TOPOLOGY MA342

Module homepage for 2016-17

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. 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.

(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 (and here is last year's handwritten version of the lecture)

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 "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). Ended up with a mention of how topology is currently being applied to the study of data. See the Ayasdi company website for more about applications of topology to data analytics.

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

Started with a "proof" that the Euler characteristic of the 2-sphere S2 is V-E+F=2. The proof used the fact that any simple closed curve on the sphere cuts the sphere into two pieces. Then observed that a simple closed loop on a möbius strip or on a torus does not necessarily cut the space 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.

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 Explained what is meant by a "topological property" of a space, and proved that connectedness is a topological property. Also gave the definition of compactness.

Lecture 9

Showed that the real line R is not compact. Showed that compactness is a topological property. Showed that [0,1] is compact. Hence [0,1] is not homeomorphic to the real line. (Exercise: Show that (0,1) is homeomorphic to the real line.)
br> Test 1

Lecture 11

Constructed a space filling curve from [0,1] to the solid equilateral triangle. But didn't establish that the curve f:=[0,1] ---> Triangle is continuous or onto.

Lecture 12

Explained why f:=[0,1] ---> Triangle is continuous. In order to establish the surjectivity of f the notions of "closed subset" and "accumulation point" were introduced.
The lecture was hampered by: 1) having to take my shoe off due to a sprained/injured foot; 2) forgetting my glasses; 3) forgetting to bring a working black pen; 4) forgetting to bring a watch.

Lecture 13

To complete the proof of the surjectivity of f:=[0,1] ---> Triangle we introduced the notion of a Hausdorff topological space, and then proved that in a Hausdorff space any compact set is closed. At the end of the lecture we introduced the notion of "general position" for vectors in Euclidean space, and we introduced the notion of a "convex subset" of Euclidean space.

Lecture 14

Introduced the notion of a "simplicial complex" and the notion of a "triangulation" of a topological space.

Lecture 15

Gave some examples (and non-examples) of triangulations. Stated the fundamental theorem which says that two simplicial complexes with homeomorphic undelying topological spaces have equal Euler characteristics. Using this theorem and the notion of triangulation we obtained the definition of the Euler characteristic of a (triangulable) topological space. Talked about the Hauptvermutung.

Lecture 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.

Lecture 17

Explained what it means for two topological spaces to be homotopy equivalent. Then stated the fundamental theorem underlying the module:

If two finitely triangulable spaces are homotopy equivalent then they have the same Euler characteristic.

Illustrated this theorem by calculating Euler characteristics of various spaces. Ended up by stating Brouwer's theorem: any continuous map f:Dn ---> Dn has at least one fixed point.

Lecture 18
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.)

Showed 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 19
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.



Lecture 20
Used the fact that the fundamental group of the circle is bijective with the integers to prove the fundamental theorem of algebra: every polynomial of degree n>0 over the complex numbers has at least one zero. [The proof did not use the group structure of the fundamental group.]

Introduced the notion of an n-player game, and defined a pure Nash equilibrium. Gave some examples

Lecture 21
Defined a mixed Nash equilibrium and stated John Nash's famous theorem: every n-played game with a finite number of strategies has at least one mixed Nash equilibrium. Gave a sketch proof of Nash's theorem based on Brouwer's theorem. This si the proof given by Nash in his PhD thesis.