Theories of Noncommutative Geometry

Question

[I have rewritten this post in a way which I hope will remain faithful to the questioner and make it seem more acceptable to the community. I have also voted to reopen it. -- PLC]

There are many ways to approach noncommutative geometry.

What are some of the most important currently known approaches? Who are the principal creators of each of these approaches, and where are they coming from? E.g., what more established mathematical fields are they using as jumping off points? What problems are they trying to solve?

Two examples:

1) The Connes school, with an approach from C$^*$-algebras/mathematical physics.

2) The Kontsevich school, with an approach from algebraic geometry.

Answer 1

In accordance with the suggestion of Yemon Choi, I am going to suggest some further delineation of the approaches to "Non-commutative Algebraic Geometry". I know very little about "Non-commutative Differential Geometry", or what often falls under the heading "à la Connes". This will be completely underrepresented in this summary. For that I trust Yemon's summary to be satisfactory. (edit by YC: BB is kind to say this, but my attempted summary is woefully incomplete and may be inaccurate in details; I would encourage anyone reading to investigate further, keeping in mind that the NCG philosophy and toolkit in analysis did not originate and does not end with Connes.)

Also note that much of what I know about these approaches comes from two sources:

1. The paper by Mahanta

2. My advisor A. Rosenberg.

Additionally, much useful discussion took place at Kevin Lin's question (as Ilya stated in his answer).

I think a better break down for the NCAG side would be:

A. Rosenberg/Gabriel/Kontsevich approach

Following the philosophy of Grothendieck: "to do geometry, one needs only the category of quasi-coherent sheaves on the would-be space" (edit by KL: Where does this quote come from?) (See Section 13.6 of Manin's book)

In the famous dissertation of Gabriel, he introduced the injective spectrum of an abelian category, and then reconstructed the commutative noetherian scheme, which is a starting point of noncommutative algebraic geometry. Later, A. Rosenberg introduced the left spectrum of a noncommutative ring as an analogue of the prime spectrum in commutative algebraic geometry, and generalized it to any abelian category. He used one of the spectra to reconstruct any quasi-separated (not necessarily quasi-compact), commutative scheme. (Gabriel-Rosenberg reconstruction theorem.)

In addition, Rosenberg has described the NC-localization (first observed also by Gabriel) which has been used by him and Kontsevich to build NC analogs of more classical spaces (like the NC Grassmannian) and more generally, noncommutative stacks. Rosenberg has also developed the homological algebra associated to these 'spaces'. Applications of this approach include representation theory (D-module theory in particular), quantum algebra, and physics.

References in this area are best found through the MPIM Preprint Series, and a large collection is linked here. Additionally, a book is being written by Rosenberg and Kontsevich furthering the work of their previous paper. Some applications of these methods are used here, here, here, and here. The first two are focusing on representation theory, the second two on non-commutative localization.

Kontsevich/Soibelman approach

They might refer to their approach as "formal deformation theory", and quoting directly from their book

The subject of deformation theory can be defined as the "study of moduli spaces of structures...The subject of this book is formal deformation theory. This means $\mathcal{M}$ will be a formal space(e.g. a formal scheme), and a typical category $\mathcal{W}$ will be the category of affine schemes..."

Their approach is related to $A_{\infty}$ algebras and homological mirror symmetry. References that might help are the papers of Soibelman. Also, I think this is related to the question here. (Note: I know hardly anything beyond that this approach exists. If you know more, feel free to edit this answer! Thanks for your understanding!)

(Some comments by KL: I am not sure whether it is appropriate to include Kontsevich-Soibelman's deformation theory here. This kind of deformation theory is a very general thing, which intersects some of the "noncommutative algebraic geometry" described here, but I think that it is neither a subset nor a superset thereof. In any case, I've asked some questions related to this on MO in the past, see this and this.

However, there is the approach of noncommutative geometry via categories, as elucidated in, for instance, Katzarkov-Kontsevich-Pantev. Here the idea is to think of a category as a category of sheaves on a (hypothetical) non-commutative space. The basic "non-commutative spaces" that we should have in mind are the "Spec" of a (not necessarily commutative) associative algebra, or dg associative algebra, or A-infinity algebra. Such a "space" is an "affine non-commutative scheme". The appropriate category is then the category of modules over such an algebra. Definitively commutative spaces, for instance quasi-projective schemes, are affine non-commutative schemes in this sense: It is a theorem of van den Bergh and Bondal that the derived category of quasicoherent sheaves on a quasi-projective scheme is equivalent to a category of modules over a dg algebra. (I should note that in my world everything is over the complex field; I have no idea what happens over more general fields.) Lots of other categories are or should be affine non-commutative in this sense: Matrix factorization categories (see in particular Dyckerhoff), and probably various kinds of Fukaya categories are conjectured to be so as well.

Anyway I have no idea how this kind of "noncommutative algebraic geometry" interacts with the other kinds explained here, and would really like to hear about it if anybody knows.)

Lieven Le Bruyn's approach

As I know nearly nothing about this approach and the author is a visitor to this site himself, I wouldn't dare attempt to summarize this work.

As mentioned in a comment, his website contains a plethora of links related to non-commutative geometry. I recommend you check it out yourself.

Approach of Artin, Van den Berg school

Artin and Schelter gave a regularity condition on algebras to serve as the algebras of functions on non-commutative schemes. They arise from abstract triples which are understood for commutative algebraic geometry. (Again edits are welcome!)

Here is a nice report on Interactions between noncommutative algebra and algebraic geometry. There are several people who are very active in this field: Michel Van den Berg, James Zhang, Paul Smith, Toby Stafford, I. Gordon, A. Yekutieli. There is also a very nice page of Paul Smith: noncommutative geometry and noncommutative algebra, where you can find almost all the people who are currently working in the noncommutative world.

References: This paper introduced the need for the regularity condition and showed the usefulness. Again I defer to Mahanta for details. Serre's FAC is the starting point of noncommutative projective geometry. But the real framework is built by Artin and James Zhang in their famous paper Noncommutative Projective scheme.

Non-commutative Deformation Theory by Laudal

Olav Laudal has approached NCAG using NC-deformation theory. He also applies his method to invariant theory and moduli theory. (Please edit!)

References are on his page here and this paper seems to be a introductory article.

Apologies

Without a doubt, I have made several errors, given bias, offended the authors, and embarrassed myself in this post. Please don't hold this against me, just edit/comment on this post until it is satisfactory. As it was said before, the nlab article on noncommutative geometry is great, you should defer to it rather than this post.

Thanks!

Answer 2

Some thoughts and links on the analysts' NCG, from someone who doesn't practice it. Caveat lector. (Some edits made to erroneous history.)

NCG a la Connes was originally non-commutative differential geometry (which is why extra structure is needed in, say, the definition of spectral triple). Having only recently looked at Connes' original long, two-part paper in the Publications of the IHES, I think this is a better place to start than the later Big-Picture-Which-Is-Really Fundable works of various people thereafter. Work of Connes & Moscovici and others on trying to generalize the Atiyah-Singer index theorem also give some indication of the original motivation. (This is where someone more expert than me should really step in and say something about the work of C & M Mischenko, Kasparov and their co-authors on the Novikov conjecture, or indeed the work of Higson et al. on the Baum-Connes conjecture.)

It was only afterwards that Connes started championing an NCG perspective on The Standard Model. (Although, if you want real connections with mathematical physics, there was some work of Jean Belissard on identifying gaps in spectra of certain operators in a quantum-mechanical model with K-theoretic invariants of associated C*-algebras. See this paper of Kaminker and Putnam and the references therein for more details.)

Personally, I am bit leery of the Big Picture motivation for noncommutative geometry, at least of this variety. The most useful variant of such motivation that I can think of, is that degenerate group actions on topological spaces give rise to better-behaved homotopy groupoids; thanks to a theorem of Rieffel (IIRC), when the group action on the space is nice, the commutative C*-algebra of the quotient space is Morita equivalent -- i.e. has "the same module theory" -- as the noncommutative C*-algebra of the homotopy groupoid.

I apologize for the rambly nature of this answer, but with all due respect to Anweshi I think his question, or at least the version of it which I can currently see, is so broad (as per Anton's original comments) that the only responses are either encyclopaedic - I haven't even had space to mention the historical role played by Brown-Douglas-Fillmore theory, for instance - or sales pitches. Nevertheless, if someone can persuade Nigel Higson to drop by, I'm sure he could give a much better answer ;)

Answer 3

There is a beautiful three-part series of lectures by Jonathan Block that introduces both the Connes and Kontsevich schools of non-commutative geometry:

http://www.math.upenn.edu/~tpantev/rtg09bc/lecnotes/block-ncg.pdf

One of the main motivating examples throughout the lectures is Lusztig's proof proof of the Novikov conjecture.

Answer 4

The nLab's page on noncommutative geometry provides a lot of information. (It's actually a good idea to look into nLab before posting any general question: I did post some when I didn't know about nLab existence and was often sent to nLab anyway, usually politely.)

Also, you should take a look at Kevin Lin's question Non-commutative algebraic geometry.

Answer 5

The physics part of the picture comes from a classic article

• String Theory and Noncommutative Geometry by Seiberg and Witten

I can't say for sure, but I think it still isn't completely transplanted to math by either Connes or Kontsevich.