Applied Algebraic Topology, WS 2022/23

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 self-study, 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 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.

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 Mayer--Vietoris 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 Gromov--Hausdorff distance, Comparing weighted barcodes: The bottleneck distance, The stability theorem]
- Application: Horizontal application
- Application: Exploring mult-dimensional data
Appendix
- Point-set topology
- Categories and functors
- Basic homological algebra

Time/Location

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

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.uni-r.de).
Step 2: register in FlexNow.

Please do not forget to also register for the Studienleistung in FlexNow.

Last change: February 10, 2023