# Tag: noncommutative

Never a dull moment with Books Ngram Viewer. Pick your favorite topic(s) and try to explain and name valleys and peaks in the Ngram.

An example. I wanted to compare the relative impact of a couple of topics I love, algebraic geometry (blue), category theory (red), representation theory (green) and noncommutative geometry (the bit of yellow in the lower right hand corner…) from 1960 onwards.

I was surprised to find out that the first three topics were almost in the same impact-league, but then Ngram-viewing can be cruel when you’re biased …

Anyone having an explanation/name for the great depressions of 1982, 1993 and 1996?

On the positive side, what happened in 1988-89 or what caused the representation-peak in 1999, or the category-delirium in 2006?

So far, I’ve only been able to pinpoint a couple of events. My favorite being the red peak in 1973, which I’d like to christen “Grothendieck’s folly”.

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…

This is a belated response to a Math-Overflow exchange between Thomas Riepe and Chandan Singh Dalawat asking for a possible connection between Connes’ noncommutative geometry approach to the Riemann hypothesis and the Langlands program.

Here’s the punchline : a large chunk of the Connes-Marcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as an exploration of the noncommutative boundary to the Langlands program (at least for $GL_1$ and $GL_2$ over the rationals $\mathbb{Q}$).

Recall that Langlands for $GL_1$ over the rationals is the correspondence, given by the Artin reciprocity law, between on the one hand the abelianized absolute Galois group

$Gal(\overline{\mathbb{Q}}/\mathbb{Q})^{ab} = Gal(\mathbb{Q}(\mu_{\infty})/\mathbb{Q}) \simeq \hat{\mathbb{Z}}^*$

and on the other hand the connected components of the idele classes

$\mathbb{A}^{\ast}_{\mathbb{Q}}/\mathbb{Q}^{\ast} = \mathbb{R}^{\ast}_{+} \times \hat{\mathbb{Z}}^{\ast}$

The locally compact Abelian group of idele classes can be viewed as the nice locus of the horrible quotient space of adele classes $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}^{\ast}$. There is a well-defined map

$\mathbb{A}_{\mathbb{Q}}’/\mathbb{Q}^{\ast} \rightarrow \mathbb{R}_{+} \qquad (x_{\infty},x_2,x_3,\ldots) \mapsto | x_{\infty} | \prod | x_p |_p$

from the subset $\mathbb{A}_{\mathbb{Q}}’$ consisting of adeles of which almost all terms belong to $\mathbb{Z}_p^{\ast}$. The inverse image of this map over $\mathbb{R}_+^{\ast}$ are precisely the idele classes $\mathbb{A}^{\ast}_{\mathbb{Q}}/\mathbb{Q}^{\ast}$. In this way one can view the adele classes as a closure, or ‘compactification’, of the idele classes.

This is somewhat reminiscent of extending the nice action of the modular group on the upper-half plane to its badly behaved action on the boundary as in the Manin-Marcolli cave post.

The topological properties of the fiber over zero, and indeed of the total space of adele classes, are horrible in the sense that the discrete group $\mathbb{Q}^*$ acts ergodically on it, due to the irrationality of $log(p_1)/log(p_2)$ for primes $p_i$. All this is explained well (in the semi-local case, that is using $\mathbb{A}_Q’$ above) in the Connes-Marcolli book (section 2.7).

In much the same spirit as non-free actions of reductive groups on algebraic varieties are best handled using stacks, such ergodic actions are best handled by the tools of noncommutative geometry. That is, one tries to get at the geometry of $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}^{\ast}$ by studying an associated non-commutative algebra, the skew-ring extension of the group-ring of the adeles by the action of $\mathbb{Q}^*$ on it. This algebra is known to be Morita equivalent to the Bost-Connes algebra which is the algebra featuring in Connes’ approach to the Riemann hypothesis.

It shouldn’t thus come as a major surprise that one is able to recover the other side of the Langlands correspondence, that is the Galois group $Gal(\mathbb{Q}(\mu_{\infty})/\mathbb{Q})$, from the Bost-Connes algebra as the symmetries of certain states.

In a similar vein one can read the Connes-Marcolli $GL_2$-system (section 3.7 of their book) as an exploration of the noncommutative closure of the Langlands-space $GL_2(\mathbb{A}_{\mathbb{Q}})/GL_2(\mathbb{Q})$.

At the moment I’m running a master-seminar noncommutative geometry trying to explain this connection in detail. But, we’re still in the early phases, struggling with the topology of ideles and adeles, reciprocity laws, L-functions and the lot. Still, if someone is interested I might attempt to post some lecture notes here.

It all started with this comment on the noncommutative geometry blog by “gabriel” :

Even though my understanding of noncommutative geometry is limited, there are some aspects that I am able to follow.
I was wondering, since there are so few blogs here, why don’t you guys forge an alliance with neverending books, you blog about noncommutative geometry anyways. That way you have another(n-category cafe) blogspot and gives well informed views(well depending on how well defined a conversational-style blog can be).

The technology to set up a ‘conversational-style blog’, where anyone can either leave twitter-like messages or more substantial posts, is available thanks to the incredible people from Automattic.

For starters, they have the sensational p2 wordpress theme : “blogging at the speed of thought”

A group blog theme for short update messages, inspired by Twitter. Featuring: Hassle-free posting from the front page. Perfect for group blogging, or as a liveblog theme. Dynamic page updates. Threaded comment display on the front page. In-line editing for posts and comments. Live tag suggestion based on previously used tags. A show/hide feature for comments, to keep things tidy. Real-time notifications when a new comment or update is posted. Super-handy keyboard shortcuts.

Next, any lively online community is open for intense debate : “supercharge your community”

Fire up the debate with commenter profiles, reputation scores, and OpenID. With IntenseDebate you’ll tap into a whole new network of sites with avid bloggers and commenters. And that’s just the tip of the iceberg!

And finally, as we want to talk math, both in posts and comments, they provide us with the WP-LaTeX plugin.

All these ingredients make up the n-geometry cafe ((with apologies to the original cafe but I simply couldn’t resist…)) to be found at noncommutative.org (explaining the ‘n’).

Anyone can walk into a Cafe and have his/her say, that’s why you’ll get automatic author-privileges if you register.

I’m well aware of the obvious dangers of non-moderated sites, but also a strong believer in any Cafe’s self-regulating powers…

If you are interested in noncommutative geometry, and feel like sharing, please try it out.

Probably the smartest move I’ve made after entering math-school was to fall in love with a feminist.

Yeah well, perhaps I’ll expand a bit on this sentence another time. For now, suffice it to say that I did pick up a few words in the process, among them : the queen bee syndrome :

women who have attained senior positions do not use their power to assist struggling young women or to change the system, thereby tacitly validating it.

A recent study by the Max Planck Institute for Human Development asserts that the QBS

likely stems from women at the top who feel threatened by other women and therefore, prefer to surround themselves with men. As a result, these Queen Bees often jeapordize the promotions of other females at their companies.

Radical feminists of the late 70-ties preferred a different ‘explanation’, clearly.

Women who fought their way to the top, they said, were convinced that overcoming all obstacles along the way made them into the strong persons they became. A variant on the ‘what doesn’t kill you, makes you stronger’-mantra, quoi. These queen bees genuinely believed it to be beneficial to the next generation of young women not to offer them any shortcuts on their journey through the glass ceiling.

By and large, the 45+generation decides about the topics that should be (or shouldn’t be) on the current math-curriculum. They also write most of the text-books and course-notes used, and inevitably, the choices they make have an impact on the new generation of math-students.

Perhaps too little thought is given to the fact that the choices we (yes, I belong to that age group) make, the topics we deem important for new students to master, are heavily influenced by our own experiences.

In the late 60ties, early 70ties, Bourbaki-style mathematics influenced the ‘modern mathematics’ revolution in schools, certainly in Belgium through the influence of George Papy.

In kintergarten, kids learned the basics of set theory. Utensils to draw Venn diagrams were as indispensable as are pocket-calculators today. In secondary school, we had a formal axiomatic approach to geometry, we learned abstract topological spaces and other advanced topics.

Our 45+generation greatly benefitted from all of this when we started doing research. We felt comfortable with the (in retrospect, over)abstraction of the EGAs and SGAs and had little difficulties in using them or generalizing them to noncommutative levels…

Bourbakism made us into stronger mathematicians. Hence, we are convinced that new students should master it if they ever want to do ‘proper’ research.

Perhaps we pay too little attention to the fact that these new students are a lot worse prepared than we were in the old days. Every revolution inevitably provokes a counter-revolution. Secondary school mathematics sank over the last two decades to a debilitating level under the pretense of ‘usability’. Tim Gowers has an interesting Ivory tower post on this.

We may deplore this evolution, we may try to reverse it. But, until we succeed, it may not be fair to freshmen to continue stubbornly as if nothing changed since our good old days.

Perhaps, Bourbakism has become our very own queen bee syndrome…