math-books
Im in the process of writing/revising/extending the course notes for next year and will therefore pack more math-books than normal.
These are for a 3rd year Bachelor course on Algebraic Geometry and a 1st year Master course on Algebraic and Differential Geometry. The bachelor course was based this year partly on Miles Reid’s Undergraduate Algebraic Geometry and partly on David Mumford’s Red Book, but this turned out to be too heavy going. Next year I’ll be happy if they know enough on algebraic curves. The backbone of these two courses will be Fulton’s old but excellent Algebraic curves. It’s self contained (unlike Hartshorne’s book that assumes a prior course on commutative algebra), contains a lot of exercises and goes on to the Brill-Noether proof of Riemann-Roch. Still, Id like to extend it with the introductory chapter and the chapters on Riemann surfaces from Complex Algebraic Curves by Frances Kirwan, a bit on elliptic and modular functions from Elliptic curves by Henry McKean and Victor Moll and the adelic proof of Riemann-Roch and applications of it to the construction of algebraic codes from Algebraic curves over finite fields by Carlos Moreno. If time allows Id love to include also the chapter on zeta functions but I fear this will be difficult.
These are to spice up a 2nd year Bachelor course on Representations of Finite Groups with a tiny bit of Galois representations, both as motivation and to wet their appetite for elliptic curves and algebraic geometry. Ive received Fearless Symmetry by Avner Ash and Robert Gross only yesterday and find it hard to stop reading. It attempts to explain Galois representations and generalized reciprocity laws to a general audience and from what I read so far, they really do a terrific job. Another excellent elementary introduction to elliptic curves and Galois representations is in Invitation to the Mathematics of Fermat-Wiles by Yves Hellegouarch. On a gossipy note, the appendix “The origin of the elliptic approach to Fermat’s last theorem” is fun reading. Finally, Ill also take Introduction to Fermat’s Last Theorem by Alf van der Poorten along simply because I love his writing style.
These are included just for fun. The Poincare Conjecture by Donal O’Shea because I know far too little about it, Letters to a Young Mathematician by Ian Stewart because I like the concept of the book and finally The sensual (quadratic) form by John Conway because I need to have at all times at least one Conway-book nearby.
Conway, differential, Galois, geometry, groups, modular, representations, Riemann, symmetry
4 comments
Posted in general
Written on Thu, 26 July 2007 at 5:23 pm
Tags: Conway, differential, Galois, geometry, groups, modular, representations, Riemann, symmetry
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July 26th, 2007 at 10:59 pm
What, no Harry Potter?
August 16th, 2007 at 3:26 am
I just completed my first read through of Ash and Gross ‘Fearless Symmetry’.
The Monster is mentioned on page 155, section, ‘Digression: The Inverse Galois Problem’, chapter 13 ‘The Galois Group of a Polynomial’, part 2 ‘Galois Theory and Representations’.
I really enjoyed this book for the superb explanation of difficult concepts. There are few proofs, but proof references are provided.
Their writing style is similar to two of the references: - Paul J Nahin, ‘An imaginary Tale: The Story of “i”; - Mario Livio, The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number.
I recall figure 122 in Livio as a contrast between ideal and perturbed planetary orbits. A similar situation may occur in nature between ideal and perturbed symmetry?
August 24th, 2007 at 4:30 pm
It seems you are back from vacation. Was your reading enjoyable?
August 24th, 2007 at 5:12 pm
A direct question deserves a similar reply. I hate to admit it but in the end I decided to take along just 3 of the books mentioned (fearless symmetry, poincare conjecture and the sensual form; i did read the stewart book before we took off). I agree with Doug (for once) that fearless symmetry is an excellent read. If Im allowed to make one critical remark, as quadratic reciprocity is so crucial in their story maybe they should have considered including the amazing proof of it contained in the appendix of Conway’s sensual form-book. So i left all three curve-books at home but took instead good-old Fulton along as well as Tsfasmann’s ‘Algebraic geometric codes’ which I recommend as a good read. In retrospect Im happy with this because sitting on a French mountain (for Gaspard, the Ardeche-Cevenolles) makes one contemplate things, so Ive changed my plans for next year courses completely as well as my overall attitude to teaching and mathematics, but maybe ill post about that another time… Apart from the math-books mentioned already I took along at the last moment Gannon’s book on Monstrous Moonshine and Hsu’s on Quilts but only read through them for one afternoon. As usual I took along some trashy novels and as the weather was bad but excellent for cycling I was too tired to do my usual amount of reading. Still, two thrillers I read through and enjoyed : “Blood memory” by Greg Iles and “Buried” by Mark Billingham. And one I stopped reading “Sign of the cross” by Chris Kuzneski for the obvious reasons. That’s about it Im afraid.
Oh, and btw. for Graham : I was in London the weekend the latest Potter was released and naturally I bought one at Waterloo station on the way back for my God child. I briefly contemplated taking it along on vacation but in the end there are only that many books you can carry…