Class book

The book Topology: beyond a first syllabus was written by students in this module.

Covid-19 update

From 6pm March 12 the lectures, continuous assessment, and summer examination for this module will be delivered online.

  • Remaining lectures

    Slides for lectures 19, 20, 22, 23, 24 will be uploaded, as usual, shortly after the lecture slot. Additionally, I will upload a recording of me delivering the lecture, as was done for Lecture 18. Though there will be no audience present when I record the remaining lectures!
  • Second in-class test

    The in-class test at 12.00pm on Monday 23 March will be delivered via Blackboard. You will need to find reliable internet access in a quiet place where you can work for 50 minutes uninterrupted; you should have pen and several sheets of white paper to hand.

    The test will consist of five equally weighted questions based on the problem sheet up to, and including, Section 12. You will be given 50 minutes to do the test, and you will be required to answer each question on a separate sheet of white paper. After the test you will be given 30 minutes to take clear and readable photopgraphs of each of your five answers and uplaod these as five files to Blackboard. The answer to each question will have to be uploaded as a single file.
  • Student paper

    Please continue to work on you student paper. The pdf file should be named MA342Surname1Surname2Surname3.pdf and emailed to graham.ellis@nuigalway.ie by 5pm on Tuesday 31 March, 2020.
  • Tutorials

    The tutorials have already covered most of the problems in the tutorial sheet so there is no need for further tutorials.
  • Semester exam

    The plan is to deliver our Maths exams more or less as usual. They'll be made available on Blackboard at the time scheduled in the official exam timetable. Students will answer them at home. They'll be given two hours to write out the solutions. Then they'll be given a short period of time (30 mins say) to scan or photograph the answers and upload them to Blackboard. The onus will be on the student to make sure that the uploaded file is easily readable by the grader.
  • Visiting students

    Visiting students who have had to go home should continue to participate in the online lectures, continuous assessment, and exam.

Module Content

On successful completion of this module you should be able to:

  • Establish the invariance of the Euler characteristic of a sphere, and compute simple Euler integrals.
  • Give the definitions of a topological space and a continuous map between topological spaces, and provide examples.
  • Understand connectedness and compactness as topological invariants.
  • Understand the concept of homeomorphism, and use topological invariants to prove that certain spaces are not homeomorphic.
  • Construct new topological spaces using the subspace and quotient constructions.
  • Understand and represent simplicial complexes and triangulated spaces.
  • Understand homotopy equivalence, and informally explain why the Euler characteristic is a homotopy invariant of a triangulated space.
  • Prove the fundamental theorem of algebra, Perron’s theorem, Brouwer’s Fixed Point Theorem.
  • Understand John Nash’s proof of the existence of Nash equilibria in game theory.
  • Understand the basic idea behind topological data analysis.

Module Coordinates

  • Lecturer: Graham Ellis
  • Lectures: take place at 12pm Monday in the McMunn Theater and 12pm Wednesday in the McMunn Theater.
  • Tutorials: take place at 2pm Wednesday in AC201 and 6pm Wednesday in AM150. The tutor is Kelvin Killeen.
  • Recomended text: The lectures will be loosely based on the first five chapters of the text: Basic Topology by M.A. Armstrong, Undergraduate Texts in Mathematics, Springer-Verlag. The book emphasizes the geometric motivations for topology and I recommend that you take a look at it. There are some copies in the library. The text: Topology and Groupoids by R. Brown is also a great book which covers similar material and should be consulted too. It is available online here.
  • Problem sheet: available here.
  • Module Website: Information and module documents will be posted to this site, which is linked from the Blackboard MA342 Topology pages. Blackboard will also be used for announcements and for posting grades.

Module Assessment

The end of semester exam will count for 65% of the assessment. The continuous assessment will count for 35% of the assessment.

The continuous assessment will consist of two in-class tests, each worth 10%, and a student paper worth 15%.

See here for details of the student paper, which must be submitted by 5pm on Tuesday 31 March.

The in-class tests are based on the homework/tutorial problems. Each test will consist of questions taken verbatim from the homework/tutorial sheet. Many questions on the end of semester exam will also be taken, almost verbatim, from the homework/tutorial sheet too.

Supplementary Material and News

Why is Topology MA342 relevant to Maths students?

Topology can be fun. It is also a major branch of mathematics, as demonstrated by the number of Fields Medals awarded to topologists such as Atiyah, Donaldson, Freedman, Jones, Milnor, Mumford, Novikov, Perelman, Quillen, Serre, Smale, Thom, Thurston, Voevodsky ... . The module MA342 tries to give students a taste of this vast subject.

Why is Topology MA342 relevant to Computer Science students?

In the last decade or so, topologists have been trying to harness the power of modern computers to apply topological ideas to problems in science and engineering. The aim is to use the deformation invariant notions of topology to provide qualitative answers to problems; see, for instance, details of the research network on Applied Computational Algebraic Topology . The module MA342 tries to hint at these applications through a discussion of Euler characteristics of digital images and Euler integration in sensor networks.

Why is Topology MA342 relevant to Financial Maths & Economics Students?

Fixed point theorems play an important role in theoretical economics; see, for instance, the textbook Fixed point theorems with applications to economics. The module MA342 provides the outline of a proof of Brouwer's fixed point theorem and an explanation of how Brouwer's theorem can be used to prove the existence of Nash equilibria. This latter notion is due to the mathematician John Nash who was awarded the Nobel Prize for Economics for his work in this area.   

Why is Topology MA342 relevant to Mathematics & Education Students?

Much of school mathematics focuses on procedural tasks: teach children the procedures for calculating answers to problems and then test their ability to do mathematics by asking them a range of problems to which the procedures can be applied. The core Maths modules in the Mathematics & Education BA programme also tend to focus to a large extent on procedural mathematics: evaluate a multiple integral; evaluate a complex integral, calculate the inverse of a matrix; determine a probability using Bayes' Rule; decipher an encrypted message by first using Euclid's algorithm to solve a system of equations; use differentiation to calculate the maximum/minimum value of some quantity; ... .  

Project Maths has been introduced into schools with the noble aim of complementing childrens' procedural knowledge of mathematics with a strong conceptual knowledge. One difficulty facing teachers of Project Maths is: how can a child's conceptual knowledge of a topic be developed, and how can it be reliably assessed?

The MA342 module is primarily concerned with developing students' conceptual knowledge of a particular area of mathematics. Even though topology, per se, is unlikely to enter into the Project Maths curriculum in the near future, the module should give students some ideas for developing and assessing conceptual mathematics.

Clicker opinion polling may be used in some lectures.

Lecture Notes

Lecture Notes
Lecture Summaries
1
Explained that topology is the study of those properties of a space that remain unchanged through a "continuous deformation" of the space, but I waived my hands a bit too much when using the term "continuous deformation". In later lectures we'll see that hand waiving can be replaced my mathematically precise definitions. I then defined the Euler characteristic (=V-E+F) of a surface such as the surface of Mars, and "observed" that this number is a topological property of the surface. I used the Euler characteristic to count the number of pentagons in a soccer ball, the number of pentagons in a fullerene molecule, and the number of pentagons in a Buckminster Fuller dome (assuming that the basement of the dome completes the sphere).

Here are some slides used during the lecture.

If you are interested in applications of topology to finance and data science take a look at the Ayasdi company website.
2
Defined the Euler integral of an integer valued weight function w:X --> Z defined on a planar region X covered by a collection of (closed) regions Ui. The weight function of main interest is defined by w(x)=|{i : x is in Ui}| (so be careful of the typo in the slides relating to this definition). Explained how a Texas farmer could use the Euler integral to count cows on the ranch. This lecture is based on the recent research article: Target enumeration via Euler characteristic integrals by Yuliy Barishnikov (Bell Labs, New Jersey) and Robert Christ (University of Pennsylvania). Unfortunately, the method could equally well be used to count tanks in a battle field in the ongoing fight against the axis of evil.
3
Started by emphasizing that, in our "proof" that the Euler characteristic of the 2-sphere S2 is V-E+F=2, we used the fact that a simple closed curve on the 2-sphere cuts the 2-sphere into two pieces. Then observed that a simple closed curve on a möbius strip or on a torus does not necessarily cut the space into two pieces. So we should really justify (i.e. prove) that a simple closed curve on the 2-sphere always cuts it into two pieces.

Gave some motivation for a more precise approach to topology and stated the Jordan Curve Theorem. Explained that the next few lectures would focus on obtaining a good understanding of the statement of this theorem and on the main features in its proof.

Ended by giving the definition of a topological space. Next lecture we'll consider lots of examples aimed at reinforcing our understanding of this definition.
4
Gave the definition of a topological space, and of a connected topological space, and illustrated these two definitions with examples.
5
Gave the definition of a topological subspace and of a connected component of a space.

Ended with a video about topology and data analysis. Data analysts are interested in connected components of topological spaces constructed from data sets (though traditional statisticians use terms such as clusters in the data).
6
Began with a brief introduction to topological data analysis. This introduction was intended to show how any set of measurements of pairwise "distances" between points in a data set gives rise to a sequence of inclusions of topological spaces. From this viewpoint, one can consider applying any notion from topology to the data set. In an example we used the notions of "subspace" and "connected component" to obtain a representation of a data set as a tree. This example was chosen to be in keeping with Charles Darwin's tree of life. For more about this approach to data see this recent book.

In the last ten minutes the definition of a continuous function between topological spaces was given.
7
Gave the definitions of continuous function and homeomorphism, and gave some examples.

In Topology two spaces are considered to be "the same" precisely when they are homeomorphic. A property is said to be topological if, whenever some space X possesses the property then so too do all spaces Y that are homeomorphic to X. (So, for instance, the "sum of the interior angles" is not a topological property of a polygon since it is 180 degrees for a triangle, and 360 degrees for a quadrangle, and yet a triangle is homeomorphic to a quadrangle.)

I also showed the classic example of a homeomorphism between a doughnut and a coffee mug.
8
Explained that the real interval (-1,1) is homeomorphic to the real line.

Proved that "connectedness" is a topological property, and used this property to prove that the real line is NOT homeomorphic to the real plane.

Finished up by justifying the use of the term "compact space" in favour of "finite space" or "bounded space", and explained one reason why we'd like to prove that "compactness" is a topological property. The precise definition of "compactness" will be given in the next lecture.
9
First in-class test on Monday 10 February 2020.

The test will consist of five questions taken from the problem sheet.
10
Gave the definition of compactness. Showed that the real line R is not compact. Showed that compactness is a topological property. Showed that the subspace [0,1] of the real line IS compact. Hence the real line is not homeomorphic to the interval [0,1].
11
Started to construct a surjective continuous function
f:=[0,1] ---> Solid triangle
from the unit interval of the real line to an equilateral triangular region of the plane. The function f is constructed as the limit of a sequence of functions f1, f2, ... . Next lecture will see why f is continuous and surjective.
12
Introduced the notion of a Cauchy sequence and then stated the theorem which says that all Cauchy sequences converge.

Using this theorem on Cauchy sequences we saw that the function
f:[0,1] ---> unit trianglular region
defined last time is indeed well-defined.

We observed that f is continuous (at least in the sense that nearby points get sent to nearby points).

In preparation for the proof that f is surjective we introduced the notion of a closed subset A in a topological space X. We also introduced the notion of a accummulation point of a subset A.

Ended by stating the theorem: a subset A is closed if and only if it contains all of its accumulation points. We'll prove this next lecture.
13
Proved that a set is closed if and only if it contains all of its accumulation points.

Explained why the proof of the surjectivity of our function
f:[0,1] ---> solid equilateral triangle
boils down to proving that Image(f) is closed. To prove this we introduced the notion of a Hausdorff topological space, and noted that Euclidean space Rn is Hausdorff. Gave an example of a space that is not Hausdorff.

Ended with a proof of the result: In a Hausdorff topological space X any compact subspace A is closed.

This Youtube video shows a range of space-filling curves (including our "triangular" one).
14
Introduced the notion of a simplicial complex K, and explained that a triangulation of a topological space X consists of a simplicial complex together with a homeomorphism h:|K| ---> X.
15
Defined the Euler characteristix χ(X) of a space X to be the Euler characteristic χ(K) of any simplicial complex K with |K| homeomorphic to X. Stated the theorem that is needed to make sure this definition is well-defined.

Gave some triangulations of a circle, torus, Möbius strip, 2-sphere and computed the Euler characteristic of these spaces from the triangulation.

Talked briefly about the Hauptvermutung.
16
Introduced the notion of "homotopy" between maps. Showed that any two maps X--->Y are homotopic when Y is a convex subset of Euclidean space. For given spaces X and Y we proved that homotopy is an equivalence relation on the collection of maps X--->Y.
17
Began by showing John Nash's PhD thesis in which he uses Brouwer's theorem to prove the existence of (what are now called) mixed Nash equilibria.

Defined what is means for two spaces to be homotopy equivalent. Showed that any two homeomorphic spaces are homotopy equivalent. Also showed that the space C\{0} of non-zero complex numbers is homotopy equivalent to a circle.

Stated a major theorem.

Theorem. Homotopy equivalent spaces have the same Euler characteristic.

Illustrated this theorem with a few examples.

Stated Brouwer's fixed-point theorem: any continuous function f:Δn ---> Δn has a fixed point.
18
A video of the lecture is available here.

Stated, illustrated, and proved Brouwer's fixed point theorem. Then used it to show that any nxn matrix of positive real numbers has at least one eignevector, and that this eigenvector can be chosen so that its entries are all non-negative.

Then showed a Holywood clip on Nash Equilibria which includes a shot of John Nash's PhD thesis in which he uses Brouwer's theorem to prove "the existence in any game of at least one equilibrium point".
19
A video of the lecture is available here. But I forgot to connect the microphone so the sound is a bit noisy.

Introduced the fundamental group of a space. Indicated why the fundamental group of the circle S1 is isomorphic to the additive group of the integers Z. This isomorphism was used to define the winding number of an arbitrary map f:S1 ---> S1.
20
A video of this lecture is available here.

Used the fundamental group of a circle to prove the Fundamental Theorem of Algebra.

Then introduced the notions of
(i) an n player game and
(ii) a pure Nash equilibrium.
We saw two examples, one with pure Nash equilibria and the other without any pure Nash equilibrium.
21

Second in-class test: 12.00pm on Monday 23 March

The in-class will be delivered through the Blackboard assignments system.
22
A video of this lecture is available here.

Defined a mixed Nash equilibrium and stated Nash's Theorem: in any game with finitely many players with finite strategy sets there exists a mixed Nash equilibrium.

Gave an outline proof of this theorem using Brouwer's fixed point theorem.
23

24