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

Question

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

• The English translation of Review of non-commutative geometry by Alain Connes,

• Surveys in non-commutative geometry, Clay mathematics proceedings, Volume 6,

• Non-commutative geometry by Alain Connes,

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?

Answer 1

$\DeclareMathOperator\coker{coker}$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 Poincaré 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.

Answer 2

My favorite example concerns the Novikov conjecture, on the homotopy invariance of higher signatures for closed manifolds with fundamental group $G$: see Novikov conjecture on Wikipedia (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, from The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space:

"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 exist control functions $\rho_{\pm}:\mathbb{R}^+\rightarrow\mathbb{R}$, with $\lim_{t\rightarrow\infty}\rho_\pm(t)=\infty$, which “control” $f$ in the sense that, for every $x,y\in G$:

$$\rho_-(\lvert x^{-1}y\rvert_S)\leq\lVert f(x)-f(y)\rVert_2\leq \rho_+(\lvert x^{-1}y\rvert_S),$$ where $\lvert{.}\rvert_S$ 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 The coarse Baum–Connes conjecture and groupoids) 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….

Answer 3

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?)

Answer 4

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 an easier way to study commutative things.

Examples 1. Consider the commutative algebra $A$ of functions on a manifold $M$ and a group $G$ acting on $M$. 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 $A\ltimes C[G]$ - the crossed-product algebra of $A$ and group algebra of $G$. In some cases it is easier to work with this crossed-product sometimes it can be described more explicitly. You may see just the first sentences in Etingof–Ginzburg's famous paper: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism.

Example 2. Quantization. Our real world is actually quantum. So physicists are interested in this. A 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 $\mathbb R^{2n}$ and quantum (i.e. just the algebra of differential operators in $n$-variables) are isomorphic. Automorphisms of the Weyl algebra by Alexei Belov-Kanel, Maxim Kontsevich.

It is somewhat related to the famous Jacobian conjecture. See Belov-Kanel and Kontsevich - The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture.

Answer 5

The spectral characterization of Riemannian manifolds is a good example in my opinion. Details are given in Connes - A unitary invariant in Riemannian geometry.

Answer 6

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:

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.

Answer 7

There is interplay among the three topics of "Hopf Algebras, Renormalization and Noncommutative Geometry" by Connes and Kreimer (1998), which are of continuing interest as illustrated by Kreimer and Yeats' "Diffeomorphisms of quantum fields" (2017), Yeats' "A Combinatorial Perspective on Quantum Field Theory" (2017), and Balduf's "Perturbation theory of transformed quantum fields" (2019). (Trees and vector fields play integral roles.)

Edit Dec. 18, 2021:

Interesting applications of NC geometry to investigations of Monstrous Moonshine can be found via the workshop synopsis "Novel approaches to the finite simple groups" by John McKay and Roland Friedrich (2012):

As an effect of the early preparation of all attendees, several conjectures previously made, in particular by J. McKay, could in fact be turned into concrete research plans during the time in Banff, one example being the application of non-commutative geometry and the Bost-Connes system to Monstrous Moonshine and replicability.

(Anyone have a copy or link to Jorge Plazas' roadmap mentioned in the synopsis?)

There are connections among noncommutative free probability theory, enumerative combinatorics (e.g. noncrossing partitions), random matrix models, NC geometry, and Monstrous Moonshine as well. See, e.g., the He and Jejalla ref in OEIS A134264.

Answer 8

In Veltman's Diagrammatica, the full Lagrangian of the standard model is spelt out. This has around a hundred terms. This is way too many for even the most dedicated physicist (or physically inclined mathematician) to work on, except by piece by piece.

In the Connes-Lott-Barrett-Chamseddine model, based on non-commutative geometry, the standard model is derived from a spectral action with a simple geometric input, spacetime is multiplied with a 'fat' non-commutative point:

$\mathbb{C} \oplus \mathbb{H}_L \oplus \mathbb{H}_R \oplus M_3(\mathbb{C})$

It is zero-dimensional, classically but has KO-theory dimension 6. This reproduces the full standard model including the Higgs and neutrino mixing. It turns out that the bimodule over the dual of this fat point, which is the sum of all irreducible irreps of odd dimension gives one full generation of the fermions. It's worth noting that the spectral action is a generalisation of the Einstein-Hilbert action.

The theory is not limited to just the standard model. In a later paper with Chamseddine, Connes shows how NCG can model a grand unification theory like the Pati-Salam SU(5) GUT.

Although non-commutative geometry is a geometry without a geometry. I mean by this that they work with a non-commutative algebra to be thought of as the algebra of functions on a non-commutative geometry which has not bern constructed yet. I'm not so sure that this will be the case in the near future (or perhaps the far near future given the density of maths required to understand and work with NCG). One of the standard examples of a non-commutative space, according to Connes, is the irrational torus where classical tools do not give any information but non-commutative tools can. However, diffeology, which is a generalisation of classical differential geometry, does and it gives roughly similar results to that of Connes.

I'd also add that Mathilde Marcolli has elaborated an explanation of the fractional quantum Hall effect and which has taken its point of departure from Bellisard, van Elst & Schulz-Baldes The Non-Commutative Geometry of the Quantum Hall Effect (1994) and which she calls the earliest work on physics in NCG. They show that the magnetic field turns the Brillioun zone into a non-commutative torus.

She has also published a recent book with Connes, titled Non-commutative Geometry, Quantum Fields and Motives. Here as a reviewer at the AMS says in 2007:

So the cat is out of the bag. What greater mathematical objective can there be to realise the RH [Riemann Hypothesis] for number fields and the RH for function fields as two sides of the same coin, the discriminators being as it were, algebraic geometry and (or versus) non-commutative geometry? This is manifestly one aspect of the rationale for what the entire sweeping programme is about, with a complementary aspect of quantum physics in its post-Feynman form.

What a wild, wild ride!

An understatement.

I imagine the latter is the reason behind another book that Marcolli has published, this time simply titled Feynman Motives.

Answer 9

Although there are more than one great answers to this question here but I am surprised no one has pointed out the greatest theme (my opinion) of Professor Connes' work. He has reproduced the Standard Model of physics from purely mathematical work which is worthy of history books. He has given a fundamentally different direction to our quest for understanding the world. Essentially, (as per my understanding), he extends the geometrical picture of gravity given by Einstein to explain all of physics as geometry. Not only that, his geometrical world also has in-built time evolution. He explains all this as starting from Heisenberg's matrix picture. Although many say Heisenberg's and Schrodinger's pictures are the same, he introduces the basic notion of non-commutativity with this matrix mechanics and uncertainty principle. He gives a whole new calculus to perform differentiation and integration etc. I don't know why his work is not talked about more.

(I would love if someone could point out if I am wrong partially or completely because I am neither a mathematician nor a physicist.)