## 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 AC215.
• 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 70% of the assessment. The continuous assessment will count for 30% of the assessment.

The continuous assessment will consist of three equally weighted in-class tests 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 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24