# TMA4190 - Introduction to Topology (Spring 2017)

This was the web page for the course; it will no longer be updated.

### News

- I will teach MA3403 - Algebraic Topology next term.

### Reference Group

Roeland ter Hoeven

Lise Millerjord

### Lecture and Lab Hours

Lecture hours:

Mondays 10:15 - 12:00 in R21

Fridays 12:15 - 14:00 in F4

Topology is not a sport, so there will not be **exercises**. In this course, we want to learn something and therefore we need **play time**. This will take place during the lab hours. Eventually, we will use computers; bring yours if you like to.

Lab hours:

Tuesdays 14:15 - 15:00 in K28

### Lecture Plan

Here is a preliminary plan of what we are going to do during the lecture slots. This is subject to change as the course progresses. In particular, there is enough time to go **slower** than indicated.

- Week 2
- 09 Jan - Introduction [Z, 1]
- 13 Jan - Topological spaces [Z, 2.1]

- Week 3 -
- 16 Jan - Manifolds [Z, 2.2]
- 20 Jan - Reading assignment, optional: [G, 1] about manifolds

- Week 4
- 23 Jan - Simplicial complexes [Z, 2.3]
- 27 Jan - Reading assignment, optional: [G, 2] about complexes

- Week 5
- 30 Jan - Reading assignment, optional: Software Suggestions
- 03 Feb - Pedagogical training

- Week 6
- 06 Feb - Pedagogical training
- 07 Feb - Lab Hour: 20170207 (Topology in Norway)
- 10 Feb - Čech complexes

- Week 7
- 13 Feb - Vietoris-Rips complexes
- 14 Feb - Lab Hour: 20170214 (Software Suggestions)
- 17 Feb - The Euler characteristic [G, 3]

- Week 8
- 20 Feb - Chain complexes
- 21 Feb - Lab Hour: 20170221 (TDA with R)
- 24 Feb - Simplicial homology

- Week 9
- 27 Feb - Examples
- 03 Mar - Reading assignment, optional: [C]

- Week 10
- 06 Mar - Pedagogical training
- 10 Mar - Review and more examples

- Week 11
- 13 Mar - Functoriality and the Mayer-Vietoris sequence
- 14 Mar - Lab Hour: 20170314 (simplicial homology with GAP and HAP)
- 17 Mar - The projective plane and its homology

- Week 12
- 20 Mar - Reading assignment, optional: [Z, 4]
- 24 Mar - Duality, cohomology, the Laplace operator, and Hodge theory

- Week 13
- 27 Mar - Kirchhoff's rules and theorem
- 28 Mar - Lab Hour: 20170328
- 31 Mar - Critical points and Morse functions [F, 2-4]

- Week 14
- 03 Apr - Gradient vector fields and the Morse complex [F, 7]
- 04 Apr - Lab Hour: Mayer-Vietoris and the homology of surfaces
- 07 Apr - Morse theory and practice [F, 9]

- Week 15
- 10 Apr - Winter/spring break
- 14 Apr - Winter/spring break

- Week 16
- 17 Apr - Winter/spring break
- 21 Apr - Pedagogical training

- Week 17
- 24 Apr - Wrap-up and review
- 25 Apr - Lab Hour: Q&A

### Reading Materials

This course is not based on a textbook. If you prefer to do some additional reading, here are some useful places to start with.

- [C] C. Curto. What can topology tell us about the neural code?

Bull. Amer. Math. Soc. 54 (2017) 63-78

- [F] R. Forman. A User's Guide to Discrete Morse Theory.

Séminaire Lotharingien de Combinatoire, B48c (2002), 35 pp.

- [G] R. Ghrist. Elementary applied topology. 2014.

www.math.upenn.edu/~ghrist/notes.html

- [EH] H. Edelsbrunner, J.L. Harer. Computational topology. 2010.

- [O] S. Oudot. Persistence Theory: From Quiver Representations to Data Analysis. 2015.

http://www.enseignement.polytechnique.fr/informatique/INF563

- [S] M. Szymik. Notes for MA3002. 2016.

szymik.ma3002v16.pdf

- [Z] A.J. Zomorodian. Topology for computing. 2005.

http://graphics.stanford.edu/courses/cs468-04-winter

### Relation to MA3002 - General Topology

This course will have very little overlap with MA3002 - General Topology as I have taught it in the years before.

**General Topology** is the theory of topological spaces and continuous maps between them. These concepts are fundamental in handling topological problems in mathematical analysis. For that reason, General Topology is often referred to as **Analytic Topology**.

However, **Topology** as a whole discipline covers much more than that and this course is intended to be an introduction to many different parts of it. For instance, we will also see other models of the idea of shape and space that are more relevant to **Topological Data Analysis**.

I will **not** assume that you have taken part in MA3002 - General Topology before this course, nor that you take part in it at the same time. All that you need to know about topological spaces and continuous maps to follow this course will be covered as part of it.

### Course Information

The official Course Description is mostly helpful to find the exam date.

As for the contents, here is what I wrote in an email to a student:

I will talk about combinatorial objects (simplicial complexes) rather than just analytic objects (manifolds) as well. This reflects the fact that topology has moved on; it does not just feed the needs of the engineering sciences, but more and more also the computing sciences. I believe that everyone needs to have a broad introduction to topology, and there will be nothing too specialized in my lectures. In the end, when I talk about Morse theory, we will see both sides (analytic and combinatorial) come together. You can have a look at these papers to see what I have in the back of my mind. E. Witten Supersymmetry and Morse theory J. Differential Geom. 17 (1982) 661-692 R. Forman Witten-Morse theory for cell complexes Topology 37 (1998) 945-979

### Code

Some GAP code that computes the homology of the skeleta of the simplices:

LoadPackage("HAP"); oneSk := function( d, n ) local C, B, M; M := MaximalSimplicesToSimplicialComplex(Combinations( [0..n], d+1 )); C := ChainComplex( M ); B := List([0..d],x->Length(Positions(Homology( C, x ),0))); Print(d," ",n," ",B,"\n"); end; skDelta := function( n ) local i; for i in [0..n] do oneSk( i, n ); od; end;

Example:

skDelta( 10 );