From: baez@galaxy.ucr.edu (John Baez)
Subject: Re: Atiyah Singer and Chern
Date: 27 Oct 2000 23:02:31 GMT
Newsgroups: sci.physics.research

In article <8t6lmg$6kt$1@nnrp1.deja.com>,  <tedsung6674@my-deja.com> wrote:

>I've heard it mentioned a few times but what is the Atiyah Singer
>(index) theorem and what relevance does it play in physics.

In a nutshell, it relates the topology of spacetime to the solutions 
of certain field equations that come up a lot in physics, like the
Dirac equation.  It turns out that you can learn a lot about topology
from  studying partial differential equations, and vice versa.

>Also, what is the importance of Chern classes? 

The importance of Chern classes is that if you don't know what they
are, you don't have a snowball's chance in hell of understanding the
Atiyah-Singer index theorem.  

>And is there any "introductory" reading on the web about these?

I think you want BOOKS.  Try these:

Charles Nash and Siddhartha Sen, Topology and Geometry for Physicists, 
Academic Press, 1983.  (This emphasizes the physics motivations... it's not
quite precise at points.) 

Mikio Nakahara, Geometry, Topology, and Physics, A. Hilger, New York, 
1990.    (More advanced, but more precise.)

Topology and analysis: the Atiyah-Singer index formula and gauge-theoretic
physics / B. Booss, D.D. Bleecker ; translated by D.D. Bleecker and A. Mader.
New York : Springer-Verlag, c1985.   (This will really give you the stuff,
but without the others you may have a bit of trouble.)  
==============================================================================

From: Chris Hillman <hillman@math.washington.edu>
Subject: Re: Atiyah Singer and Chern
Date: 28 Oct 2000 15:48:12 GMT
Newsgroups: sci.physics.research

On Wed, 25 Oct 2000 tedsung6674@my-deja.com wrote:

> I've heard it mentioned a few times but what is the Atiyah Singer
> (index) theorem and what relevance does it play in physics.
> Also, what is the importance of Chern classes?  And is there any
> "introductory" reading on the web about these?

I can suggest two print resources: vol. 5 of Spivak's differential
geometry textbook, and Frankel, The Geometry of Physics, Cambridge U
Press, 1999.

Chris Hillman

Home Page: http://www.math.washington.edu/~hillman/personal.html