Applied Algebraic Topology, WS 2022/23

Prof. Dr. Clara Löh / Matthias Uschold

News

Applied Algebraic Topology

The tools and methods from algebraic topology have a wide range of applications to other fields and to real-world problems.

Graphs are combinatorial structures that can be used to model connectivity of various kinds, e.g., connections between people in social networks, genetic proximity in biology, or dependencies between software components. Simplicial complexes are a higher-dimensional generalisation of graphs and thus allow for more fine-grained models, e.g., for discrete approximations of geometric shapes, connectivity of high-dimensional data, decentralised computations in sensor networks, configuration spaces for robots, or dependencies between agents in distributed systems.

Algebraic topology is mainly concerned with geometric features that are invariant under a special type of deformations: homotopies and homotopy equivalences. A classical (and computable) example of a homotopy invariant of simplicial complexes is simplicial homology.

Moreover, the language of homotopy theory also found a completely different type of applications in the foundations of mathematics and computer science.

In this course, we will learn the basics of simplicial complexes, simplicial homology, and homotopy invariance. We will explore modelling and real-world applications of these notions and invariants. Whenever feasible, we will also look at implementation matters.

The exact contents of the course will be adapted to the background of the participants.

If all participants agree, this course can be held in German; solutions to the exercises can be handed in in German or English.

Time/Location

Tuesday, 8:30--10, M 104
Fridday, 8:30--10, M 104

On demand, we can switch to a hybrid format (with zoom live streaming); please contact Clara Löh by email. The access data is available on GRIPS.

Exercise classes

Time/Location: tba

Exercise sheets

Solutions can be submitted in English or German via GRIPS and in teams of up to two people. Please do not forget to add your name to all your submissions!

Problems marked as 'Quick check' are not to be submitted and will not be graded; these problems will be discussed in the exercise classes.

Literature

A guide to the literature will be provided at the beginning of WS 2022/23. There will be lecture notes.

Prerequisites

All participants should have a firm background in Analysis I/II (in particular, basic point set topology, e.g., as in Analysis II in WS 2011/12), in Linear Algebra I/II, and basic knowledge in group theory (as covered in the lectures on Algebra).
Experience in Geometry/Topology or Algebraic Topology is helpful, but not really necessary.
Programming experience is helpful, but not necessary.

Exams

tba

Last change: June 27, 2022