The search for discrete differential geometry

January 26, 2004

The search for discrete differential geometry

Posted by Urs Schreiber

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At first it may look like a trivial problem whose solution should have been known for ages, being used all over the place for applications in mathematics, theoretical physics and engineering. But surprisingly, it has apparently not fully been understood yet. I am talking about the adaption of the full machinery of differential geometry to discrete spaces. In particular, I am being told that the metric aspects of such a theory still puzzle many researchers, the Hodge star operator, for instance, being notoriously hard to come by on general discrete spaces.

A while ago Eric Forgy has convinced me that it may be worthwhile to think about these issues, and after a very intesive collaboration we came up with what looks like an interesting approach to discrete metric differential geometry to us. Now we are trying to communicate our results with other researchers in the field.

Since most of this exchange is currently going on by e-mail and since this puts severe restrictions on the amount of true interaction and maybe cross-fertilization as soon as more than two people are involved, I was wondering if maybe we’d need some sort of discussion forum. This blog entry is supposed to be the entry point to such a discussion. To get started, I list some relevant liks to the current literature below. The list won’t be comprehensive at all at the moment, but I am planning to update it as we go along.

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Ok, so here are some links to people and texts currently dealing with discrete differential geometry. This is just a very rough first approximation to a comprehensive link list. I am going to flesh this out as soon as I find the time.

First of all, Dimakis and Mueller-Hoissen have written many papers on dicrete differential geometry. For the moment I’ll just cite this list.

Jenny Harrison says she has been developing a new discrete theory of exterior calculus using chainlets. I don’t know if the new results are already published.

There is an interesting talk by Sullivan about his ideas on discrete geometry.

Alain Bossavit is an expert on discrete electromagnetism using finite elements and Whitney forms.

A. Sitarz has papers on Noncommutative Geometry apporaches to discrete spaces, for instance Noncommutative Geometry of Finite Groups.

In Berlin there is a ‘Sonderforschungsbereich’ concerned with Discrete Differential Geometry, Quantum Field Theory, and Statistical Mechanics. See their list of publications.

Our own notes (which are still in a draft stage) are

E. Forgy, U. Schreiber Discrete Differential Geometry on $n$ -Diamonds.

More links, some of which need to be incorporated into this list, can be found here of course.

Posted at January 26, 2004 11:07 PM UTC

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Re: The search for discrete differential geometry

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Hi Urs! :)

The relationship between string theory and discrete differential geometry is not obvious to me, but a lot of the papers by Dimakis and Mueller-Hoissen mention the word now and then :)

I hope that the other hosts do not mind if we use this powerful forum to discuss our work out in the open. If anyone besides us ends up reading this, let me just quickly say that Urs and I typically end up writing several emails to each other each day. It is usually me coming up with some whacky idea and then he’ll turn it into something fairly intelligible. It’s great fun! :)

Urs is a graduate student studying string theory. I somehow managed to get him hooked on discrete differential geometry, and he is now set on applying it toward string theory. So this is not really as off topic as it might seem.

Until any of the other gracious hosts object, we’ll probably try this forum out as a way to communicate our ideas to one another and to anyone else who might find what we have to say interesting.

It should go without saying that we welcome any and all comments along the way. There is a great potential to have some really nice discussion here.

Best wishes,

Eric

Posted by: Eric on January 27, 2004 2:56 AM | Permalink | Reply to this

Re: The search for discrete differential geometry

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I will sketch some of the central ideas of the approach that Eric and I have been following. The basic idea is quite simple and involves essentially a straightforward merging of the concepts of Noncommutative Geometry (NCG) with that of abstract differential calculus. The result is indeed nothing but an NCG defined by an $N = 2$ spectral triple, i.e. one which has not only one but two (anticommuting) Dirac operators.

So let $𝒜$ be an associative algebra of functions characterizing a topological space. In the present case this will be the group-valued algebra of functions on a denumerable set of points of a discrete space.

Over $𝒜$ one can consider differential calculi $Ω (𝒜, d)$ whose elements are generated by elements $a$ of grade 0 and elements $d a$ of grade 1 (with $a \in 𝒜$ ), where $d$ is a formal exterior derivative which satisfies $d^{2} = 0$ as well as the graded Leibnitz rule. Elements of grade $p$ in this algebra are the analoga of differential $p$ -forms.

In order to obtain a substitute for the Hodge inner product $〈 α ∣ β 〉 = \int α \land ⋆ β$ known from the continuum, define on $Ω (𝒜, d)$ (regarded as a vector space) a non-degenerate (sesquilinear) inner product

(1)

〈 \cdot ∣ \cdot 〉 : Ω \times Ω \to C .

The elements $a \in 𝒜$ of the original algebra as well as $d$ are represented on the resulting inner product space $ℋ (Ω, 〈 \cdot ∣ \cdot 〉)$ as operators in the obvious way. Taking adjoints of these operators with respect to $〈 \cdot ∣ \cdot 〉$ in particular yields $d^{†}$ , the analogue of the exterior coderivative.

The information about the metric geometry of the discrete space is encoded in $〈 \cdot ∣ \cdot 〉$ and hence inherited by $d^{†}$ . The connection to the familiar formulation of Noncommutative Geometry is obtained by noting that $d \pm d^{†}$ are two odd-graded Dirac operators on $ℋ$ , so that $(𝒜, ℋ, d \pm d^{†})$ is a spectral triple.

Now denote by $⋆$ an operator on $ℋ$ (if it exists) which satisfies

(2)

d^{†} = \pm ⋆ d ⋆^{- 1} .

Define a volume-like form to be an element $∣ vol 〉$ of $Ω (𝒜, d)$ that is of maximal grade and annihilated by $d^{†}$ : $d^{†} ∣ vol 〉 = 0$ .

With these definitions the following fact holds: Given the inner product space $ℋ (Ω, 〈 \cdot ∣ \cdot 〉)$ we have

(3)

⋆ exists \Leftrightarrow ∣ vol 〉 exists

and they are related by

(4)

∣ vol 〉 = ⋆ ∣ 1 〉

and

(5)

⋆ ∣ a_{0} d a_{1} \dots d a_{p} 〉 = {(a_{0} d a_{1} \dots d a_{p})}^{†} ∣ vol 〉 .

It follows in particular that $⋆$ exists iff there is a solution $∣ vol 〉$ to the equation $〈 d α ∣ vol 〉 = 0, \forall α \in Ω (𝒜, d) .$ In the case of discrete geometry on topologically hypercubic graphs and complex valued $a \in 𝒜$ this condition can be solved explicitly for $∣ vol 〉$ . The above equation then defines the Hodge star operator on these spaces.

This is spelled out in detail in section 4 of our notes.

Posted by: Urs Schreiber on January 27, 2004 6:54 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

It seems like there is quite some excitement about Thiemann’s paper. Sounds interesting.

One quick comment…

One neat consequence of what Urs outlined was that if elements of the algegra are self-adjoint, then the space is “flat”. This was surprising to both of us and explained one of the reasons I had been struggling over the years. I was trying to have both non-flat spaces and self-adjoint algebras. This was a crucial insight.

Gotta run. More later…

Eric

Posted by: Eric on January 27, 2004 8:13 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

Hi Eric,

I need help in finding the reference to the approach to discrete diff geometry by means of using Whitney forms, finite elements, interpolation, and so on. Thanks!

Posted by: Urs Schreiber on January 28, 2004 11:27 AM | Permalink | Reply to this

Re: The search for discrete differential geometry

Hi Urs,

I don’t think there exists the biblical reference on discrete methods for which you are looking :)

I particularly like the “Japanese Papers” by Alain Bossavit, whose reference can be found on that link you provided to his web page

http://www.lgep.supelec.fr/mse/perso/ab/bossavit.html

I see his book is also available for download, which is a good reference.

This

http://www.math.unm.edu/~stanly/mimetic.html

is probably the best resource for finding “who’s who” in discrete methods. The list of references (in .bib format and everything! :)) there is pretty helpful.

You might, in particular, like to follow Tonti’s link (if it works, my proxy is complaining).

You’ll find all you ever wanted to know about (applied) discrete methods by browsing the list of people and their web pages.

Where does the time go…

Eric

Posted by: Eric on January 28, 2004 10:23 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

Deformed differential geometry related to finite differences has been developed by Majid ten
years ago, as an example to his q-group approach. Possibly the right setting for finite
differences would be the tangent groupoid, as it includes the possibility of a continous limit.

Speaking of continous limit, Brouder hep-th/9904014 shows that Butcher trees, which
classify finer and finer Runge-Kutta methods, can be related to Connes-Kreimer trees, which
control the renormalization group (hep-th/9904044 etc). Besides, CK trees seem to be related to
the group of diffeomorfisms of a geometrical object.

As for lattices, let me point that commutative spectral triples over discrete lattices have
been classified by Alain Connes (example at page 15 of hep-th/9603053), by
T. Krajewsky (hep-th/9701081) and by M. Paschke and A. Sitarz (q-alg/9612029)

See also hep-th/0312276 and also the big series of papers and preprints by Jian Dai.

A non-go result for Poincare Duality on naive discrete dirac operators was given
by M. Gokeler and T. Schucker in hep-th/9805077. A workaround has been proposed
by A. Rivero ;-) at math-ph/0203024, hep-th/0204238.

I have not read “Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory”,
which seems closer to string theory or at least to the stringy view of NCG. To me, string theory is
about having distances and energy-momentum adscribed to the same physical entity, while field theory
(and very specifically fields in the discrete lattice) are about having distances and momentum adscribed to
different entities, sometimes called vacuum and particles respectively.

Posted by: alejandro rivero on February 6, 2004 3:04 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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Hi Alejandro,

Thanks for your post and the references. I wasn’t aware of those two most excellent workaround papers from last year :) I’ll have to check them out.

By “Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory”, are you referring to this

Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory

http://arxiv.org/abs/hep-th/0004147

By the way, Urs and I would definitely value any feedback you might have regarding our paper. I believe that although what we have can probably be consumed under the huge umbrella that Connes’ constructed, the specific model we have seems to be particularly nice for doing concrete applications. I haven’t seen any reference with the same emphasis that developed from our work (e.g. the need for n-diamond complexes).

Best wishes,

Eric

PS: I am fairly certain the Poincare duality is trivial on an n-diamond complex. I could be wrong.

Posted by: Eric on February 6, 2004 4:40 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

Hi guys,

Mi experience is that poincare duality is never trivial with spectral triples; I will bet that my own work have some failure in it that my friends have been nice enought to no report to me, so I will not depress… Probably the same can be said about the enormous effort of the two australian students, Rennie and Lord, who took upon them the effort to reproduce completely the commutative continuous example.

Yep, the other paper was J. Ambjorn et al. I do not know what to think about all the NCG variants outside the -almost extinct- first mainstream.

I am printing your text to take a look during the weekend. On my own, I believe that any discretization should be measured in units of plank constant, so I do not believe in naive discrete geometry over the space, nor the space-time.

Posted by: alejandro rivero on February 6, 2004 7:22 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

Hi Alejandro -

many thanks for all these references! I haven’t yet had the time to look in any detail at all this information. But here are some questions:

Could we perhaps clarify what we mutually mean by Hodge duality versus Poincare duality?

Could you try to explain the basic assumptions and ideas behind this no-go theorem that you mentioned? Also, what are the assumptions and ideas behind the workarounds that you are proposing?

Posted by: Urs Schreiber on February 6, 2004 7:12 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

Hi Urs and Alejandro,

When I said Poincare duality was trivial with diamond complexes, what I really meant was that constructing a Poincare dual grid was trivial. This I guess isn’t really what is usually meant by Poincare duality. For that, we would need to define a product between elements of the primary grid with elements of the Poincare dual grid. We have not done this. We have talked about it, but it was not obvious how to proceed. In that sense, Poincare duality is not trivial in our formalism either. Then again, we may have a slightly unorthodox way to view Poincare duality on a grid. This is due to the fact that our Hodge star does not map p-forms to (n-p)-forms on a dual lattice. Rather, we get (n-p)-form on the SAME lattice. Therefore, we could probably define a meaningful notion of Poincare duality on n-diamond complexes although it might not be exactly what one might usually think of (which is a good thing in my opinion).

I should say that I am also very much against ad hocness and naive discretizations. That is the primary motivation for this entire effort in the first place. So although Planck’s constant has not “emerged” from our work from some primordial concept, I’d really hesitate to say that what we did is naive (not that that is what I think you were implying).

I’ll return the favor and print out your papers for the weekend. I look forward to discussing them here.

Best regards,
Eric

Posted by: Eric on February 6, 2004 7:37 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

Well Poincare duality, as used in NCG and other mathematical field, maps –note super and sub-indices– from H^p to H_{n-p}; on the contrary, I understand that the usual Hodge operator maps from p-forms to n-p forms.

My better grasping of Poincare Duality comes from third volume of Dubrovin-Fomenko-Novikov. Poor Fomenko crossed the border of madness sometime, according a website around. But anyway, the point to grasp is the role of the intersection or cap product. It is the operation that let us to reduce the number of variables in a integral, ie to do separate integration of (along) a single coordinate. Thus a fixed volume form plus an intersection product give us the tools to define Poincare Duality.

Posted by: Alejandro Rivero on February 7, 2004 10:53 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

So Hodge duality plus a way to send p-forms to p-chains should give us also Poincare-duality, right?

Posted by: Urs Schreiber on February 9, 2004 11:30 AM | Permalink | Reply to this

Re: The search for discrete differential geometry

I would expect so. In fact my main worry about Froelich et al approach to Hodge operator is that it is added to the spectral triple, while I expected at least one such operator to come canonically from the volume form and the intersection product. Note that in NCG axioms the intersection product is required, and it should be. It is a important point, to be able to do calculus.

Posted by: arivero on February 9, 2004 1:10 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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Whether or not to include the Hodge star, if it exists, in the spectral ‘triple’ is just a matter of notation, I’d say.

But yes, as I have mentioned before the existence of a volume form is equivalent to the existence of a Hodge star (at least if by volume form we mean a top form which is annihilated by the adjoint of $d$ ).

Posted by: Urs Schreiber on February 9, 2004 1:55 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

And on other hand the existence of such volume form is already required in the axioms for spectral triples, so the Hodge should follow sort of canonical way, that was my doubt against Froelich’s two papers; they seem to work with an ad-hoc Hodge.

Posted by: alejandro rivero on February 9, 2004 2:22 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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Hi Alejandro -

when you say ‘volume form’ you are not referring to the same thing that I am! :-) Let’s try to get in sync. If you don’t mind, let’s start from the very beginning:

We are given an associative algebra $𝒜$ , a Hilbert space $ℋ$ on which $𝒜$ is represented and which supports a grading operator $γ$ with $γ^{2} = 1$ , $[γ, 𝒜] = 0$ and an operator $D$ with ${D, γ} = 0$ . When all this is collected in an ordered tuple $(𝒜, ℋ, D)$ we speak of a spectral triple.

Now what is it that you are calling a ‘volume form’ in this construction?

What I would like to refer to as a volume form is the following: I consider the special case where there is a second odd-graded operator $D_{2}$ on $ℋ$ such that $D^{2} = D_{2}^{2}$ and ${D, D_{2}} = 0$ . Then I set

(1)

d : = \frac{1}{2} (D + i D_{2}) .

(2)

d^{†} = (d)^{†} = \frac{1}{2} (D - i D_{2}) .

As discussed by Froehlich, the existence of the second $D_{2}$ refines the grading of the Hilbert space form $Z_{2}$ to $Z$ . What I am calling a volume form is this: A non-vanishing element $∣ ψ 〉 \in ℋ$ which is of maximal grade and satisfies

(3)

d^{†} ∣ ψ 〉 = 0 .

This use of the term ‘volume form’ in NCG is my own invention and if it collides with any established convention I need to change my nomenclature.

Posted by: Urs Schreiber on February 9, 2004 2:43 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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The point is that $γ$ contains already the info for oriented volumes, lackin only of the metric information coming from $D$ . So it can be told that $γ$ is the volume form. It is not standard use, anyway, because really one needs not only the representative $γ$ but the Hochschild n-cycle $c$ such that $π (c) = γ$ . The situation is explicitly stated, for the commutative case, in page 12 of one of the fundational papers, hep-th/9603053

Posted by: Alejandro Rivero on February 9, 2004 5:31 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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Alejandro, many thanks for your help! Apparently it is important to say that what Eric and I have done is NCG-inspired rather than true NCG, because I am not sure that our construction satisfies all the axioms on pp. 13 of that paper.

I have a couple of further questions:

1) Assume in an even number of dimensions that a $γ$ with $γ = γ^{*}$ , $γ^{2} = 1$ , $[γ, a] = 0$ , ${γ, D} = 0$ exist on $ℋ$ but that it does not come from any Hochschild cycle. What construction would fail? I.e., what is it good for to know the Hochschild cycle that $γ$ comes from?

2) What is the purpose of axioms (2’) and (7’) on pages 13,14? What constructions would fail if (2’),(7’) were not satisfied?

This is probably crucial for our approach. I can currently not see how in a discrete differential calculus equipped with an inner product (2’) should hold.

Thanks for your help!

Posted by: Urs Schreiber on February 9, 2004 6:42 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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I am sorry I am been already pushed into my limit. Lets see. As for (1), I would say that you need the cycle because of the same reasons you wanted a volume form defined from a operator d. Note for instance that page 8 uses that $π (c) = 1$ (odd case, here) to get the equation actually defining the circle $S^{1}$ . In fact all the plethora of papers on NCG spheres are extensions of this example.

As for question (2), it is not easy to get it if you have not read the redbook, last chapter. There He :-) used two different algebras to define all the duality plays. When joining them in a single selfdual algebra, this leads to a new operator, J, keeping memory of the two previuosly separated algebras so that. The axiom 2, which makes sure that D is first-order (I was told), must be modifyed also to keep its previous significance. The new concept of using $J$ appeared in the preprint “Non Commutative Geometry and Reality”. It is not at the arxiv, but it is scanned at KEK database.

Aside… did you enjoyed my planck length==>compton length charade? I am happy to see, even if naively, that there is some quantum mechanics going on inside quantum gravity.

Posted by: alejandro rivero on February 9, 2004 7:32 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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it is not easy to get it if you have not read the redbook, last chapter.

Thanks, I’ll see if that helps. So do I understand correctly that there was a certain evolution of Connes’ axioms over the years?

Aside… did you enjoyed my planck length==>compton length charade? I am happy to see, even if naively, that there is some quantum mechanics going on inside quantum gravity.

I assume that you are referring to this? A nice observation. I am not quite sure what to make of it, though. I think area quantization rests on shaky grounds, but of course the dimensional analysis you do is suggestive even without strict area quantization.

Posted by: Urs Schreiber on February 10, 2004 9:49 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

Hi Alejandro,

Well your paper math-ph/0203024 certainly looks interesting at first glance :) I am now collecting all of the papers but I like the abstract of hep-th/9805077

We are unable to formulate lattice gauge theories in the framework of Connes’
spectral triples.

I think we also might have a counterexample :)

Off to my reading! :)

Best regards,
Eric

Posted by: Eric on February 6, 2004 7:53 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

Hi again

About your paper: the main problem, when glancing over it, is a sense of deja-vu. I mean, people looking at the question of relating discrete derivations and non commutative geometry has already heard this music, and you need some catching notes to convince them to put attention. Of course a sound counterexample to Goelecker-Shucker should be a good thing.

Another interesting point I see is the calculation of divergences. Here it seems you have gone a little step forward than another papers, and it could be interesting if you were able to propose a couple of problems (classical EM fields or something so, with Gauss theorem and all that) and solve them by discrete differential geometry.

The relationship with stochastic calculus has also been invoked in Dimakis-Tzanatkis work, q-alg/960601, math-ph/9912016 … Now here I believe your paper is in position to do deeper things,in considering both the osmotic and symmetric derivative or the ambiguity in combining left and right derivatives. One expects that a whole new set of differential equations could be defined if one gets to save this ambiguity in the continuous limit without increasing the order of the differential equation.

(Actually, time ago I conjectured that a try to save this ambiguity in second order partial differential equations and beyond should force the definition of a sort of cabbibo angles and mass matrices)

Posted by: alejandro rivero on February 9, 2004 12:09 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

I am now in NY for a conference visiting Dennis Sullivan, Jozef Dodziuk, and Jenny Harrison. It is painfully obvious how inferior my knowledge in mathematics is :)

Eric

Posted by: Eric on February 9, 2004 2:27 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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Hello,

Urs mentioned that it might be good to write a short summary of my experience at the conference at CUNY.

I went for two meetings. The first was a conference honoring the retirement of Professor Edgar Feldman.

Geometric Analysis and its Applications:
A Conference in Honor of Edgar Feldman
February 7-8, 2004
Science Center, Fourth Floor
CUNY Graduate Center
365 Fifth Avenue, New York, NY

What drew me to the conference was the convergence of three people I admire very much: Professor Dennis Sullivan, Professor Jozef Dodziuk, and Professor Jenny Harrison.

For me, meeting and getting a chance to discuss research with Professor Sullivan was not far from what I might experience meeting someone like Einstein or Dirac. He’s clearly so much more knowledgeable than I am that it was almost as if we spoke two different languages. At one point he said, “Mathematics has its own grammar not unlike any other language. What you are saying to me is like a sentence without a verb. What is the statement?!” Yes, I felt like an idiot, but I’d much prefer having someone be mercilessly honest rather than politely nodding, while quietly thinking the same thing :) He certainly had a lot of patience because he kept at me until I (kicking and screaming) was finally able to make a coherent statement about the work Urs’ and I have been doing. This was frustrating because I really am happy with our work and wanted to communicate the ideas better. I wish Urs was there to help! :)

The conference itself for the most part blew right over my head except for two talks: one by Professor Mark Hillery on quantum walks on graphs, the other by Professor Sullivan. Professor Jenny Harrison was scheduled to talk (and was one of the reason I attended), but she couldn’t make it due to her son coming down with a cold.

Since I am technically an engineer, Professor Hillery’s work reminded me a LOT of transmission line theory. Motivated by the fact that random walks are of significance for many algorithms in classical computing, he is hoping that the quantum mechanical version of a random walk will be of significance for developing algorithms in quantum computing. In this approach, you begin with some directed graph where each edge of the graph is replaced by two oppositely directed edges and an edge from node $j$ to node $j + 1$ is denoted $∣ j, j + 1 〉$ . He then defined a unitary time evolution operator

(1)

U ∣ j - 1, j 〉 = t ∣ j, j + 1 〉 + r ∣ j, j - 1 〉

(2)

U ∣ j + 1, j 〉 = t^{*} ∣ j, j - 1 〉 - r^{*} ∣ j, j + 1 〉

where $t$ is basically a transmission coefficient and $r$ is a reflection coefficient. I am about 99% sure that this is essentially casting standard transmission line theory in a different language.

Professor Sullivan’s talk was interesting, but if you had followed Professor Baez’s quantum gravity seminar notes there were quite a lot of similarities. He was discussing algebra and coalgebras via diagrams and the algebraic relations there can also be viewed as simple topological relations of diagrams. It was fun.

However, the exciting part of the trip for me and what made the whole thing worth it was the Einstein Chair Seminar on the following Tuesday. The Einstein Chair Seminar is usually a single talk where Professor Sullivan basically has free reign to grill the speaker for as long as he wishes (according to Professor Baez :)). They usually last from 4-6 hours (or until the speaker passes out). The speaker was to be Professor Jenny Harrison talking about her work on “Discrete Exterior Calculus.” This Seminar was a little special because Professor Harrison doesn’t like to speak for more than 45 minutes, so instead of the usual grilling, they had a handful of people talk for 45 minutes each. Kind of like a mini afternoon conference. The theme of the afternoon was

DISCRETE DIFFERENTIAL GEOMETRY

Woohoo! :)

The first speaker was Professor Jozef Dodziuk. His talk was essentially a review of his Ph.D. thesis, which was WAY ahead of its time as far as applied finite-element methods are concerned (although that probably wasn’t the intended use he had in mind)

Finite difference approach to the Hodge theory of harmonic forms
Amer. J. Math., 98 (1976), 79-104.

I like this paper a lot. In it, he defines an inner product

(3)

〈 a, b 〉 = \int_{M} Wa \land ⋆ Wb

of cochains via the inner product of forms and the Whitney map $W : C (K) \to Ω (M)$ from cochains on a simplicial approximation $K$ to differential forms on a smooth manifold $M$ . With this inner product he defines a discrete adjoint exterior derivative $δ : C^{p} (K) \to C^{p - 1} (K)$ and Hodge decomposition falls out naturally. This is one of my all time favorite papers and it was a treat to see him present it in person.

The next speaker was Scott Wilson who I think was a student of Professor Sullivan’s. If not a student, Professor Sullivan seemed very much aware of his work as if he were an advisor. This paper hit on some topics I tried to ask Professor Dodziuk about regarding algebraic properties of the Whitney map. For one thing, the Whitney map is not an algebra morphism, which I think is at the heart of some of the issues addressed throughout the day. His talk began with a

Definition:
We say $\cup : C^{p} (K) \otimes C^{q} (K) \to C^{p + q} (K)$ is an approximation of $\land$ if $\forall f, g \in Ω (M)$

(4)

∣ ∣ W (Rf \cup Rg) - f \land g ∣ ∣ \leq O (mesh)

where $f, g$ are forms, $R : Ω (M) \to C (K)$ is the de Rham map, and O(mesh) is some constant that depends on the largest cell in mesh.

He makes the statement, “If $\cup$ is associative, then $\cup$ is not an approximation of $\land$ .” He doesn’t prove this, so I am not sure if it is true or not. I think both he and Sullivan are pretty convinced it is true. This is somewhat odd because Urs and I DO have an associative product that DOES approximate the wedge product. However, our definition of “approximate” hasn’t been stated quite as formally as they did. This made me think that if the statement was true, then it could be because the definition of “approximate” is too tight. I suggested the alternative

Definition:
We say $\cup : C^{p} (K) \otimes C^{q} (K) \to C^{p + q} (K)$ is an approximation of $\land$ if $\forall a, b \in C (K)$

(5)

∣ ∣ W (a \cup b) - Wa \land Wb ∣ ∣ \leq O (mesh) .

For some reason, I don’t think Scott was very interested in listening to what I had to say. Oh well. This alternative definition is pleasant in the sense that it says that you want your product to be such that the Whitney map is an algebra morphism. I have to admit that it would be kind of fun to disprove their statement :)

The next talk was by a friend of mine, Professor Robert Kotiuga. This wasn’t so much about discrete methods as it was about applied electromagnetics and I’m running out of energy, so I’ll skip his talk although it was definitely interesting.

The next talk was Professor Harrison. Unfortunately, her talk was a bit over my head so I still have lots of questions about it. Professor Sullivan seemed to be very interested in it and asked her a ton of questions. If was great to observe the two of them going back and forth. If only I had a brain and could have understood what they were talking about :) Anyway, I will try to see if Professor Harrison can maybe explain a bit about what that was all about.

The final talk was by Professor Sullivan. It’s kind of fun to watch him give a talk. It is almost as if he is just thinking out loud about some problem. I don’t even know if he prepares before standing up there. Apparently he has been working with a physicist Jae Suk Park. Together they are attempting to define exactly what a quantum field theory is in such a way that a mathematician can understand it. The talk was essentially constructing some discrete model of Poincare duality and cell complexes that sort of fit into Jae Suk Park’s framework. One interesting thing is that I thing Urs and my framework actually does fit into Jae Suk Park’s work. I hope to get a preprint from him so we can compare notes. I gave him a copy of our paper.

That’s it for now! Time to go home!

Ciao,
Eric

Posted by: Eric on February 12, 2004 10:30 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

Hi Eric -

many thanks for this most interesting report. It is very interesting to see that the questions the mathematicians find most interesting are so radically remote from the questions that I, as a physicist, consider interesting.

For instance I think it is trivial to see that what we have done has the correct continuum limit. (What I find much more interesting is how we can define nice action principles in the discrete framework, for instance.) But of course we should try to make that formally explicit if we want to communicate this idea. In order to do so, we somehow need to adapt the mathematician’s language. You have already made an attempt in this direction by reformulating that definition about continuum limits. I surely think that your remark is legitimate. But before I can do anythink to help I need to resolve the following problem:

Both definitions, the one used by Sullivan and your’s, involve the Whitney map. This, again, is only defined for simplices. But we would rather formulate the continuum limit in terms of hypercubes instead of simplices. None of these two definitions can directly be checked in our framework, because we don’t even have a Whitney map. Right?

Posted by: Urs Schreiber on February 13, 2004 8:23 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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Hi Urs,

None of these two definitions can directly be checked in our framework, because we don’t even have a Whitney map. Right?

Well, the Whitney map is defined originally for simplices, but if you understand what the Whitney map really says you will see that you can construct it for topological n-cubes easily enough. Don’t worry :)

Here is a conjecture of mine that would be great if we could prove.

If a manifold $M$ can be triangulated, then $M \times R$ can be diamonated.

If this is true, which I think it is, then we are in business. We simply need to write down the Whitney map for n-cubes. If this hasn’t been done, then it will be original, but I think it has been :)

I stated this during Dodziuk’s talk and everyone jumped on me. Revenge will be sweet :)

Eric

Posted by: Eric on February 13, 2004 8:41 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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Eric Forgy is continuing to discuss issues that have been raised here with Dr. Bossavit, who made the following important remark:

I’m not inclined to doubt what Wilson says (and that Sullivan told me nine years ago): There seems to be a strong no-go result that says that you can’t have a discrete triple $(⋆, d, \land)$ with all the desirable properties. How we cope with that is a serious issue.

Unfortunately I am not yet familiar with the assumptions that enter this result and the list of ‘all desirable properties’. Perhaps it all depends on the nature of our desires!

Eric knows a little more about this result. He writes:

What Wilson and Sullivan say that I do not find controversial (because it doesn’t contradict what Urs and I constructed) is that you can not have an algebra that is both associative and graded commutative that will approximate the continuum theory. This statement seems fairly clear to me. What I find controversial is their statement that you must have nonassociativity. Our work (as well as that of Alain Connes, since our work fits into his framework, sort of) is a counterexample to this. You can have an associative but non-graded commutative algebra that does approximate the continuum.

I fully agree with what Eric writes here. I would like to clarify this issue in detail. Even though there are about 80 pages in our notes, the crucial idea by which we get a triple $(⋆, d, \land)$ with some desirable properties can be sketched in just a few sentences. This is already attempted in the introduction to our notes on discrete differential geometry.

Let me recall the basic steps in just a few words:

- Consider an associative algebra $𝒜$ which characterizes a generalized manifold.

- Consider a differential calculus $Ω (𝒜, d)$ over $𝒜$ .

- Equip $Ω (𝒜, d)$ , regarded as a vector space, with an non-degenerate sesqui-linear inner product $〈 \cdot ∣ \cdot 〉$ , thus promoting it to the $Z$ -graded inner product space $ℋ : = (Ω (𝒜, d), 〈 \cdot ∣ \cdot 〉)$ .

- The differential $d$ and the elements $a \in 𝒜$ are represented as operators $\hat{d}$ and $\hat{a}$ on $ℋ$ in the obvious way.

- Let ${\hat{d}}^{†}$ be the adjoint of $\hat{d}$ with respect to $〈 \cdot ∣ \cdot 〉$ .

- Let $∣ vol 〉 \in Ω (𝒜, d)$ be an element of top grade that is in the kernel of ${\hat{d}}^{†}$ .

- Denote by $⋆$ an operator on $ℋ$ such that

(1)

⋆ \hat{d} = \pm {\hat{d}}^{†} ⋆ .

- Given all this it is very easy to check that the following is true:

a) $⋆$ exists iff $∣ vol 〉$ exists.

b) If they exist they are related by $∣ vol 〉 = ⋆ ∣ 1 〉$ and

(2)

⋆ ∣ a_{0} d a_{1} \dots d a_{p} 〉 = ({\hat{a}}_{0} [\hat{d}, {\hat{a}}_{1}] \dots [\hat{d}, {\hat{a}}_{p}])^{†} ∣ vol 〉 .

If we identify $\land$ with the product in $Ω (𝒜, d)$ the above gives a triple $(⋆, d, \land)$ for all cases where the above $∣ vol 〉$ exists. In our notes we show that it does exist in particular on topologically hypercubic graphs and that the resulting $⋆$ approximates the usual continuum Hodge star operator.

What are the properties of the triple that we construct?

- The only property of $⋆$ that is guaranteed is $⋆ \hat{d} = \pm {\hat{d}}^{†} ⋆$ .

- The only properties of $\hat{d}$ that are guaranteed are nilpotency and the graded Leibnitz propery.

- The product $\land$ is associative but non-commutative.

These properties are good from our point of view because they are sufficient to apply the formalism all over the place in physics. The construction is very similar to Connes’s NCG which also features non-commutativity and preserves associativity, being modeled after quantum mechanics.

(As Alejandro Rivero has kindly pointed out, some NCG papers state a couple of auxiliary axioms which we currently see no need for.)

To my mind the construction of the Hodge star in our formalism is a mere afterthought. (What I find more interesting is what we have to say about deformations of the inner product $〈 \cdot ∣ \cdot 〉$ and how these can describe metric geometry on generalized spaces as well as other ‘backgrounds’.) But still, seeing that a lot of people have thought about the construction of discrete Hodge star operators, I would very much like to try to understand how the above construction fits in with other approaches. Probably it is missing properties that are ‘desirable’ elsewhere.

Posted by: Urs Schreiber on February 23, 2004 5:10 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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Hello,

Looking back at my notes from Scott’s talk, I see he has their product of cochains expressed in an alternative form via

(1)

a \tilde{cup} b : = R [(Wa) \land (Wb)],

where $R$ is the de Rham map and $W$ is the Whitney map. This essentially verifies my suspicion that their algebra involves an antisymmetrization (coming from $\land$ ).

This gives a deformation $\tilde{cup}$ of the existing associative cup product of cochains (as defined in Munkres). In my opinion, this implies a prejudice toward the continuum theory and tries to construct a discrete theory from that.

On the other hand, we could be prejudiced toward the discrete theory and take the associative cup product of cochains as being fundamental and then construct a modified continuum product $\tilde{\land}$ defined by

(2)

f \tilde{\land} g : = W [(Rf) cup (Rg)]

Note that $\tilde{cup}$ is nonassociative, but commutative, whereas $\tilde{\land}$ is associative, but noncommutative. Note also the symmetry between the two definitions. Kind of suggestive, isn’t it?

Hence, it is becoming more clear to me that Urs and I do have something quite distinct from Wilson and Sullivan, but each approach should agree in the continuum limit.

Personally, I think losing associativity is more of a sin than losing commutativity. In fact, losing noncommutativity turns out to be a virtue here because it draws close parallels to noncommutative geometry and quantum theory. Quantum theory is kind of dead without associativity in some form or another.

Another interesting aspect of $\tilde{\land}$ is that the Whitney forms close as an algebra, i.e. if $f = Wa$ is a Whitney $p$ -form and $g = Wb$ is a Whitney $q$ -form, then $f \tilde{\land} g$ is a Whitney $(p + q)$ -form. Whitney forms do not constitute an algebra under the usual $\land$ -product, i.e. the $\land$ -product of a Whitney $p$ -form and a Whitney $q$ -form is NOT a Whitney $(p + q)$ -form. Whitney forms do constitute an algebra under the $\tilde{\land}$ -product.

Eric

PS: This is probably obvious, but I will point it out that since $d$ commutes with both $R$ and $W$ , we have

(3)

d (a \tilde{cup} b) = (da) \tilde{cup} b + (- 1)^{∣ a ∣} a \tilde{cup} (db)

and

(4)

d (f \tilde{\land} g) = (df) \tilde{\land} g + (- 1)^{∣ f ∣} f \tilde{\land} (dg)

so that in each case we satisfy both the graded Leibniz rule and nilpotency of $d$ .

Posted by: Eric on February 23, 2004 9:51 PM | Permalink | Reply to this

Re: The search for discrete differential geometry

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Hi Eric -

great, this looks like considerable progress in comparing the different approaches!

Let me see if I follow what you said:

It is obvious that $\tilde{cup}$ is graded commutative and that $\tilde{\land}$ is not.

For associativity one needs to know to which degree $W$ is the inverse of $R$ . I recall that you once said

(1)

RW = 1

(2)

WR \sim 1 .

I assume that the $\sim$ means ‘in the continuum limit’?

Ok, using this it is immediate that $\tilde{\land}$ is associative and $\tilde{cup}$ is so only in the continuum limit.

Yes, this looks good. As far as the algebraic product in our approach can be identified with the $cup$ product (which I think it can) $\tilde{\land}$ is certainly the natural thing to consider.

Can you show that for your generalization of the Whitney map to hypercubes we still have $RW = 1$ ?

Posted by: Urs Schreiber on February 24, 2004 11:33 AM | Permalink | Reply to this

Re: The search for discrete differential geometry

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Good morning Urs (and anyone else who may be reading this)!

What I wrote in that post wasn’t really anything new. I simply put down some thoughts that I had been aware of for a long time. As far as I know, $\tilde{\land}$ is my invention, but it wouldn’t surprise me to find it in some tome from 100 years ago :) I think this does point out the differences between our work and their’s.

Can you show that for your generalization of the Whitney map to hypercubes we still have $RW = 1$ ?

I certainly hope so since this is basically the definition of a Whitney form :)

I suspect that there is even some theorem somewhere saying that if you have any map $W : C (K) \to Ω (M)$ satisfying $RW = 1$ , then you will have $WR ~ 1$ (in the continuum limit). Take this with the usual grain of salt though.

To make the analogy between $\tilde{cup}$ and $\tilde{\land}$ complete, I won’t really be satisfied until we can make a concrete connection between our algebra and that of the usual cup product of cochains. Of course it will be a little different, but I suspect all the pieces should fit together nicely. I made some progress toward this last night (which is just a continuation of some of the stuff I’ve been doing over the last couple of weeks).

Last night I picked up an old book on introductory abstract algebra (Abtract Algebra, Kuczkowski) and reviewed the isomorphism theorems. It goes something like this:

Given a homomorphism $f : G \to H$ with kernel $K$ , then the groups $f (G)$ and $G / K$ are isomorphic.

This essentially says that the image of any homomorphism is equivalent to a quotient space. This helps make precise some of the half baked ideas I’ve been discussing with Urs behind the scenes.

Let me try to fit all this together now (while my energy lasts).

Definition: A directed $n$ -graph $G$ consists of $n + 1$ countable sets $G_{0}, . . ., G_{n}$ together with surjective source and target maps $s, t : G_{p} \to G_{p - 1}$ satisfying $st = ts$ and $s e_{i} = t e_{i} = 0$ for all $e_{i} \in G_{0}$ (0 is the emptyset).

The elements of $G_{p}$ are referred to as elementary $p$ -paths and are often denoted by a $(p + 1)$ -tuple of indices, e.g. $e_{i_{0} . . . i_{p}} \in G_{p} .$ The source of an elementary $p$ -path is then obtained by deleting the rightmost index, e.g. $s e_{i_{0} . . . i_{p}} = e_{i_{0} . . . i_{p - 1}}$ while the target of an elementary $p$ -path is obtained by deleting the leftmost index, e.g. $t e_{i_{0} . . . i_{p}} = e_{i_{1} . . . i_{p}} .$ Pictorially, an elementary $p$ -path may be thought of a $p$ -dimensional arrow extending from the source to the target, e.g. $e_{i_{0} . . . i_{p}} : e_{i_{0} . . . i_{p - 1}} \to e_{i_{1} . . . i_{p}} .$ (This is supposed to remind you of $n$ -categories.)

Now for another

Definition: The space of paths $P (G)$ on a directed $n$ -graph $G$ is a free abelian group $P (G) = ⨁_{r = 0}^{n} P_{r} (G),$ where $P_{r} (G)$ is generated by the set of elementary $r$ -paths $G_{r}$ .

Having defined $P (G)$ , we can now define the all-important boundary map $\partial : P_{p} (G) \to P_{p - 1} (G)$ by defining how it acts on the bases, namely $\partial e_{i_{0} . . . i_{p}} : = \sum_{r = 0}^{p} (- 1)^{r} e_{i_{0} . . . \hat{i_{r}} . . . i_{p}} .$ This probably looks familiar to anyone who knows about simplices and simplicial complexes. It is basically the same expression as the boundary of a $p$ -simplex. Let me point out the crucial difference. On a simplicial complex, every term on the right hand side is present, by definition. However, on a directed $n$ -graph $G$ , there may be terms on the right-hand side that are not present in $G$ . We handle this by setting these terms to zero. This has the nontrivial affect of destroying the nilpotent property of the boundary map. For example consider the directed $2$ -graph consisting of the six elementary paths ${e_{ijk}, e_{ij}, e_{jk}, e_{i}, e_{j}, e_{k}} .$ The boundary of $e_{ijk}$ is then given by $\partial e_{ijk} = e_{jk} - e_{ik} + e_{ij} = e_{jk} + e_{ij}$ because $e_{ik} = 0$ . The boundary of this is given by $\partial^{2} e_{ijk} = \partial e_{jk} + \partial e_{ij} = e_{k} - e_{i},$ which is not zero so that the boundary map is not nilpotent on the space of paths. Since one of my all-time favorite mottos is

“The boundary of a boundary is zero”

then I use this as kind of a guiding principle. Therefore, I refer to the space of paths suggestively as being “pre-geometric.” The next step is to look at “geometric” objects. These are the objects for which the boundary of a boundary is zero. More mathematically speaking, here is another

Definition: The kernel of $\partial^{2}$ is referred to as the space of chains and is denoted $C (G) = ⨁_{r = 0}^{n} C_{r} (G),$ where $C_{r} (G)$ is the subgroup of $P_{r} (G)$ for which $\partial^{2} = 0$ .

In other words, $c \in C_{p} (G) \Leftrightarrow \partial^{2} c = 0 .$ Now we are in business because the isomorphism theorem is within sight. Before getting to that, let me give another example that is less trivial (the space of 2-chains on that previous 6 element graph vanishes because there are no 2-paths for which $\partial^{2} = 0$ ). Consider the directed 2-graph whose elementary paths are given by ${e_{ijl}, e_{ikl}, e_{ij}, e_{jl}, e_{ik}, e_{kl}, e_{i}, e_{j}, e_{k}, e_{l}} .$ As before, we have $\partial^{2} e_{ijl} = e_{l} - e_{i}$ and $\partial^{2} e_{ikl} = e_{l} - e_{i}$ so that neither of these are 2-chains. However, the linear combination $\partial^{2} (e_{ijl} - e_{ikl}) = 0$ is a 2-chain. This element plays such a vital role in all of Urs and my work that we gave it a special name: the 2-diamond. The space of 2-chains on this directed 2-graph is then nontrival, i.e. it is generated by the 2-diamond $e_{ijl} - e_{ikl}$ .

The next step is to define a projection map $A : P (G) \to C (G),$ which is the analog of antisymmetrization in the continuum. In other words, the space of paths $P (G)$ is analogous to the space of tensors and $C (G)$ is analogous to the space for differential forms (currents actually). Athough the map $A$ plays a role similar to antisymmetrization and I suggestively use the symbol “ $A$ ”, it is important to realize that it is not antisymmetrization in the usual sense.

For example, with the previous directed 2-graph, there is only one nonvanishing 2-chain so we have $A (e_{ijl}) = 1 / 2 (e_{ijl} - e_{ikl})$ and $A (e_{ikl}) = - 1 / 2 (e_{ijl} - e_{ikl}),$ where the factor of $1 / 2$ is required since we must have $A^{2} = A$ . It is worth mentioning now that for any directed $n$ -graph $G$ , we always have $C_{1} (G) = P_{1} (G)$ and $C_{0} (G) = P_{0} (G)$ because $\partial^{2}$ vanishes on any 0- or 1-path so that $A (e_{ij}) = e_{ij}, \forall e_{ij} \in P_{1} (G)$ and $A (e_{i}) = e_{i}, \forall e_{i} \in P_{0} (G) .$ We can now bring in the isomorphism theorem. Since $C (G)$ is a subgroup of $P (G)$ and $A : P (G) \to C (G)$ is a homomorphism, that means that $C (G) \sim P (G) / \ker (A) .$ In other words, we can write things like $A (e_{ijk}) \sim$

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January 26, 2004