# Tag: Connes

Last time we did recall Manin’s comparisons between some approaches to geometry over the absolute point $\pmb{spec}(\mathbb{F}_1)$ and trends in the history of art.

In the comments to that post, Javier Lopez-Pena wrote that he and Oliver Lorscheid briefly contemplated the idea of extending Manin’s artsy-dictionary to all approaches they did draw on their Map of $\mathbb{F}_1$-land.

So this time, we will include here Javier’s and Oliver’s insights on the colored pieces below in their map : CC=Connes-Consani, Generalized torified schemes=Lopez Pena-Lorscheid, Generalized schemes with 0=Durov and, this time, $\Lambda$=Manin-Marcolli.

Durov : romanticism

In his 568 page long Ph.D. thesis New Approach to Arakelov Geometry Nikolai Durov introduces a vast generalization of classical algebraic geometry in which both Arakelov geometry and a more exotic geometry over $\mathbb{F}_1$ fit naturally. Because there were great hopes and expectations it would lead to a big extension of algebraic geometry, Javier and Oliver associate this approach to romantism. From wikipedia : “The modern sense of a romantic character may be expressed in Byronic ideals of a gifted, perhaps misunderstood loner, creatively following the dictates of his inspiration rather than the standard ways of contemporary society.”

Manin and Marcolli : impressionism

Yuri I. Manin in Cyclotomy and analytic geometry over $\mathbb{F}_1$ and Matilde Marcolli in Cyclotomy and endomotives develop a theory of analytic geometry over $\mathbb{F}_1$ based on analytic functions ‘leaking out of roots of unity’. Javier and Oliver depict such functions as ‘thin, but visible brush strokes at roots of 1’ and therefore associate this approach to impressionism. Frow wikipedia : ‘Characteristics of Impressionist paintings include: relatively small, thin, yet visible brush strokes; open composition; emphasis on accurate depiction of light in its changing qualities (often accentuating the effects of the passage of time); common, ordinary subject matter; the inclusion of movement as a crucial element of human perception and experience; and unusual visual angles.’

Connes and Consani : cubism

In On the notion of geometry over $\mathbb{F}_1$ Alain Connes and Katia Consani develop their extension of Soule’s approach. A while ago I’ve done a couple of posts on this here, here and here. Javier and Oliver associate this approach to cubism (a.o. Pablo Picasso and Georges Braque) because of the weird juxtapositions of the simple monoidal pieces in this approach.

Lopez-Pena and Lorscheid : deconstructivism

Torified varieties and schemes were introduced by Javier Lopez-Pena and Oliver Lorscheid in Torified varieties and their geometries over $\mathbb{F}_1$ to get lots of examples of varieties over the absolute point in the sense of both Soule and Connes-Consani. Because they were fragmenting schemes into their “fundamental pieces” they associate their approach to deconstructivism.

Another time I’ll sketch my own arty-farty take on all this.

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‘.

Sunday january 2nd around 18hr NeB-stats went crazy.

Referrals clarified that the post ‘What is the knot associated to a prime?’ was picked up at Reddit/math and remained nr.1 for about a day.

Now, the dust has settled, so let’s learn from the experience.

A Reddit-mention is to a blog what doping is to a sporter.

You get an immediate boost in the most competitive of all blog-stats, the number of unique vistors (blue graph), but is doesn’t result in a long-term effect, and, it may even be harmful to more essential blog-stats, such as the average time visitors spend on your site (yellow graph).

For NeB the unique vistors/day fluctuate normally around 300, but peaked to 1295 and 1733 on the ‘Reddit-days’. In contrast, the avg. time on site is normally around 3 minutes, but dropped the same days to 44 and 30 seconds!

Whereas some of the Reddits spend enough time to read the post and comment on it, the vast majority zap from one link to the next. Having monitored the Reddit/math page for two weeks, I’m convinced that post only made it because it was visually pretty good. The average Reddit/math-er is a viewer more than a reader…

So, should I go for shorter, snappier, more visual posts?

Let’s compare Reddits to those coming from the three sites giving NeB most referrals : Google search, MathOverflow and Wikipedia.

This is the traffic coming from Reddit/math, as always the blue graph are the unique visitors, the yellow graph their average time on site, blue-scales to the left, yellow-scales to the right.

Here’s the same graph for Google search. The unique visitors/day fluctuate around 50 and their average time on site about 2 minutes.

The math-related search terms most used were this month : ‘functor of point approach’, ‘profinite integers’ and ‘bost-connes sytem’.

More rewarding to me are referrals from MathOverflow.

The number of visitors depends on whether the MathO-questions made it to the front-page (for example, the 80 visits on december 15, came from the What are dessins d’enfants?-topic getting an extra comment that very day, and having two references to NeB-posts : The best rejected proposal ever and Klein’s dessins d’enfant and the buckyball), but even older MathO-topics give a few referrals a day, and these people sure take their time reading the posts (+ 5 minutes).

Other MathO-topics giving referrals this month were Most intricate and most beautiful structures in mathematics (linking to Looking for F-un), What should be learned in a first serious schemes course? (linking to Mumford’s treasure map (btw. one of the most visited NeB-posts ever)), How much of scheme theory can you visualize? (linking again to Mumford’s treasure map) and Approaches to Riemann hypothesis using methods outside number theory (linking to the Bost-Connes series).

Finally, there’s Wikipedia

giving 5 to 10 referrals a day, with a pretty good time-on-site average (around 4 minutes, peaking to 12 minutes). It is rewarding to see NeB-posts referred to in as diverse Wikipedia-topics as ‘Fifteen puzzle’, ‘Field with one element’, ‘Evariste Galois’, ‘ADE classification’, ‘Monster group’, ‘Arithmetic topology’, ‘Dessin d’enfant’, ‘Groupoid’, ‘Belyi’s theorem’, ‘Modular group’, ‘Cubic surface’, ‘Esquisse d’un programme’, ‘N-puzzle’, ‘Shabat polynomial’ and ‘Mathieu group’.

What lesson should be learned from all this data? Should I go for shorter, snappier and more visual posts, or should I focus on the small group of visitors taking their time reading through a longer post, and don’t care about the appallingly high bounce rate the others cause?

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.

No christmas- or new-years family party without heated discussions. Often on quite silly topics.

For example, which late 19th-century bookcharacter turned out to be most influential in the 20th century? Dracula, from the 1897 novel by Irish author Bram Stoker or Sir Arthur Conan Doyle’s Sherlock Holmes who made his first appearance in 1887?

Well, this year you can spice up such futile discussions by going over to Google Labs Books Ngram Viewer, specify the time period of interest to you and the relevant search terms and in no time it spits back a graph comparing the number of books mentioning these terms.

Here’s the 20th-century graph for ‘Dracula’ (blue), compared to ‘Sherlock Holmes’ (red).

The verdict being that Sherlock was the more popular of the two for the better part of the century, but in the end the vampire bit the detective. Such graphs lead to lots of new questions, such as : why was Holmes so popular in the early 30ties? and in WW2? why did Dracula become popular in the late 90ties? etc. etc.

Clearly, once you’ve used Books Ngram it’s a dangerous time-waster. Below, the graphs in the time-frame 1980-2008 for Alain Connes (blue), noncommutative geometry (red), Hopf algebras (green) and quantum groups (yellow).

It illustrates the simultaneous rise and fall of both quantum groups and Hopf algebras, whereas the noncommutative geometry-graph follows that of Alain Connes with a delay of about 2 years. I’m sure you’ll find a good use for this splendid tool…