# Tag: noncommutative

The lecturers, topics and dates of the 6 mini-courses in our ‘advanced master degree 2011 in noncommutative algebra and geometry’ are :

February 21-25
Vladimir Bavula (University of Sheffield) :
Localization Theory of Rings and Modules

March 7-11
Hans-Jürgen Schneider (University of München) :
Nichols Algebra and Root Systems

April 11-12
Bernhard Keller (Université Paris VII):
Cluster Algebra and Quantum Cluster Algebras

April 18-22
Jacques Alev (Université Reims):
Automorphisms of some Basic Algebras

May 3-8
Goro Kato (Cal Poly University, San Luis Obispo, US):
Sheaf Cohomology and Zeta – Functions

May 9-13
Markus Reineke (University of Wuppertal):
Moduli Spaces of Representatives

More information can be found here. I’ve been told that some limited support is available for foreign graduate students wanting to attend this programme.

Mathblogging.org is a recent initiative and may well become the default starting place to check on the status of the mathematical blogosphere.

Handy, if you want to (re)populate your RSS-aggregator with interesting mathematical blogs, is their graphical presentation of (nearly) all math-blogs ordered by type : group blogs, individual researchers, teachers and educators, journalistic writers, communities, institutions and microblogging (twitter). Links to the last 7 posts are given so you can easily determine whether that particular blog is of interest to you.

The three people behind the project, Felix Breuer, Frederik von Heymann and Peter Krautzberger, welcome you to send them links to (micro)blogs they’ve missed. Surely, there must be a lot more mathematicians with a twitter-account than the few ones listed so far…

Even more convenient is their list of latest posts from their collection, ordered by date. I’ve put that page in my Bookmarks Bar the moment I discovered it! It would be nice, if they could provide an RSS-feed of this list, so that people could place it in their sidebar, replacing old-fashioned and useless blogrolls. The site does provide two feeds, but they are completely useless as they click through to empty pages…

While we’re on the topic of math-blogging, the results of the ‘What should we write about next?’-poll that ran the previous two days on the entry page. Of all people visiting that page, 2.6% left suggestions.

The vast majority (67%) wants more posts on noncommutative geometry. Most of you are craving for introductions (and motivation) accessible to undergraduates (as ‘it’s hard to find quality, updated information on this’). In particular, you want posts giving applications in mathematics (especially number theory), or explaining relationships between different approaches. One person knew exactly how I should go about to achieve the hoped-for accessibility : “As a rule, I’d take what you think would be just right for undergrads, and then trim it down a little more.”

Others want rather specialized posts, such as on ‘connection and parallel transport in noncommutative geometry’ or on ‘trees (per J-L. Loday, M. Aguiar, Connes/Kreimer renormalization (aka Butcher group)), or something completely other tree-related’.

Fortunately, some of you told me it was fine to write about ‘combinatorial games and cool nim stuff, finite simple groups, mathematical history, number theory, arithmetic geometry’, pushed me to go for ‘anything monstrous and moonshiney’ (as if I would know the secrets of the ‘connection between the Mathieu group M24 and the elliptic genus of K3’…) or wrote that ‘various algebraic geometry related posts are always welcome: posts like Mumford’s treasure map‘.

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.

In view or recents events & comments, some changes have been made or will be made shortly :

categories : Sanitized the plethora of wordpress-categories to which posts belong. At the moment there are just 5 categories : ‘stories’ and ‘web’ (for all posts with low math-content) and three categories ‘level1’, ‘level2’ and ‘level3’, loosely indicating the math-difficulty of a post.

MathJax : After years of using LatexRender and WP-Latex, we’ll change to MathJax from now on. I’ll try to convert older posts as soon as possible. (Update : did a global search and replace. ‘Most’ LaTeX works, major exceptions being matrices and xymatrix commands. I’ll try to fix those later with LatexRender.)

theme : The next couple of days, the layout of this site may change randomly as I’ll be trying out things with the Swift wordpress theme. Hopefully, this will converge to a new design by next week.

name : Neverendingbooks will be renamed to something more math-related. Clearly, the new name will depend on the topics to be covered. On the main index page a pop-up poll will appear in the lower right-hand corner after 10 seconds. Please fill in the topics you’d like us to cover (no name or email required).

This poll will close on friday 21st at 12 CET and its outcome will influence name/direction of this blog. Use it also if you have a killer newname-suggestion. Among the responses so far, a funnier one : “An intro to, or motivation for non-commutative geometry, aimed at undergraduates. As a rule, I’d take what you think would be just right for undergrads, and then trim it down a little more.”

guest-posts : If you’d like to be a guest-blogger here at irregular times, please contact me. The first guest-post will be on noncommutative topology and the interpretation of quantum physics, and will appear soon. So, stay tuned…

SocialMention gives a rather accurate picture of the web-buzz on a specific topic. For this reason I check it irregularly to know what’s going on in noncommutative geometry, at least web-wise.

Yesterday, I noticed two new kids on the block : Jason and David. Their blogs have (so far ) 44 resp. 27 posts, this month alone. My first reaction was: respect!, until I glanced at their content…

David of E-Infinity

Noncommutative geometry has a deplorable track record when it comes to personality-cults and making extra-ordinary claims, but this site beats everything I’ve seen before. Its main mission is to spread the gospel according to E.N.

A characteristic quote :

“It was no doubt the intention of those well known internet thugs and parasites to distract us from science and derail us from our road. This was the brief given to them by you know who. Never the less we will attempt to give here what can only amount to a summary of the summary of what E. N. considers to be the philosophical background to his theory.”

Jason of the E.N. watch

The blog’s mission statement is to expose the said prophet E.N. as a charlatan.

The language used brings us back to the good(?!) old string-war days.

“This is amusing because E. N.’s sockpuppets go on and on about E. N. being a genius polymath with an expert grasp of science, art, history, philosophy and politics. E. N. Watch readers of course know that his knowledge in all areas comes primarily from mass-market popularizations.”

As long as the Connes support-blog and the Rosenberg support-blog remain silent and the Jasons and Davids of this world run the online ncg-show, it is probably better to drop the topic here.