# Module homepage for 2014-15

## 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.

If you'd ike to do a final-year project in topology then please get in touch.

## 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.

If you'd ike to do a final-year project in computational topology then please get in touch

## 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.

If you'd ike to do a final-year project on applications of fixed pont theory in economics then please get in touch.

## 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.

 Text:  The lectures will be 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.  (Another good topology book is Undergraduate Topology - A Working Textbook by Aisling McCluskey and Brian McMaster, Oxford University Press . This text places more emphasis on the importance of topology in analysis. It is available here and I think there are a few copies in the library.) Assessment: The end of semester exam will count for 70% of the assessment. The continuous assessment will count for 30% of the assessment. Continuous 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. Homework sheet will be available here. Lectures take place at 12 pm Monday in the McMunn Theater and 12pm Wednesday in AC215. The lecturer is Graham Ellis. Tutorials take place at 1pm Tuesday in AC214 and Wednesdays at 2pm in AC201. The tutor is Adib Makroon. Student feedback on this module will be posted here. Previous years: The material covered in the module this year is a bit different to previous years. For that reason the exam questions will be very closely based on the homework/tutorial sheet.

Lectures: I plan to write everything in lectures on old fashioned overhead transparencies and then place copies of these transparencies below after each lecture.

Lecture 1

Lecture 2

Lecture 3 and research article by Y. Baryshnikov and R. Ghrist.

Lecture 4

Lecture 5

Lecture 6

Lecture 7 and here is the link to the topological data analysis company Ayasdi. The company provides topological analysis for many areas, and in particular for financial services.

Lecture 8

Lecture 9 and a link to a popular animated gif file showing how a coffee cup is homeomorphic to a doughnut.

Lecture 10

The first in-class test will take place on Monday 16 Ferbruary.

Lecture 11

Lecture 12 and a link to an easy proof of the Jordan Curve Theorem for piecewise linear curves.

Lecture 13

Lecture 14

Lecture 15

Lecture 16

Lecture 17 and a youtube clip on Nash Equilibria and a link to John Nash's PhD thesis. The main result in the thesis is the existence in any game of at least one equilibrium point.

Lecture 18

The second in-class test will take place on Monday 23 March.

Lecture 19

Lecture 20 and here is a link to a more complete account of Nash's Theorem.

Third Test