Applied Algebraic Topology, WS 2022/23
Prof. Dr. Clara Löh
If you plan to write a bachelor thesis under my supervision in SS 2023 (in
Topology/Geometry), you should participate in a seminar in the Global Analysis
and Geometry group before SS 2023.
Applied Algebraic Topology
The tools and methods from algebraic topology have
a wide range of applications to other fields and to
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.
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.
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.
A guide to the literature will be provided at the beginning of WS 2022/23.
There will be lecture notes.
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.
Last change: June 27, 2022