543
These are lecture notes for the course on geometry
and topology in physics which I teach along with Charles Nash. My
half ment a series of nine two hour lectures intended to cover the
more algebraic topological part of geometry and topology in physics. I
have now finished my half of the course and will be giving no more
lectures.
Lecture 1
Introduction to manifolds and
homotopy.
| Lecture 2
Homotopy,
definitions and theorems, winding number as momentum: T-duality.
| Lecture 3
Winding number as particle
number: the sine-Gordon kink. Triangulation and calculating the
fundamental group.
| Lecture 4
Boundary operators, the definition of the homology groups.
| Lecture 5
Calculating homology groups, first part of exact sequences.
| Lecture 6
Exact sequences continued including examples. The Mayer-Vietoris sequences. Singular homology.
| Lecture 7
Higher homotopy.
| Lecture 8
Introduction to differential geometry: vectors, forms and the exterior derivative.
| Lecture 9
Introduction to de Rham cohomology.
All these lectures are also available as pdf:Lecture 1 Lecture 2
Lecture 3 Lecture 4 Lecture 5
Lecture 6
Lecture 7 Lecture 8 Lecture 9. The source files are available: LaTeX and eps and fig and all material is released under the GFDL.
Text books.The course is based on the text book Nash and Sen
and the notes above are often very close to that book. The book
Nakahara is very similar but with less homology and cohomology; it
does not mention the important topic of exact sequences. It does have
more physical examples. Spivak is of course a classic and has some
intersection with the material I am trying to cover. The book by Hatcher
is very good too, it covers a lot more than this course does and in
more detail, sometimes the presentation is formidable. Flanders is
good on forms and vectors. There are nice soliton collision
animations to be found on the web.
The future.
Since the notes were written as I went along, they have become a bit
incoherent. I will rectify this next year (2001/2002) if I am
teaching the course again. I also intend to make some changes of
material, I will do less homotopy and more differential geometry. In
particular, I will not discuss the calculation of homotopy next year,
I will do higher homotopy earlier and leave out relative homotopy. I
will try and do homology in two lectures. I will then add Lie
derivatives and some Lie groups and algebras and maybe a bit of
algebraic geometry. I am quite keen on any comments on the notes,
please email me.
Back to home page.
The picture was
taken from http://www.britmovie.co.uk/studios/ealing/filmography/64i.html
Last updated 12 February, 2001.
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