## What is the significance of non-commutative geometry in mathematics?

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This is a question that has been winding around my head for a long time and I have not found a convincing answer. The title says everything, but I am going to enrich my question by little more explanations.

As a layman, I have started searching for expositories/more informal, rather intuitive, also original account of non-commutative geometry to get more sense of it, namely, I have looked through

Nevertheless, I am not satisfied with them at all. It seems to me, that even understanding a simple example, requires much more knowledge that is gained in grad school. Now for me, this field merely contains a lot of highly developed machineries which are more technical (somehow artificial) than that of other fields.

The following are my questions revolving around the significance of this field in Mathematics. Of course, they are absolutely related to my main question.

1. How can a grad student be motivated to specialize in this field? and

2. What is (are) the well-known result(s), found solely by non-commutative geometric techniques that could not be proven without them?

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I think Alain Valette's answer to your Q2 is a very good example. As for Q1, I have found that some combination of "there is some beautiful mathematics here" and "there might be a better chance of getting a grant or a postdoc if you do this, than if you do something idiotic like the cohomology of Banach algebras", quite forceful. – Yemon Choi Feb 11 2012 at 8:46
Fist of all, it's easy to understand simple example at grad level (at least a decent grad level) : it's \$C*-algebra theory for example, or even easier, it's algebraic geometry. Both are foundations and motivate NCG. Second, as for the contribution to existing maths of purely NCG theory, I'm not a specialist at all, but if I remember well what Connes said at a talk (but I didn't get much of it honestly), one achievement would be to prove Riemann's hypothesis in terms of the NCG analog model of the Weil proof of Riemann's hypothesis. – Amin Feb 11 2012 at 8:49

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Personally, I have come to believe more in good mathematics than in important mathematics, but one could perhaps attribute that to sour grapes... – Yemon Choi Feb 11 2012 at 8:58

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While the original question was quite good, I have to say that these comments are descending very rapidly into "subjective and argumentative" territory. MO is absolutely not about "ranking" subjects etc. Also, shouldn't this be community wiki... – Matthew Daws Feb 11 2012 at 14:59

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Dear @Yemon Choi: In fact, I don't intend to think like that. I wanted to know what the reactions are when I say the words that I was told some years ago about this field. This might be another subject for discussion. BTW, I've intended to work in a field which has common features of NCG, in some sense, regarding the grant, postdoc, etc. and now, is not very supported and funded, but it appeals me the most, is still growing and good works are being done. – Ehsan M. Kermani Feb 11 2012 at 18:50
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## 6 Answers

I think I'm in a pretty good position to answer this question because I am a graduate student working in noncommutative geometry who entered the subject a little bit skeptical about its relevance to the rest of mathematics. To this day I sometimes find it hard to get excited about purely "noncommutative" results, but the subject has its tentacles in so many other areas that I never get bored.

Before saying anything further, I need to say a few words about the Atiyah-Singer index theorem. This theorem asserts that if $D$ is an elliptic differential operator on a manifold $M$ then its Fredholm index $dim(ker(D)) - dim(coker(D))$ can be computed by integrating certain characteristic classes of $M$. Non-trivial corollaries (obtained by "plugging in" well-chosen differential operators) include the generalized Gauss-Bonnet formula, the Hirzebruch signature theorem, and the Hirzebruch-Riemann-Roch formula. It was quickly realized (first by Atiyah, I think) that the proof of the theorem can be viewed as a statement about the Poincare duality pairing between topological K-theory and its associated homology theory (these days called K-homology).

I wasn't around, but I'm told that people were very excited about Atiyah and Singer's achievement (understandably so!) People quickly began trying to generalize and strengthen the theorem, and my claim is that noncommutative geometry is the area of mathematics that emerged from these attempts. Saying that marginalizes the other important reasons for developing the subject, but I think it was Connes' main motivation and in any event it is a convenient oversimplification for a MO answer. It also helps me answer your first question by playing to my personal biases: when I was choosing an area of research I told my adviser that I was interested in learning more about that Atiyah-Singer index theorem and I was led inexorably toward the tools of noncommutative geometry.

The origin of the relationship between NCG and Atiyah-Singer lies in equivariant index theory. Atiyah and Singer realized from the start that if $M$ admits an action by a compact Lie group $G$ and $D$ is invariant under the group action then it is better to think of the index of $D$ as a virtual representation of $G$ (i.e. an element of the $G$-equivariant K-theory of a point) rather than as an integer. If $G$ is not compact then this doesn't really work, but the noncommutative geometers realized that $D$ does have an index in the K-theory of the reduced group C$^\ast$-algebra $C_r^\ast(G)$. Indeed, to a noncommutative geometer equivariant index theory is all about a map $K_\ast(M) \to K_\ast(C_r^\ast(M)$ where $K_\ast(M)$ is the K-homology of $M$; in the case where $M$ is the universal classifying space of $G$, Baum and Connes conjectured that this map is an isomorphism. Proving this conjecture for more and more groups and understanding its consequences motivates a great deal of the development of NCG to this day.

The conjecture is interesting in its own right if you already care about index theory, but even if you don't injectivity of the Baum-Connes map implies the Novikov conjecture (see Alain Valette's answer) and surjectivity is related to the Borel conjecture. It has numerous other applications, for example to the theory of positive scalar curvature obstructions in Riemannian geometry or to the Kadison-Kaplansky conjecture in functional analysis (which would follow from surjectivity). Recently there has been a lot of interest in connections between the Baum-Connes conjecture and representation theory; the Baum-Aubert-Plyman conjecture in p-adic representation theory has its origins in these sorts of considerations.

Much of the rest of NCG can also be traced back to index theory. Kasparov's KK-theory arose as a way to understand maps and pairings between K-theory and K-homology, motivated in part by index theory. Connes' work on noncommutative measure theory arose from his work on index theory for measurable foliations (with applications to dynamical systems). Cyclic (co)homology was invented in part to gain access in a noncommutative setting to the Chern character map from K-theory to cohomology which translates the K-theoretic formulation of the index theorem into a cohomological formula. Connes' theory of spectral triples and noncommutative Riemannian geometry is based on the theory of Dirac operators which was invented by Atiyah and Singer to prove the index theorem. I guess my point with all of this is that all the esoteric machinery of NCG seems less artificial when viewed through the lens of index theory.

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Just to add up something to this nice answer I'd like to mention that (exactly through NC index theorems) NCgeometry contributed much in understanding foliations. Much work is by now devoted to the non commutative geometry of foliations. The idea is that this direction will lead to a better understanding also of singular foliations. – Nicola Ciccoli Feb 11 2012 at 16:48

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@Suvrit: I should have made a disclaimer at the outset of my answer that I am only informed about the analytic and topological aspects of NCG (a la Connes) and not NCAG. My understanding is that NCAG developed largely separately and for different reasons, though I think there are some like Jonathan Block and Ryszard Nest who straddle the line. That said, both areas use a lot of noncommutative algebra; in Connes' approach it enters via Hochschild and cyclic (co)homology. – Paul Siegel Feb 11 2012 at 20:08

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"Before saying anything further, I need to say a few words about the Atiyah-Singer index theorem." Paul, if I had a dollar for every time I've heard you preface something with that sentence, I'd be a very rich man. – Vaughn Climenhaga Feb 13 2012 at 1:31

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You're just jealous because you haven't thought of a better pick-up line. :) – Paul Siegel Feb 13 2012 at 9:36

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Just seen this conversation. Well, Paul, I think we know how you started or will start your marriage proposal... – Yemon Choi Feb 26 2012 at 12:08
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The spectral characterization of Riemannian manifolds is a good example in my opinion. Details are given in the following article: http://www.alainconnes.org/docs/ckm1.pdf .

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Connes has on his home page a very nice downloadable survey article "A view of mathematics" which also gives a good idea of the role of groupoids in this area, and links with many topics mentioned by others above.

I have also raised the question of the problem of applying these or related techniques to algebraically structured groupoids:

http://mathoverflow.net/questions/86617/convolution-algebras-for-double-groupoids

Since "motivating graduate students" was part of the original question, someone needs to point out at least one thing that has not been done. It can be useful to explain the "exterior" of a subject: unknowns, known unknowns, etc! One would also like the experts to point out anomalies in this area: I won't try to define this term, but they can lie around unrecognised.

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There are much better answers above than this one, but:

If you believe fiber bundles are important to classical mathematics, then you probably believe fibrations are, and maybe foliations are, as well. If you don't, note that a foliation of a smooth manifold is a decomposition of the manifold into integral submanifolds (roughly, solutions to differential equations). You can't get much more classical than this. In his book Noncommutative Geometry Connes tried to make it clear that to understand the leaf space of a foliation, more is needed than the classical quotient construction, groupoids and noncommutative geometry give more information about a patently classical "space". You probably say: So what? There are other ways. Connes tries then to show us that there is a connection between a fundamental von Neumann algebra invariant (the flow of weights) and one of the key invariants for a codimension 1 foliation (the Godbillon-Vey class), which appears in the first chapter on many introductory accounts of foliations. I find it hard to believe that this is coincidental. For me, this warrants closer investigation.

The index theorem for measured foliations discussed above perhaps grew from a seed like the above mentioned connection. (I wonder what we need to do to get Connes to weigh-in over here at MO?)

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And Kasparov, Moscovici, Baum, ... – Yemon Choi Feb 12 2012 at 19:45
Thanks, Yemon. I was thinking of Connes's book, and neglected to type that. – Jon Bannon Feb 12 2012 at 22:54

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We should at least be thankful that we have Alain Valette! – Paul Siegel Feb 13 2012 at 0:03

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I second that, Paul! – Jon Bannon Feb 13 2012 at 1:01

Alain's answer is the best application I heard about. I cannot add something comparable. Just two "motivations" which are somewhat nice to me. However they does not answer yours questions, sorry.

General claim - studying non-commutative objects is useful for understanding commutative ones.

Subclaim - non-commutative algebras can be equivalent (Morita, Koszul dual or whatever) to non-commutative ones, however non-commutative "models" can provide more easy way to study commutative things.

Examples 1. Consider commutative algebra A of functions on manifold M and group G. You may be interested in factor $M/G$ which is related to invariants $A^G$.

Claim. Under certain conditions COMMUTATIVE $A^G$ is Morita equivalent to NON-COMMUTATIVE $A\times C[G]$ - cross-product algebra of $A$ and group algebra of $G$. In some cases it is more easy to work with this cross-product sometimes it can be described more explicitly. You may see just the first sentences in Etingof Ginzburg famous paper: http://arxiv.org/abs/math/0011114

Example 2. Quantization. Our real world is actually quantum. So physicists are interested in this. Mathematical way to understand quantization is a procedure to construct the non-commutative algebras from commutative ones. The big mathematical challenge is to understand how to relate properties on non-commutative quantum algebras to properties of commutative ones. Probably the most striking and most simple formulated is the conjecture that automorphisms group of classical symplectic R^2n and quantum (i.e. just the algebra of differential operators in n-variables) are isomorphic. http://arxiv.org/abs/math/0512169 Automorphisms of the Weyl algebra Alexei Belov-Kanel, Maxim Kontsevich

It is somewhat related to the famous Jacobian conjecture. See http://arxiv.org/abs/math/0512171

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My favorite example concerns Novikov conjecture, on the homotopy invariance of higher signatures for closed manifolds with fundamental group $G$: see http://en.wikipedia.org/wiki/Novikov_conjecture (note that this Wikipedia entry rather stupidly says that it has been proved for finitely generated abelian groups: that's correct, but it was proved for MANY more groups, e.g. hyperbolic groups, countable subgroups of $GL_n(\mathbb{C})$, etc...). I think we agree that this is a conjecture in topology.

Now, look at this remarkable result by Guoliang Yu ( http://www.kryakin.com/files/Invent_mat_(2_8)/139/139_21.pdf )

"If the group $G$ admits a coarse embedding into Hilbert space, then it satisfies the Novikov conjecture".

A coarse embedding is a map $f:G\rightarrow L^2$ for which there exists control functions $\rho_{\pm}:\mathbb{R}^+\rightarrow\mathbb{R}$, with $\lim_{t\rightarrow\infty}\rho_\pm(t)=\infty$, which control'' $f$ in the sense hat, for every $x,y\in G$:

`$$\rho_-(|x^{-1}y|_S)\leq\|f(x)-f(y)\|_2\leq \rho_+(|x^{-1}y|_S),$$` where $|.|_S$ denotes word length with respect to some finite generating subset $S$ in $G$. The existence of a coarse embedding is a weak metric condition (actually we know of basically just one class of groups which do not admit such an embedding, the Gromov monsters''). And this weak metric condition, quite surprisingly, implies a strong consequence in topology.

Now my point is that the two known proofs of Yu's result (the original one, and the one by Skandalis-Tu-Yu, see http://www.math.univ-metz.fr/~tu/publi/coarse.pdf) both appeal in a fundamental way to the tools of non-commutative geometry: $C^*$-algebras, $K$-theory, groupoids, Kasparov's $KK$-theory (to be precise: equivariant $KK$-theory for groupoids'').

Now to answer your first question: how to motivate a graduate student? Well, the subject mixes classical geometry, algebraic topology, non-commutative algebra, functional analysis, so it is one of those subjects that give you a feeling of the unity of mathematics...

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Cool, I didn't know about that ! – Amin Feb 11 2012 at 8:52

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Just as an addendum, and with caveat that AV can explain this better than I can. There's an article by Connes, late 80s or early 90s, where he mentions some of the work on the Novikov conjecture done using the early incarnations of the machinery mentioned. Unfortunately my copy is buried somewhere in my office, but I vaguely recall it used index theory for $C(T^n)=C*(Z^n)$ to give a proof of Novikov's conjecture for abelian groups, as motivation for the need to do differentiyal geometry on noncommutative spaces, viz. K-theory of C*-algebras, cyclic cohomology & Chern character, etc – Yemon Choi Feb 11 2012 at 8:56

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I'd also put in a word for the Kadison-Kaplansky conjecture: in this case the problem originates in analysis rather than topology, but it seems that some of the most significant progress has either used NCG or used tools which received a lot of impetus from work in NCG – Yemon Choi Feb 11 2012 at 9:25

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"note that this Wikipedia entry rather stupidly says..." Why not edit it then? – David Corfield Feb 11 2012 at 14:46

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@Dear Alain: many thanks for your explanations. One of the reasons that I chose the Paul's answer is the sense that I got from the way he tried to relate important (classical) results that I've at least heard of many times to the known facts developed by NCG's method. As for my Q1, I'd say, unity of math sounds heroic and I don't really consider it as a motivation. – Ehsan M. Kermani Feb 14 2012 at 2:55
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