dr. Fabian Hebestreit: Higher Categories II (Type II)
apl. Prof. Dr. Michael Joachim: Stable homotopy theory (Type II)
Dr. Bakul Sathaye: Geometric Group Theory (Type I, II)
Prof. Dr. Michael Weiss: Topology II (Type I, II)

Winter semester 2022/23

Dr. Martin Bays: Geometric Group Theory I (Type II)
Prof. Dr. Michael Weiss: Topology III (Type I, II)

Summer semester 2023

Prof. Dr. Michael Weiss: Topology IV