Via Peter Woit, I learn that Alain Connes has made a download of the full text of his book Noncommutative Geometry available from his website.

This is a subject I know very little about, despite my interest in geometry of all sorts. My general impression is that it's a very abstract algebraic approach to the subject, viewing spaces in terms of the commutative ring of polynomials over coordinates for the spaces, and extending that viewpoint to noncommutative rings. But apparently it can be used to describe concrete geometric objects such as Penrose tilings. I've considered buying this book for a long time, but had been stopped by the likelihood that the mismatch between the symbolic reasoning needed for this sort of algebra and the more visual way I prefer to think would mean I would find it very tough going, and that I'd end up feeling like I'd wasted my money; now that barrier is gone. So I'll be interested to take a look at the book and see how far through it I can get.




Comments:

mattimooz:
2006-05-23T12:55:08Z

Hi

My name is Martha I'm a third year Mathematics student at Birmingham in the UK.

I've been looking at the links you've left as I find them very interesting and am currently constructing an essay for my exam in the history of mathematics next week. I cannot however find an answer to a sub question:

Does the ellipsoid method solve the travelling saleman problem?

Any ideas on where I can find the answer to this?

Thanks in advance :)

11011110:
2006-05-23T15:24:51Z

Short answer: no.

Longer answer: the ellipsoid method is good for linear programs, and convex programs, in which one is trying to find a real-valued vector subject to linear constraints that optimizes a linear or convex objective function. The TSP can be set up to look similar to one of these problems, but with an integer-valued vector as the desired answer. So if you try to use the ellipsoid method to solve it, it will tell you an answer involving using e.g. 1/2 of one edge and 1/2 of another instead of just a set of edges forming the desired cycle. Linear program solvers can be useful as subroutines in solving the TSP (e.g. the Concorde TSP solver) but they don't do it by themselves.