What is An Invitation to Noncommutative Geometry and why should you read it?

[marcus]

The Berkeley Math department has TWO seminars on NCG this semester. One is run by graduate students on their own. I went to it Monday. It was good. I'm too cheap to buy the book they are using. Alain Connes has his book called "Noncommutative Geometry" free online at his website. I might be able to follow a few more. tomorrow I'll try the faculty-run seminar.

It's hot. Everybody knows this. The graduate students are looking to do their research in it. Last year both John Barrett and Alain Connes got the Standard Model out of NCG plus some predictions.

There are people here like Alejandro who have studied NCG for years and were at Cambridge Newton Institute listening when Connes was there last year.

It would be nice if some people with NCG savvy would explain some things to the rest of us.

Why I'm especially anxious to learn some in a hurry is this: Jasper Grimstrup and Hans Aastrup have found a way to mate NCG with a new version of LQG that they are working out.
I have seen only their preliminary 2005 and 2006 papers. If things happen as I expect there will be some 2007 papers and some kind of NCG+LQG crossbreed will be a quantum theory of gravity that gives you the Standard Model.

Smolin has been working on another way of doing this (as Carl Brannan and others have discussed here at PF) just since 2005.

Garrett Lisi has a way of doing it which he described at Loops 07.

all this is happening very fast. But Alain Connes approach is longstanding, it goes back to a 1996 paper (or maybe Alejandro knows some earlier). All these other approaches may turn out to work and may be really extraordinary and elegant eventually, while Connes with the help of Grimstrup has the air of working NOW.

 

[arivero]

all this is happening very fast. But Alain Connes approach is longstanding, it goes back to a 1996 paper (or maybe Alejandro knows some earlier).

http://dftuz.unizar.es/~rivero/research/ncactors.html

Connes-Lott model was published in
Nuclear Physics B - Proceedings Supplements
Volume 18, Issue 2, January 1991, Pages 29-47

1996 is the first evolution, disposing of bivectors fields and using real (or Real) structure. 2006 is a second evolution changing the topological dimension of the 0-dimensional part.

 

[arivero]

Note also

Noncommutative Differential Geometry and New Models of Gauge Theory.
Michel Dubois-Violette (Orsay, LPT) , Richard Kerner (Paris U., VI-VII) , John Madore (Orsay, LPT) . LPTHE-ORSAY-88/58, (Received Jan 1989). 26pp.
Published in J.Math.Phys.31:323,1990.Noncommutative geometry and theoretical physics
R. Coquereaux*
Journal of Geometry and Physics
Volume 6, Issue 3, 1989, Pages 425-490

 

[nitin]

"An Invitation to NCG" out this month

I didn't want to start a new thread for this announcement, so I hope no one objects to I mentioning it here.

"An Invitation to Noncommutative Geometry" http://www.worldscibooks.com/physics/6422.html" (MPI für Mathematik, Bonn) is scheduled to be out this month. The lectures that figure in this volume (except for the one by Grosse and Wulkenhaar) have been on the arXiv for some time now, so I shall list-link them here:

1. A Walk in the Noncommutative Garden (A. Connes & M. Marcolli) http://arxiv.org/abs/math.QA/0601054"
2. Renormalization of Noncommutative Quantum Field Theory (H. Grosse & R. Wulkenhaar)
3. Lectures on Noncommutative Geometry (M. Khalkhali) http://front.math.ucdavis.edu/math.QA/0702140"
4. Noncommutative Bundles and Instantons in Tehran (G. Landi & W. D. van Suijlekom) http://front.math.ucdavis.edu/0603.3053"
5. Lecture Notes on Noncommutative Algebraic Geometry and Noncommutative Tori (S. Mahanta) http://arxiv.org/abs/math/0610043"
6. Lectures on Derived and Triangulated Categories (B. Noohi) http://arxiv.org/abs/0704.1009"
7. Examples of Noncommutative Manifolds: Complex Tori and Spherical Manifolds (J. Plazas) http://arxiv.org/abs/math.QA/0703849v1"
8. D-Branes in Noncommutative Field Theory (R. J. Szabo) http://arxiv.org/abs/hep-th/0512054"

Tonight's bedtime reading (if you haven't digested it already): An Interview with Alain Connes http://www.ipm.ac.ir/IPM/news/connes-interview.pdf", featuring A. Connes, M. Khalkhali and G. B. Khosrovshahi.