neverendingbooks

Guest post by Fred Van Oystaeyen.

In my book “Virtual Topology and Functorial Geometry” (Taylor and Francis, 2009) I proposed a noncommutative version of space-time ; it is a toy model, but mathematically correct and I included a few philosophical remarks about : “What if reality is noncommutative ?”.

I want to add a few ideas about how “strange” ideas in quantum mechanics all fit naturally in the noncommutative world. First let us talk about noncommutative geometry in an intuitive way.

Then noncommutative space may be thought of as a set of noncommutative places but these noncommutative places need not be sets, in particular they are not sets of points. There is a noncommutative join $\vee$ and a noncommutative intersection $\wedge$, and they satify the axioms (very natural ones) of a noncommutative topology.

The non-commutativity is characterized by the existence of non $\wedge$-idempotent places, i.e. places with a nontrivial self intersection. This allows the $\wedge$ to be noncommutative. From algebraic geometric it follows that one may be interested to let $\vee$ be an abelian operation (hence defining a virtual topology) so let us assume this from hereon.

The set of $\wedge$-idempotent noncommutative-places forms the “commutative shadow” of the noncommutative space; it has operations $\vee$ and $\mathop{\wedge}\limits_{\bullet}$ which are abelian and $\sigma \mathop{\wedge}\limits_{\bullet}\tau$ may be thought of as the largest $\wedge$-idempotent smaller than $\sigma$ and $\tau$ in the partial ordering of the noncommutative space.

The $\wedge$-idempotent noncommutative places are sets in a commutative topology and these are the observable places in the noncommutative space. In the book I present a dynamic (time !) model allowing further elaboration on the noncommutative space but for now let us stick to the intuitive model and assume that space is in fact noncommutative with commutative shadow built upon our space time of physics.

In fact all observations, measurings and predictions made in physics are not about reality but about our observations of reality, so it may be an eternal fact that our observations of reality are described in our brains by commutative methods. Nevertheless we can observe effect of objects existing at noncommutative places in “neighboring” $\wedge$-idempotents sets or observable places.

First if an object exists at a noncommutative place it also exists at all subplaces (a harmless assumption not really essential for the rest). So if there is a noncommutative place, where some object exists, parts of this object may be observed at idempotent subplaces of the noncommutative place. These may even be disjoint in the commutative shadow, not “too far apart” as one object exists on the total noncommutative space.

Since only a part of the noncommutative object is observed on the $\wedge$-idempotent subplace it is not clear that one may actually recognize the observations at different commutative places as belonging to the same noncommutative object. Once one observes one observable place that object seems to exist only on that (commutative) place. Hence a quantum particle can be thought of as existing on several “places” but once observed it looks like it only exists there. This is a first natural phenomenon reflecting “strange” quantum mechanical principles.

Secondly let us look at the double slit experiment. The slits correspond to commutative places $\sigma_1$ and $\sigma_2$ and $\sigma_1 \mathop{\wedge}\limits_{\bullet}\sigma_2=\emptyset$, however in the noncommutative world $\sigma_1\wedge\sigma_2$ need not be empty, only it has no $\wedge$-idempotent subplaces !

Therefore if a photon is defined on a noncommutative place with “light”-effect on observable places “near enough” to it (in a neighborhood small enough to have an observable effect say) then the photon may pass though both slits without splitting or without splitting reality (parallel universes) but just moving into the noncommutative space inside $\sigma_1$ and $\sigma_2$ !

The observable effect at the slits will appear in commutative places near enough (for example, intersecting) to $\sigma_1$ or to $\sigma_2$. As the photon moves on, observable effects will appear in commutative places intersecting the one near to $\sigma_1$ or the one near to $\sigma_2$ and these may themselves have nonempty intersections.

At the moment the effect via $\sigma_1$ interacts with the effect via $\sigma_2$. As the photon progresses in its observed direction other $\wedge$-idempotents showing observable effects may meet and so several interactions between observable effects (via $\sigma_1$ and $\sigma_2$) build a picture of interference.

The symmetry of this picture actually suggests a symmetric arrangement of commutative places around a noncommutative place. So the noncommutativity of space may explain this phenomenon without holographic principle or parallel universes.

In a similar way dark mass may well be mass existing in a non-observable noncommutative place (i.e. containing no observable places). If a lot of mass is in a non-observable noncommutative place its gravity may pull matter from surrounding observable places into the noncommutative place and this may explain black holes. All kinds of problems relating to black holes may have natural non commutative solutions, e.g. information may pass from observable places to noncommutative places and is not lost, only non-observable.

In fact is the definition of information not depending on the nature and capability of the recipient ? There are many philosophically interesting ramifications of these ideas, for example every chemical or neurochemical activity should also be placed in the noncommutative space.

In the book I mentioned how “free will” could be a noncommutative space aspect of the brain activity. I also mention a possible relations with string theory. I am not a specialist in all these things but now I reached the point that I “feel” noncommutative space is a better approximation of the reality and one should investigate it further.

is decided : I’ll keep maintaining this URL until new-year’s eve. At that time I’ll be blogging here for 5 years…

The few encounters I’ve had with architects, taught me this basic lesson of life : the main function of several rooms in a house changes every 5 years (due to children and yourself getting older).

So, from january 1st 2009, I’ll be moving out of here. I will leave the neverendingbooks-site intact for some time to come, so there is no need for you to start archiving it en masse, yet.

Previously I promised to reconsider this blog’s future over a short vacation, but as vacation is looking to be as illusory as the 24-dimensional monster-manifold, I spend my time throwing up ideas into thin and, it seems, extremely virtual air.

Some of you will think this is a gimmick, aiming to attract more comments (there is no post getting more responses than an imminent-end-to-this-blog-post) but then I hope to have settled this already. Neverendingbooks will die on 31st of december 2008. The only remaining issue being : do I keep on blogging or do I look for another time-consumer such as growing tomatoes or, more probably, collecting single malts…

For reasons I’ve stated before, I can see little future in anything but a conceptual-, group- blog. The first part I can deal with, but for the second I’ll be relying on others. So, all I can do is offer formats hoping that some of you are willing to take the jump and try it out together.

Such as in the bloomsday-post where I sketched the BistroMath blog-concept. Perhaps you thought I was just kidding, hoping for people to commit themselves and them calling “Gotcha…”. Believe me, 30 years of doing mathematics have hardwired my brains such that I always genuinely believe in the things I write down at the moment I do (but equally, if someone offers me enough evidence to the contrary, I’ll drop any idea on the spot).

I still think the BistroMath-project has the potential of leading to a bestseller but Ive stated I was not going to pursue the idea if not at least 5 people were willing to join and at least 1 publisher showed an interest. Ironically, I got 2 publishers interested but NO contributors… End of that idea.

Today I offer another conceptual group-blog : the Noether-boys seminar (with tagline ; _the noncommutative experts’ view on 21st century mathematics_). And to make it a bit more concrete Ive even designed a potential home-page :

So, what’s the deal? In the 1930-ties Emmy Noether collected around her in Goettingen an exceptionally strong group of students and collaborators (among them : Deuring, Fitting, Levitski, Schilling, Tsen, Weber, Witt, VanderWaerden, Brauer, Artin, Hasse, MacLane, Bernays, Tausky, Alexandrov… to name a few).

Collectively, they were know as the “Noether-boys” (or “Noether-Knaben” or “Trabanten” in German) and combined seminar with a hike to the nearby hills or late-night-overs at Emmy’s apartment. (Btw. there’s nothing sexist about Noether-boys. When she had to leave Germany for Bryn Mawr College, she replaced her boys to form a group of Noether-girls, and even in Goettingen there were several women in the crowd).

They were the first generation of mathematicians going noncommutative and had to struggle a bit to get their ideas accepted.
I’d like to know what they might think about the current state of mathematics in which noncommutativity seems to be generally accepted, even demanded if you want to act fashionable.

I’m certain half of the time they would curse intensely, and utter something like ‘steht shon alles bei Frau Noether…’ (as Witt is witnessed to have done at least once), and about half the time they might get genuinely interested, and be willing to try and explain the events leading up to this to their fellow “Trabanten”. Either way, it would provide excellent blog-posts.

So I’m looking for people willing to borrow the identity of one of the Noether-boys or -girls. That is, you have to be somewhat related to their research and history to offer a plausible reaction to recent results in either noncommutative algebra, noncommutative geometry or physics. Assuming their identity you will then blog to express your (that is, ‘their’) opinion and interact with your fellow Trabanten as might have been the case in the old days…

I’d like to keep Emmy Noether for the admin-role of the blog but all other characters are free at this moment (except I’m hoping that no-one will choose my favourite role, which is probably the least expected of them anyway).

So please, if you think this concept might lead to interesting blogging, contact me! If I don’t get any positives in this case either, I might think about yet another concept (or instead may give up entirely).

Referring to the triple of exceptional Galois groups $L_2(5),L_2(7),L_2(11) $ and its connection to the Platonic solids I wrote : “It sure seems that surprises often come in triples…”. Briefly I considered replacing triples by trinities, but then, I didnt want to sound too mystic…

David Corfield of the n-category cafe and a dialogue on infinity (and perhaps other blogs I’m unaware of) pointed me to the paper Symplectization, complexification and mathematical trinities by Vladimir I. Arnold. (Update : here is a PDF-conversion of the paper)

The paper is a write-up of the second in a series of three lectures Arnold gave in june 1997 at the meeting in the Fields Institute dedicated to his 60th birthday. The goal of that lecture was to explain some mathematical dreams he had.

The next dream I want to present is an even more fantastic set of theorems and conjectures. Here I also have no theory and actually the ideas form a kind of religion rather than mathematics.
The key observation is that in mathematics one encounters many trinities. I shall present a list of examples. The main dream (or conjecture) is that all these trinities are united by some rectangular “commutative diagrams”.
I mean the existence of some “functorial” constructions connecting different trinities. The knowledge of the existence of these diagrams provides some new conjectures which might turn to be true theorems.

Follows a list of 12 trinities, many taken from Arnold’s field of expertise being differential geometry. I’ll restrict to the more algebraically inclined ones.

1 : “The first trinity everyone knows is”

but I would like to alter it into

where $\mathbb{H} $ are the Hamiltonian quaternions. The trinity on the left may be natural to differential geometers who see real and complex and hyper-Kaehler manifolds as distinct but related beasts, but I’m willing to bet that most algebraists would settle for the trinity on the right where $\mathbb{O} $ are the octonions.

2 : The next trinity is that of the exceptional Lie algebras E6, E7 and E8.

with corresponding Dynkin-Coxeter diagrams

Arnold has this to say about the apparent ubiquity of Dynkin diagrams in mathematics.

Manin told me once that the reason why we always encounter this list in many different mathematical classifications is its presence in the hardware of our brain (which is thus unable to discover a more complicated scheme).
I still hope there exists a better reason that once should be discovered.

Amen to that. I’m quite hopeful human evolution will overcome the limitations of Manin’s brain…

3 : Next comes the Platonic trinity of the tetrahedron, cube and dodecahedron

Clearly one can argue against this trinity as follows : a tetrahedron is a bunch of triangles such that there are exactly 3 of them meeting in each vertex, a cube is a bunch of squares, again 3 meeting in every vertex, a dodecahedron is a bunch of pentagons 3 meeting in every vertex… and we can continue the pattern. What should be a bunch a hexagons such that in each vertex exactly 3 of them meet? Well, only one possibility : it must be the hexagonal tiling (on the left below). And in normal Euclidian space we cannot have a bunch of septagons such that three of them meet in every vertex, but in hyperbolic geometry this is still possible and leads to the Klein quartic (on the right). Check out this wonderful post by John Baez for more on this.

4 : The trinity of the rotation symmetry groups of the three Platonics

where $A_n $ is the alternating group on n letters and $S_n $ is the symmetric group.

Clearly, any rotation of a Platonic solid takes vertices to vertices, edges to edges and faces to faces. For the tetrahedron we can easily see the 4 of the group $A_4 $, say the 4 vertices. But what is the 4 of $S_4 $ in the case of a cube? Well, a cube has 4 body-diagonals and they are permuted under the rotational symmetries. The most difficult case is to see the $5 $ of $A_5 $ in the dodecahedron. Well, here’s the solution to this riddle

there are exactly 5 inscribed cubes in a dodecahedron and they are permuted by the rotations in the same way as $A_5 $.

7 : The seventh trinity involves complex polynomials in one variable

the Laurant polynomials and the modular polynomials (that is, rational functions with three poles at 0,1 and $\infty $.

8 : The eight one is another beauty

Here ‘numbers’ are the ordinary complex numbers $\mathbb{C} $, the ‘trigonometric numbers’ are the quantum version of those (aka q-numbers) which is a one-parameter deformation and finally, the ‘elliptic numbers’ are a two-dimensional deformation. If you ever encountered a Sklyanin algebra this will sound familiar.

This trinity is based on a paper of Turaev and Frenkel and I must come back to it some time…

The paper has some other nice trinities (such as those among Whitney, Chern and Pontryagin classes) but as I cannot add anything sensible to it, let us include a few more algebraic trinities. The first one attributed by Arnold to John McKay

13 : A trinity parallel to the exceptional Lie algebra one is

between the 27 straight lines on a cubic surface, the 28 bitangents on a quartic plane curve and the 120 tritangent planes of a canonic sextic curve of genus 4.

14 : The exceptional Galois groups

explained last time.

15 : The associated curves with these groups as symmetry groups (as in the previous post)

where the ? refers to the mysterious genus 70 curve. I’ll check with one of the authors whether there is still an embargo on the content of this paper and if not come back to it in full detail.

16 : The three generations of sporadic groups

Do you have other trinities you’d like to worship?

You may not have noticed, but I’m in a foul mood for weeks now because of comments and reactions to the last line of the post on Finding Moonshine. I wrote

Du Sautoy is a softy! I’d throw such students out of the window…

and got everyone against me for this (first floor) defenestration threat…

That’s OK! I sometimes post what’s on my mind and if you don’t like it you are free to leave a comment, and, usually I won’t even bother to reply to it. But occasionally, stuff is bottling up un-healthily.

So, I thought it was a good idea to have a prolonged easter-vacation, somewhere in the south of France. The weather, food, rest, drinks, company and all that were just gorgeous

but …

A quick recap. Here’s the relevant section in duSautoy’s book again :

One of my graduate students has just left my office. He’s done some great work over the past three years and is starting to write up his doctorate, but he’s just confessed that he’s not sure that he wants to be a mathematician. I’m feeling quite sobered by this news. My graduate students are like my children. They are the future of the subject. Who’s going to read all the details of my papers if not my mathematical offspring? The subject feels so tribal that anyone who says they want out is almost a threat to everything the tribe stands for.
Anton has been working on a project very close to my current problem. There’s no denying that one can feel quite disillusioned by not finding a way into a problem. Last year one of my post-docs left for the City after attempting to scale this mountain with me. I’d already rescued him from being dragged off to the City once before. But after battling with our problem and seeing it become more and more complex, he felt that he wasn’t really cut out for it.
What is unsettling for me is that they both questioned the importance of what we are doing. They’ve asked that ‘What’s it all for?’ question, and think they’ve seen the Emperor without any clothes.
Anton has questioned whether the problems we are working on are really important. I’ve explained why I think these are fundamental questions about basic objects in nature, but I can see that he isn’t convinced. I feel I am having to defend my whole existence. I’ve arranged for him to join me at a conference in Israel later this month, and I hope that seeing the rest of the tribe enthused and excited about these problems will re-inspire him. It will also show him that people are interested in what he is dedicating his time to.

For starters, I’m getting old so I’m from those long-forgotten days when you had to do a Ph.D. to prove that you could conduct research independently.

A fortiori this meant that the topic of your thesis was your own choice and interest. The role of your Ph.D. advisor was to get you going and, occasionally, to warn you when you were straying too far afield but that was it.

You, and only you, were responsible to get the thesis finished and defended.

Today, the Ph.D. is just another item on the market to be consumed.

Graduate students shop around looking for the advisor having the best sales-pitch, offers the best deal and, if possibly, the best after-phd service aka the promise of an academic career.

Topic and main outline of the proofs are provided by the advisor and an exceptionally good student today means that (s)he proved a few results along the way on her/his own.

University policy and the promotion-rat-race appear to make the Ph.D. more important to the advisor than to the defendent.

Independence of research today means that after your PhD is obtained, you ditch your advisor and try to get into the slipstream of another more powerful guru, having better after-phd service prospects…

For those who stick with their old advisor, the moment of truth comes when they fail to get a renewal of their grant or a permanent position.

At that time one can hear complaints such as : “That persons’ student got ranked ahead of me and I always thought you were better than that person?” or “The better ranked people for the position are all doing that topic instead of ‘ours’, so I guess your topic isn’t so important after all!”. duSautoy’s captures it all in this one key sentence :

They’ve asked that ‘What’s it all for?’ question, and think they’ve seen the Emperor without any clothes.

As if, failing to get a permanent position is the advisors fault, more than yours…

Just for once, try to be honest to yourself : count the number of hours a day your brain-power gets you over 120 IQ. Substract from this the number of hours a day lost surfing the web idly, trying to read unreadable hep-th papers, socializing, kissing asses, socializing, doing fun things with you fellow graduate students, socializing, working on a relation, chatting, texing, emailing insults but softening it all with a closing smily , socializing, etc… (you know the daily-drill of a 20-30-something phd-student a lot better than I do)

I’ll be damned if you get a positive outcome. But if you do, I’ll be happy to take you on as a PhD student…

Well, it’s no threat, it’s a promise : the first ex-student who gets me into a ‘why was it all good for?’ discussion will experience first floor defenestration! (provided I’ll get my window open in time)

And, to soften it all, I’ll add the obligatory

The next thing on my tech-to-do-list : learn all about Yahoo Pipes :

Pipes is a powerful composition tool to aggregate, manipulate, and mashup content from around the web. Like Unix pipes, simple commands can be combined together to create output that meets your needs. Here are a few popular ways the service can be used:
– create your ultimate custom feed by combining many feeds into one, then sorting, filtering and translating them.
– geocode your favorite feeds and browse the items on an interactive map.
– remix your favorite data sources and use the Pipe to power a new application.
– build custom vertical search pages that are impossible with ordinary search engines.
– power widgets/badges on your web site.
– consume the output of any Pipe in RSS, JSON, KML, and other formats.

I’ve posted before on setting up your own lifestream, or your own planet, or scraping feeds, or subscribing to my brain, or … whatever. The good news is : all these ideas are now superseded by Pipes!

Pipes is a free online service that lets you remix popular feed types and create data mashups using a visual editor. You can use Pipes to run your own web projects, or publish and share your own web services without ever having to write a line of code. You make a Pipe by dragging pre-configured modules onto a canvas and wiring them together in the Pipes Editor. Once you’ve built a Pipe, you’ll be able save it on our server and then call it like you would any other feed. Pipes offers output in RSS 2.0, RSS 1.0 (RDF), JSON and Atom formats for maximum flexibility. You can also choose to publish your Pipe and share it with the world, allowing other users to clone it, add their own improvements, or use it as a subcomponent in their own creations.

This is the essential message to get : yahoo-pipes allows you to remix the web, filtering out all noise! And the good news is

1. There are plenty of public pipes around to get you going, and
2. Pipes has an iTouch-friendly interface (see above left). All you have to do is to Safari to iphone.pipes.yahoo.com and use them.

Here are a few public-pipes you can use out of the box!

• iPhone / iPod Touch: The Most Comprehensive Feed Ever!, doing what it promises : giving you the best iTouch-posts without having to roam for them.
• JSON Geocoder, returning lat/lon/address info from the the given address.
• Uber Blog Search, Search all the blogosphere with one query. Hits Google, Ask, Technorati, and icerocket then returns the unique results. Below the web-interface giving the results for ‘noncommutative’…

• Meta Search Alerts if you want a less ‘neverendingbooks’-based reply

and finally, one of my favorites, implementing to some extend the Lifestream-idea (iTouch-interface above left)

• lifefeed – virable, Easily Aggregate your social whereabouts great for blogs profiles and more! Aggregates Your Feeds From: -Digg -Last.fm -Twitter -Flickr -Del.icio.us and your very own blog Adopt and Improve, enjoy!

I’ll promise to spend some time soon to set up my very own pipes and make them available…