Questions regarding non-commutative geometry

[tom.stoer]

I tried to understand Connes' approach several times but eventually I got stuck all the time. Does anybody know an introduction / review paper which explains the basic ideas, results and open issues?

 

[arivero]

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

First you would get the general idea from old papers. There was some from Gracia Bondia, Varilly, Coquereaux, Landi, to name a few. Got the point of the duality between a commutative algebra and a manifold?

 

[skippy1729]

I like "Introduction to Noncommutative Spaces and their Geometry" by Giovanni Landi, arxiv hep-th/9701078. Lots of examples and a large appendix with results from functional analysis and topology for background info. I too get stuck. Put it away and start over in a week or so. I usually get a little further before getting stuck again.

Skippy

 

[paolo_th]

Some introductory papers on A.Connes' non-commutative geometry:

R.Coquereaux, Noncommutative Geometry and Theoretical Physics, J.Geom.Phys. 6, 425-490 (1989).

R.Coquereaux, Noncommutative Geometry: a Physicist’s Brief Survey, J.Geom.Phys. 11, 307-324 (1993).

J.Varilly, J.Gracia-Bondia, Connes’ Noncommutative Differential Geometry and the Standard Model, J.Geom.Phys. 12, 223-301 (1993).

J.Varilly, An Introduction to Noncommutative Geometry, Summer School “Noncommutative Geometry and
Applications” Lisbon (1997). http://arxiv.org/abs/physics/9709045"

M.Khalkhali, Very Basic Noncommutative Geometry (2004). http://arxiv.org/abs/math/0408416"

M.Khalkhali, Lectures on Noncommutative Geometry (2007). http://arxiv.org/abs/math/0702140"

Intoductory books:

G.Landi, An Introduction to Noncommutative Spaces and Their Geometry, Springer Verlag (1997). http://arxiv.org/abs/hep-th/9701078"

J.M.Gracia-Bondia, H.Figueroa, J.C.Varilly, Methods of Noncommutative Geometry, Birkhauser (2001).

More advanced books:

A.Connes, Noncommutative Geometry, Academic Press (1994). http://www.alainconnes.org/docs/book94bigpdf.pdf"

A.Connes, M.Marcolli, Noncommutative Geometry, Quantum Fields and Motives, AMS (2007). http://www.alainconnes.org/docs/bookwebfinal.pdf"

For general background on Operator Algebras, among several books/notes:

N.Landsman, Lecture Notes on C*-algebras Hilbert C*-modules and Quantum Mechanics, http://arxiv.org/abs/math-ph/9807030"

R.V.Kadison, J.R.Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. I-II, AMS (1997).

G.J.Murphy, C*-Algebras and Operator Theory, Academic Press (1990).

O.Bratteli, D.W.Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol.I-II, Springer (1987-1997).

M. Takesaki, The Theory of Operator Algebras, Vol. I-II-III, Springer (2001-2002).

B. Blackadar, Operator Algebras, Springer (2006).

For K-theory and Cyclic Cohomology:

N.E.Wegge-Olsen, K-Theory and C*-Algebras a Friendly Approach, Oxford University Press (1993).

J.Brodzki, An Introduction to K-Theory and Cyclic Cohomology (1996). http://arxiv.org/abs/funct-an/9606001"

Fast introductions to Functional Analysis (as used in Operator Algebras):

V.S.Sunder, Functional Analysis: Spectral Theory, Birkhauser (1997),

G.K.Pedersen, Analysis Now, Springer (1995).

For general background in Differential Geometry, Clifford Algebras and Dirac Operators (among several books):

M.Nakahara, Geometry, Topology and Physics, Institute of Physics Publishing (1990).

L.Nicolaescu, Lectures on the Geometry of Manifolds, World Scientific (1996).

N.Berline, E.Getzler, M.Vergne, Heat Kernels and Dirac Operators, Springer Verlag (1992).

H.B.Lawson, M.L.Michelsohn, Spin Geometry, Princeton University Press (1989).

For students, I gave a few years ago a very elementary introductory seminar to some ideas in NCG that might be helpful for really absolute beginners http://math.science.cmu.ac.th/docs/chiang-mai.pdf"

Hope it might help ...

 

[tom.stoer]

wow - thanks