# Module homepage for 2015-16

## 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 and last 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
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 of a surface such as the surface of Mars, and "proved" that this number is a topological property of the surface. Ended up by using 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).

Lecture 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. 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.

Lecture 3
Gave some motivation for a more precise approach to topology! Then stated the Jordan Curve Theorem and started towards explaining the terms in its statement.

Lecture 4
Gave the definition of a topological space, and of a connected topological space.

Lecture 5
Gave the definition of a topological subspace and of a connected component of a space. Ended up showing a four-minute video by Gunnar Carlsson of Stanford University on Topological Data Analysis -- he mentions connected components in the video. The video clip can be found on the Ayasdi website -- this is the company founded by Carlsson and others to help companies investigate large, high-dimensional data sets.

Lecture 6
Explained how connected components of a sequence of spaces give rise to a dendragram. An early example of a dendragram is the "tree of life" introduced by Charles Darwin. Ended the lecture with some more mainstream material: the definition of a continuous function.

Lecture 7
Gave the definitions of continuous function and homeomorphism. Gave some examples, including the classic doughnut being homeomorphic to a coffee mug, and ended up by proving that a composite of continuous functions is continuous.

Lecture 8
Described a continuous function f from the unit interval [0,1] to a triangular region in the plane. The definition of f relied on the fact that any Cauchy sequence of points in the Euclidean plane has a limit. The proof that f is surjective was deferred: it requires the notion of compactness (see the textbook).

Lecture 9 First Test

Lecture 10
Showed that connectedness is a topological property (i.e. a homeomorphism invariant). As a consequence we showed that [0,1] is not homeomorphic to R2. Introduced the notion of compactness.

Lecture 11
Proved that compactness is a topological property. Proved that [0,1] is a compact subspace of the real line. Introduced the notion of a closed subset. Introduced the notion of an accumulation point (also called a limit point).

Lecture 12
Proved that a set is closed if and only if it contains its accumulation points. Introduced the notion of a Hausdorff space. Proved that a compact subspace of a Hausdorff space is closed. Finally observed that our continuous map f:[0,1] --> (triangular subregion of the plane) really is surjective. (We've proved earlier that no such map f can be a homeomorphism.)

Lecture 13
Introduced simplicial complexes and gave the definition of a triangulable space.

Lecture 14
Gave some examples of triangulated spaces. Talked around "die Hauptvermutung".

Lecture 15
Introduced the notion of homotopic maps. Gave some examples, including the fact that any two maps X-->Y are homotopic if Y is convex. Proved that homotopy is an equivalence relation.

Lecture 16
Defined the winding number of a map S1 --> S1. The definition is based on the following proposition for which we gave a careful proof:

Proposition
Any map F:[0,1] --> S1 lifts to a map f':[0,1] --> R such that exp(f'(t)) = f(t) for all t in [0,1].

Here exp is the function exp:R--> S1, x--> e2x pi i. An analogous result, with similar proof, was stated without proof:

Proposition
If F:[0,1]x[0,1] --> S1 is a map such that F(0,t)=F(1,t) = 1 for all t in [0,1], then there is a unique map F':[0,1]x[0,1] --> R which satisfies exp(F'(t)) =F and F'(0,t)=0 for all t in [0,1].

Lecture 17
Proved that [S1, S1] is bijective with Z. Then used this bijection to prove the Fundamental Theorem of Algebra.

Lecture 18
Explained what it means for two topological spaces to be homotopy equivalent. Then stated a major theorem: homotopy equivalent triangulated spaces have the same Euler charactersitic. Illustrated this theorem on a few examples. The proof of this theorem is beyond the scope of our course (but a proof is given in the text book). Ended with the the statement of Brouwer's theorem: any continuous map f:Dn --> Dn on the n-disk has at least one fixetd point. Illustrated the theorem for n=1.

Lecture 19
Proved Brouwer's Theorem. Then used it to prove that any square matrix with only positive entries has at least one positive eigenvalue. (Google Perron's Theorem to find other proofs of this result.)

Lecture 20
Introduced the notions of pure strategy game, a mixed strategy game, a pure Nash equilibrium and a mixed Nash equilibrium. Stated John Nash's theorem:

In any game with finitely many players, and finitely many pure strategy sets Si, there exists at least one mixed Nash equilibrium.

Lecture 21
Gave an example of a mixed Nash equilibrium. Then used the Brouwer fixed-point theorem to prove Nash's theorem.

There are many examples of the use of homotopy theory in economics. For one fairly recent example see S. Weinberger's article On the topological social choice model published in the Journal of Economic Theory, (2004).

Lecture 21
Described some computational applications of topology.

Third Test