Applied Algebraic Topology, WS 2022/23
Prof. Dr. Clara Löh
/
Matthias Uschold
News

Exam dates: Wed 15.02.; Wed 22.03.; Wed 29.03
(see below for the registration details).

Exercise sheet 14 is online: pdf
Optional submission before 10.02.2023, 8:30 (via GRIPS).
All credits are bonus credits!

Exercise sheet 15 is online: pdf
No submission
Blorx needs your help!

Organisational matters

According to current plans (13.10.2022): This course will
be taught on campus in person. On request, I could turn
this into a hybrid format (with live streaming). Please
note that there will be no recordings of the lectures. The
lectures are a precious opportunity for live interaction
and I want to keep the atmosphere as casual and
unintimidating as possible. For asynchronous selfstudy,
lecture notes will be made available.
Please let me know (by email) in case there is a need for
the hybrid option!

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
realworld 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 higherdimensional
generalisation of graphs and thus allow for more
finegrained models, e.g., for discrete approximations
of geometric shapes, connectivity of highdimensional
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 realworld 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.
Lecture notes:
Topics covered so far:

Literature
 Introduction

Homotopy
 Homotopy and homotopy equivalence
 Application: Basic motion planning
 Homotopy invariance

Simplicial complexes
 Graphs
 Application: The seven bridges of Königsberg
 Simplicial complexes and simplicial maps
[Simplicial complexes,
simplicial maps,
basic constructions,
simplicial homotopy]
 Modelling: Consistency relations
 Modelling: Complexes from point clouds

Geometric realisation
[Geometric realisation,
subdivision,
simplicial approximation]
 Modelling: Approximation by simplicial structures
 Implementation: Simplicial complexes

Simplicial homology

The construction of simplicial homology
[The simplicial chain complex,
simplicial homology of simplicial complexes]

Computations: Divide and conquer
[The MayerVietoris sequence,
The long exact sequence of pairs,
Simplicial homotopy invariance]

Implementation: Simplicial homology
Python source code:
homology_examples.py

Homotopy invariance
[Simplicial homology and barycentric subdivision,
Simplicial homology and simplicial approximation,
Topological homotopy invariance,
Simplicial homology of triangulable spaces]

The Brouwer fixed point theorem
[The Lefschetz and Brouwer fixed point theorems,
Sperner's lemma]

Application: Consensus in distributed systems

Application: Social choice

Application: Nash equilibria

Application: Sensor network coverage

Persistent homology

Persistent homology
[Filtrations,
Persistence objects,
Persistent homology and persistent Betti numbers]

The structure theorem for persistent homology
[Persistence modules and polynomial rings,
The structure theorem,
Barcodes]

Implementation: Computing barcodes
[Homogeneous matrix reduction,
Persistent homology via matrix reduction,
A sparse implementation]
Haskell source code for sparse homogeneous matrix reduction:
HomogMatrixReduction.hs

The stability theorem for persistent homology
[A basic experiment,
Comparing point clouds: The GromovHausdorff distance,
Comparing weighted barcodes: The bottleneck distance,
The stability theorem]

Application: Horizontal application

Application: Exploring multdimensional data
 Appendix
 Pointset topology
 Categories and functors
 Basic homological algebra
Time/Location
Tuesday, 8:3010, M 104
Friday, 8:3010, 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

Please register before October 19, 10:00, via GRIPS for the exercise classes!

The exercise classes start in the second week.

Organisational matters
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.
Sheet 1,

of October 21, 2022,

submission before October 28, 2022 (08:30)

will be discussed in the exercise classes in the second/third week

Sheet 2,

of October 28, 2022,

submission before November 4, 2022 (08:30)

will be discussed in the exercise classes in the third/fourth week

Sheet 3,

of November 4, 2022,

submission before November 11, 2022 (08:30)

will be discussed in the exercise classes in the fourth/fifth week

Sheet 4,

of November 11, 2022,

submission before November 18, 2022 (08:30)

will be discussed in the exercise classes in the fifth/sixth week

Sheet 5,

of November 18, 2022,

submission before November 25, 2022 (08:30)

will be discussed in the exercise classes in the sixth/seventh week

Sheet 6,

of November 25, 2022,

submission before December 2, 2022 (08:30)

will be discussed in the exercise classes in the seventh/eighth week

Sheet 7,

of December 2, 2022,

submission before December 9, 2022 (08:30)

will be discussed in the exercise classes in the eighth/ninth week

Sheet 8,

of December 9, 2022,

submission before December 16, 2022 (08:30)

will be discussed in the exercise classes in the ninth/tenth week

Sheet 9,

of December 16, 2022,

submission before December 23, 2022 (08:30)

will be discussed in the exercise classes in the tenth/11th week

Sheet 10,

of December 23, 2022,

submission before January 13, 2023 (08:30)

will be discussed in the exercise classes in the 11th/12th week

Sheet 11,

of January 13, 2023,

submission before January 20, 2023 (08:30)

will be discussed in the exercise classes in the 12th/13th week

Sheet 12,

of January 20, 2023,

submission before January 27, 2023 (08:30)

will be discussed in the exercise classes in the 13th/14th week

Sheet 13,

of January 27, 2023,

submission before February 3, 2023 (08:30)

will be discussed in the exercise classes in the 14th/15th week

Sheet 14,

of February 3, 2023,

optional submission before February 10, 2023 (08:30)

will be discussed in the exercise classes in the 15th week

Sheet 15,

of February 10, 2023,

no submission


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
Dates for the
oral exams (25 minutes):

Wed, February 15,

Wed, March 22,

Wed, March 29.
Registration details:

Step 1: register for a time slot with Ms Bonn (you can visit her in person in M217 or send an email to rosina.bonn@mathematik.unir.de).

Step 2: register in FlexNow.
Please do not forget to also register for the Studienleistung in FlexNow.
Last change: February 10, 2023