“Lecture Geometry and topology”

What?

The aim of the unit is to expand visual/geometric ways of thinking. The Geometry section is concerned mainly with transformations of the Euclidean plane (that is, bijections from the plane to itself), with a focus on the study of isometries (proving the classification theorem for transformations which preserve distances between points), symmetries (including the classification of frieze groups) and affine transformations (transformations which map lines to lines). The basic approach is via vectors and matrices, emphasizing the interplay between geometry and linear algebra. The study of affine transformations is then extended to the study of collineations in the real projective plane, including collineations which map conics to conics. The Topology section considers graphs, surfaces and knots from a combinatorial point of view. Key ideas such as homeomorphism, subdivision, cutting and pasting and the Euler invariant are introduced first for graphs (1-dimensional objects) and then for triangulated surfaces (2-dimensional objects). Topics include the classification of surfaces, map coloring, decomposition of knots and knot invariants.

Contact

Daniel Tubbenhauer email

Please put [Lecture Geometry and topology] as the subject.

Who?

Second semester students in Mathematics interested in a mixture of (linear) geometry and discrete mathematics, but everyone is welcome.

Where and when?

The lecture.
- Monday 10:00-11:00, Thursday 09:00-10:00, Friday 09:00-10:00.
- Starting 7th week, ending 12th week.
- Carslaw Lecture Theatre 275 (Monday) and 373 (Thursday and Friday) and online via zoom.
My tutorial.
- Friday 13:00-14:00.
- Starting 8th week, ending 13th week.
- Carslaw Seminar Room 350.
Material for the lecture.
- There is a script [Hi11] available via Canvas that the lecture will follow. Additional literature (not mandatory but recommendations only). The recommended literature from the course outline is [Ad94], [Bl67] and [FiGa91]. The lecture sometimes takes a different perspective and I sometimes borrow the exposition from [Ba10], [BoMu08], [BrHa12], [We96] or [Wi96] for graphs, and from [A+21], [FaSt96] and [Ka93] for surfaces and knots
- Prerecorded lectures on the “What is...algebraic topology?” and “What is...geometric topology?” (twice: as mini lectures and the real lectures) playlists here: Click
Summary: Click
Mini presentation: Click
The detailed plan: Click
All slides: Click or Click

Schedule (there might be some fluctuations around the begin/end of every week)

Speaker: Daniel Tubbenhauer, Topic: Basics about graphs - Graphs, subdivision, trees, Eulerian circuits Slides: Click
Speaker: Daniel Tubbenhauer, Topic: Surfaces I - Various surfaces, homeomorphism, Euler characteristic Slides: Click
Speaker: Daniel Tubbenhauer, Topic: Surfaces II - Invariance under subdivision, cutting and pasting, orientation Slides: Click
Speaker: Daniel Tubbenhauer, Topic: Surfaces III - Classification of surfaces Slides: Click
Speaker: Daniel Tubbenhauer, Topic: Graphs and surfaces - Graphs on surfaces, planar graphs Slides: Click
Speaker: Daniel Tubbenhauer, Topic: Knots - Knots diagrams, knot coloring, Seifert surfaces Slides: Click
Speaker: Daniel Tubbenhauer, Topic: Geometric topology - recap Slides: Click

Exercises

The exercises correspond 1:1 to the talks in the list above.

Exercise 7, Click, Solutions: Click
Exercise 8, Click, Solutions: Click
Exercise 9, Click, Solutions: Click
Exercise 10, Click, Solutions: Click
Exercise 11, Click, Solutions: Click
Exercise 12, Click, Solutions: Click

References

Here are a few references used in this lecture:

[Ad94] C.C. Adams. The knot book. An elementary introduction to the mathematical theory of knots. Revised reprint of the 1994 original. American Mathematical Society, Providence, RI, 2004. xiv+307 pp.
[A+21] Edited by C. Adams, E. Flapan, A. Henrich, L.H. Kauffman, L.D. Ludwig and S. Nelson. Encyclopedia of knot theory. CRC Press, Boca Raton, FL, [2021], @2021. xi+941 pp.
[Ba10] R.B. Bapat. Graphs and matrices. Universitext. Springer, London; Hindustan Book Agency, New Delhi, 2010. x+171 pp.
[Bl67] D.W. Blackett. Elementary topology. A combinatorial and algebraic approach. Academic Press, New York-London 1967 ix+224 pp.
[BoMu08] J.A. Bondy, U.S.R. Murty. Graph theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. xii+651 pp.
[BrHa12] A.E. Brouwer, W.H. Haemers. Spectra of graphs. Universitext. Springer, New York, 2012. xiv+250 pp.
[FaSt96] D.W. Farmer, T.B. Stanford. Knots and surfaces. A guide to discovering mathematics. Mathematical World, 6. American Mathematical Society, Providence, RI, 1996.
[FiGa91] P.A. Firby, C.F. Gardiner. Surface topology. A guide to discovering mathematics. Second edition. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood, New York; distributed by Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. 220 pp.
[Hi11] J. Hillman. Topology. Lecture notes for the Topology component of Geometry and Topology. Available via Canvas.
[Ka93] L.H. Kauffman. Knots and physics. Second edition. World Scientific Publishing Co., Inc., River Edge, NJ, 1993. xiv+723 pp.
[We96] D.B. West. Introduction to graph theory. Prentice Hall, Inc., Upper Saddle River, NJ, 1996. xvi+512 pp.
[Wi96] R.J. Wilson. Introduction to graph theory. Fourth edition. Longman, Harlow, 1996. viii+171 pp.