What will I study in this module?

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.

How will the module be assessed?

There will be:

  • 15% for online homeworks.
    There will be three online Oksuon homeworks, each worth 5%.
  • 30% for class tests.
    There will be three 50-minute class tests, each worth 10%. These will be delivered as Blackboard assignments. The three tests will be based on the tutorial problem sheet available here.
  • 25% for project work.
    Students will be asked to work in pairs, and asked to submit a project. The project will count for 25% of the assessment and both students in a pair will receive the same score. I'll post more details about this around Week 6.
  • 30% for end of semester exam.
    I'll post more details here later in the semester.

Live lectures, tutorials, textbook, and contact

  • Lecturer: Graham Ellis.
  • Lectures: Students can participate in live Zoom lectures at 12 noon Mondays and 12 noon Wednesdays. The link will be posted on Blackboard and circulated by email to registered students prior to the lectures.

    Lectures will be recorded and the videos will be uploaded to Youtube at the links posted below. If you happen to miss a live lecture you can always watch it dead on Youtube. Only the material on my white board and oral conversations will be recorded. The chat won't be recorded and your images won't be recorded. Please do try to keep your cameras on during the lectures and use the chat or the microphone to make comments/ask questions. Facial feedback and live chat/questions usually have a significant positive impact on my delivery of lecturers -- so if your cameras are all turned off, and the chat is silent, then don't blame me for my boring delivery!
  • Workshops: Weekly tutorials will be delivered by Kelvin Killeen, starting in Week 2. I'll post details here as soon as I have them.
  • Recomended text: The module aims to cover a selection of material from

    Basic Topology by M.A. Armstrong (Springer New York)


    The book is available online through the James Hardiman Library and I'll do my best to 'force' students to read parts of it.
  • Module Website: Information and module documents will be posted to this site, which is linked from the Blackboard Topology MA342 (2020-21) pages. Blackboard will also be used for announcements, quizzes, second semester exams, and for posting grades.

Homework sheets and Deadlines

The online Okuson homework sheets are available here.

The tutorial problem sheet is available here.

Deadlines

Friday 26 February, 5pm First Okuson homework sheet
Wednesday 10 March, 12 noon First class test
Friday 26 March, 5pm Second Okuson homework sheet
Wednesday 14 April, 12 noon Second class test
Friday 23 April, 5pm Third Okuson homework sheet
Wednesday 05 May, 12 noon Third class test

Recorded lectures and notes

Lecture Notes

click number to view notes
Lecture Videos, Summaries and Other Material
Lecture 1
Video of the lecture.
Waiting room video: interview with Edward Frenkel.

The lecture was based on Section 1.1 and 1.2 of Basic Topology by M.A. Armstrong.

I explained that
topology is the study of those properties of a space that remain unchanged throughout any continuous deformation of the space.
The usual map of the London Underground is a continuous deformation of an accurate geographic map; the usual deformed map retains topologocal properties sufficient for passengers to plan their journey.

I then defined the Euler characteristic (=V-E+F) of a spherical surface such as the surface of Mars or the surface of a soccer ball, and informally proved 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 any fullerene molecule, and the number of pentagons in any 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.
Lecture 2
Video of the lecture.
Waiting room video: interview with Sylvia Serfaty.

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}|. 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
Video of the lecture.
Waiting room video: interview with Dusa McDuff

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 concepts in its proof. Eventually you'll be able to read page 112 of Armstrong's book for more details on the Jordan Curve Theorem -- but there are some definitions and tools you need to acquire first.

Ended by giving the definition of a topological space (Definition 2.1 on page 28 of Armstrong's book). Next lecture we'll consider more examples aimed at reinforcing our understanding of this definition.
Lecture 4
Video of the lecture.
Waiting room lecture: interview with Curti McMullen

Gave the definition of a topological space, and of a connected topological space, and illustrated these two definitions with examples.
Lecture 5
Video of the lecture.
Waiting room video: interview with Olga Paris-Romaskevich.

Gave the definitions and supporting examples of:
a subspace topology and a topological subspace;
a connected component of a topological space;
a continuous function between two topological spaces.
Lecture 6
Video of the lecture.
Waiting room video: interview with Roger Penrose.

Tried to explain the basics of how topology is transforming certain aspects of data analysis. Started by talking about the successful Ayasdi company which uses topological methods to investigate financial and other data. Then talked about phylogenetic trees and the related notion of "barcodes". This paper by Gunnar Carlsson and others uses topological data analysis to study the evolution of viruses in pandemics.
Lecture 7
Video of the lecture.
Waiting room video: Who cares about topology

Introduced and illustrated the notion of homeomorphic topological spaces. A topological property is a property which holds for a space X only if it holds for all spaces homeomorphic to X.
Lecture 8
Video of the lecture
Waiting room video: The Borsuk-Ulam Theorem and stolen necklaces

Showed that the interval (-1,1) is homeomorphic to the real line R and that the real line R is not homeomorphic to the plane R2. For the latter we had to prove that connectedness is a topological property.
Lecture 9

Lecture 10

Lecture 11

Lecture 12

Lecture 13

Lecture 14

Lecture 15

Lecture 16

Lecture 17

Lecture 18

Lecture 19

Lecture 20

Lecture 21

Lecture 22

Lecture 23

Lecture 24

Supplementary material

STUDENT FEEDBACK

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