Michaelmas 2024: Algebraic topology (Part II)

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Resources for Part II Differential Geometry: 

Lecture notes:

• I recommend Prof Paternain's excellent typed notes here. 

• Lent 2022: handwritten lecture notes are available in a OneNote folder. They should match the lecture recordings, and follow Paternain's notes reasonably closely, with lots of added pictures.  

If you would like access to these, please email me (amk50) from your university account. You are also welcome to get access if you are taking or considering taking Part II Differential Geometry in a subsequent year.

Minimal surfaces (Chap 3): this is some really beautiful mathematics! There's lots of resources online to help you visualise them, for instance:

• Database with lots of minimal surfaces: here. (There's also out-of-date mathematica code for them, but it's easy to use it to just get the functions.)

• Students can get free mathematica licenses through the university (instructions here).  Folder with mathematica code for the surfaces we saw in lectures (+ some screenshots of them).

• Movie transforming the helicoid to the catenoid: https://minimal.sitehost.iu.edu/archive/Classical/Classical/AssociateCatenenoid/web/qt.mov 

• Soap films: clip from the Maths installation at the Museum of Science in Boston here (start at 25 seconds). [There's a lot of other great maths exhibits there too!] Soap helicoid in this article, and soap catenoid here. If you start googling it you can quicky unearth lots of others!

Articles for some of the theorems towards the end of the course (you will need to be on the university network or to login to mathscinet first)

• • Fáry–Milnor theorem: Milnor, On the total curvature of knots (1950); Fáry, Sur la courbature total d'une courbe gauche faisant un noeud (1949)

  • Fenchel's theorem: orignal article (1929, in German), English version in the Bulletin of the American Math society (1951)

  • Xavier article proving that a complete minimal surface has a Gauss map omiting at most six points. (The optimal result, namely at most four points, is due to Fujimoto, and a lot more involved.)

  • Gauss' original book is called `Disquisitiones generales circa superficies curvas'. An English translation (together with a transctiption of the original) is available here. For a digitalisation of the original book, see e.g. here. (There are other papers of his in the same file. Even if you don't know a word of latin, you can scroll through and recognise some friends, for instance, the first fundamental form (with the notation using E, F, G); the area of a geodesic triangle at the start of p. 35; change of variables formulae in section 21; geodesic polars in section 22.) 

Past teaching includes:

LT 2021, 2022: Part II Differential Geometry

MT 2021: Topics in Symplectic Topology (graduate course)

LT 2019, 2020, 2021: Part III Symplectic Topology

MT 2017, 2018, 2019: IB Linear Algebra