dg.differential-geometrymp.mathematical-physicsnoncommutative-geometry
Are there "elementary" resources on Penrose Tilings in relation to noncommutative geometry? It's all a big blur to me. There are two transformations S and T that can grow the tilings and every tiling corresponds to a sequence of S's and T's. Somehow, there is a C* algebra related to this.
Can the elements of this algebra be interpreted as observables in a quantum mechanical system? What is the "geometry" of this noncommutative space?
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:
The paper by Mahanta
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:
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?)
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.
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][22] and [this][23].
However, there is the approach of noncommutative geometry via categories, as elucidated in, for instance, [Katzarkov-Kontsevich-Pantev][24]. 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][25] (see in particular [Dyckerhoff][26]), 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.)
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.
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][16] paper introduced the need for the regularity condition and showed the usefulness. Again I defer to [Mahanta][17] 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][18].
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][19] and [this][20] paper seems to be a introductory article.
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][21] article on noncommutative geometry is great, you should defer to it rather than this post.
Thanks!
[16]: https://books.google.com/books?hl=en&lr=&id=_BnSoQSKnNUC&oi=fnd&pg=PA33&dq=%252522Artin%252522+%252522Some+algebras+associated+to+automorphisms+of+elliptic+curves%252522+&ots=hRXnP7udMW&sig=t77CnWnsYPHhuonQQffrSXedyj0#v=onepage&q="Artin" "Some algebras associated to automorphisms of elliptic curves"&f=false [17]: https://arxiv.org/abs/math/0501166 [18]: https://web.archive.org/web/20121023193142/http://www.ingentaconnect.com/content/ap/ai/1994/00000109/00000002/art01087 [19]: https://web.archive.org/web/20181103123848/http://folk.uio.no:80/arnfinnl/ [20]: https://web.archive.org/web/20080425144650/http://folk.uio.no/arnfinnl/Noncom.alg.geom.pdf [21]: https://ncatlab.org/nlab/show/noncommutative%20geometry [22]: What is a deformation of a category? [23]: Deformation theory and differential graded Lie algebras [24]: https://arxiv.org/abs/0806.0107 [25]: Matrix factorizations and physics [26]: https://arxiv.org/abs/0904.4713
I don't know of the Connes calculus, but the others (including nonstandard analysis à la Robinson) have been brought under a common framework using models of synthetic differential geometry. However: it is important to point out that the infinitesimals used in algebraic geometry (for jet bundles, etc.) are nilpotent infinitesimals, whereas the infinitesimals used in nonstandard analysis are invertible. So in a sense the answer to the question is, "yes and no", but I'm going to concentrate here on the "yes".
This is explored in Models for Smooth Infinitesimal Analysis by Moerdijk and Reyes, which I recommend. This book can be read as "applied sheaf theory" or "applied Grothendieck topos theory", where the art is to choose a site (small category + covering sieves) judiciously to achieve several aims at once. In many of the models, one takes the underlying category of the site to be something like affine spectra of commutative rings, except one is not dealing with commutative rings exactly, but with richer algebraic structures called $C^\infty$-rings. The formal definition of these is in terms of a Lawvere algebraic theory which allows one to apply not just polynomial operations but more general operations based on $C^\infty$ functions. So the underlying category of the site in these models is the opposite of finitely generated $C^\infty$-rings, which Moerdijk and Reyes call $\mathbb{L}$ (for "locus").
The representing object which gives the locus of invertible infinitesimals is the spectrum of the $C^\infty$-ring given by $C^\infty$ functions $\mathbb{R} - \{0\} \to \mathbb{R}$, modulo the ideal of functions that vanish on some neighborhood of $0$, aka the $C^\infty$ ring of germs at $0$. This is really not much different from nonstandard infinitesimals: usually infinitesimal elements are thought of in some way as germs of functions at infinity, i.e., of functions $\mathbb{R} \to \mathbb{R}$ modulo those which vanish for sufficiently large $x$ (compare here the infinitesimals of Du Bois-Reymond and Hardy). Here Moerdijk and Reyes use $0$ instead of $\infty$. Either way, there are nonarchimedean elements, i.e., nonzero elements less than any $1/n$ in absolute value.
[In nonstandard analysis, one typically refines this idea by considering germs of functions $\mathbb{R} \to \mathbb{R}$ at an "ideal point at infinity", i.e., at a non-principal ultrafilter $U$ on $\mathbb{R}$, or alternatively germs of functions $\mathbb{N} \to \mathbb{R}$ at a non-principal ultrafilter $U$ on $\mathbb{N}$. The more familiar buzzword here is "ultrapower", but see this MO answer by François Dorais, where the implicit message is that an ultrapower along $U$ is really the same as taking a stalk at $U$. (I call $U$ an "ideal point at infinity" because we can think of a non-principal ultrafilter $U$ on $\mathbb{N}$ as a point in the fiber over $\infty$ with respect to the canonical continuous map $\beta(\mathbb{N}) \to \mathbb{N} \cup \{\infty\}$, from the Stone-Cech compactification of $\mathbb{N}$ to the one-point compactification of $\mathbb{N}$.)]
On the other hand, a typical representing object for nilpotent infinitesimals is the spectrum of the $C^\infty$-ring of functions $\mathbb{R} \to \mathbb{R}$ modulo the ideal of squares of elements which vanish at $0$. The "internal hom" represented by this spectrum gives the tangent bundle functor, and other jet bundles can be similarly represented, by using $C^\infty$-rings with different types of nilpotent elements.
The tricky part of all this is to get the right notion of covering sieves, i.e., of sheaves w.r.t. a Grothendieck topology, to achieve disparate aims. One aim would be to embed the usual category of manifolds fully and faithfully in the category of sheaves, so as to preserve "good colimits", such as a manifold $M$ obtained as a colimit along an open covering of $M$. A different aim would be to arrange the topology so that the locus of invertible infinitesimals, as a presheaf on the site category $\mathbb{L}$, is a sheaf w.r.t. the topology. In summary, both aims can be achieved simultaneously so as to accommodate both nilpotent and invertible infinitesimals.
Best Answer
I don't have any magical references for you, nor do I understand the NCG point of view on the Penrose tiling all that well. I learned just enough about this to convince myself that I didn't need to learn more, and I'll try to convey the solace that I achieved.
First, let me say a few words about the philosophy of Connes' NCG. The basic premise is that sometimes when you have a space equipped with an equivalence relation it is not necessarily a good idea to pass to the space of equivalence classes. The best-advertised justifications for this premise are examples where the space is nice and the equivalence relation is nice but the quotient is miserable, but I want to remark before going on that the tools of NCG are still extremely useful when the quotient is nice (such as when a manifold is viewed as the quotient of its universal cover by its fundamental group).
Here is how the philosophy plays out for Penrose tilings. One regards a Penrose tiling as a tiling of the plane by isometric copies of two specific triangles that are only allowed to connect in a few very specific ways. See page 181 of Connes' Book for pictures (and a more detailed, but elementary, description of what I am about to say). We declare two Penrose tilings to be equivalent if there is an isometry of the plane which carries one to the other. There is a more down to earth way to express the space of Penrose tilings modulo this equivalence relation by interpreting it combinatorially. It is the space of sequences $(a_n)$ of $0$'s and $1$'s with the property that $a_{n+1} = 0$ whenever $a_n = 1$ modulo the equivalence relation of eventual equality: $a_n = b_n$ for $n$ sufficiently large. What isn't necessarily obvious at the outset is that this space has absolutely no sensible local structure. In terms of the Penrose tilings, this can be expressed by observing that any finite patch in one tiling appears infinitely often in any other tiling by the same tiles.
There is by now a standard way of fitting such a setup into the machinery of NCG. Any equivalence relation on a space gives rise to a certain groupoid whose objects are points in the space and whose morphisms are determined by the equivalence relation. The idea is supposed to be that the groupoid keeps track of which points in the space are equivalent as well as the reason why they are equivalent, rather than violently collapsing each equivalence class down to a single point. In most cases it is possible to equip the groupoid with a compatible system of measures and then use integration with respect to that system to define a convolution product on a suitable space of functions on the groupoid (generally the original space and the groupoid have a topology and the space of functions is the space of continuous functions). The C* algebra of the groupoid is defined as a certain completion of this convolution algebra.
Notice that none of that last paragraph involves the details of the Penrose tiling construction in any essential way: it is a purely mechanical procedure which starts with an equivalence relation and spits out a C* algebra. One can think of this as analogous to the procedure of replacing a function with an operator (in this case convolution against the function) which flies under the moniker "quantization" in physics. Indeed, various sorts of quantization in physics can be realized via convolution algebras on groupoids - though the groupoids are generally more complicated than those which arise from an equivalence relation. As far as I know, the relationship between Penrose tilings and physics ends with this analogy. I could be tragically wrong.
So when you ask about the "geometry" of the noncommutative space of Penrose tilings, you are really asking about the structure of the C* algebra spat out by the machine described above. What is the structure of this C* algebra? I don't really know. Connes claims that it has trivial center and a unique trace, which has consequences on its "measure theoretic" structure. It was at this point in the story that I noticed that I'm more interested in the geometry of manifolds and decided to move on. Still, it wouldn't surprise me if there turns out to be way more to this story than meets the eye.