
Supernatural numbers also appear in noncommutative geometry via James Glimm’s characterisation of a class of simple $C^*$algebras, the UHFalgebras. A uniformly hyperfine (or, UHF) algebra $A$ is a $C^*$algebra that can be written as the closure, in the norm topology, of an increasing union of finitedimensional full matrix algebras $M_{c_1}(\mathbb{C}) \subset M_{c_2}(\mathbb{C}) \subset … \quad… Read more »

F_un Mathematics Hardly a ‘new’ blog, but one that is getting a new life! On its old homepage you’ll find a diagonal banner stating ‘This site has moved’ and clicking on it will guide you to its new location : cage.ugent.be/~kthas/Fun. From now on, this site will be hosted at the University of Ghent and… Read more »

In preparing for next year’s ‘seminar noncommutative geometry’ I’ve converted about 30 posts to LaTeX, centering loosely around the topics students have asked me to cover : noncommutative geometry, the absolute point (aka the field with one element), and their relation to the Riemann hypothesis. The idea being to edit these posts thoroughly, add much… Read more »

Previously, we have recalled comparisons between approaches to define a geometry over the absolute point and arthistorical movements, first those due to Yuri I. Manin, subsequently some extra ones due to Javier Lopez Pena and Oliver Lorscheid. In these comparisons, the art trend appears to have been chosen more to illustrate a key feature of… Read more »

Penrose tilings are aperiodic tilings of the plane, made from 2 sort of tiles : kites and darts. It is well known (see for example the standard textbook tilings and patterns section 10.5) that one can describe a Penrose tiling around a given point in the plane as an infinite sequence of 0’s and 1’s,… Read more »

The lecturers, topics and dates of the 6 minicourses in our ‘advanced master degree 2011 in noncommutative algebra and geometry’ are : February 2125 Vladimir Bavula (University of Sheffield) : Localization Theory of Rings and Modules March 711 HansJürgen Schneider (University of München) : Nichols Algebra and Root Systems April 1112 Bernhard Keller (Université Paris… Read more »

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 RSSaggregator with interesting mathematical blogs, is their graphical presentation of (nearly) all mathblogs ordered by type : group blogs, individual researchers, teachers and educators, journalistic writers,… Read more »

Guest post by Fred Van Oystaeyen. In my book “Virtual Topology and Functorial Geometry” (Taylor and Francis, 2009) I proposed a noncommutative version of spacetime ; 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… Read more »

In view or recents events & comments, some changes have been made or will be made shortly : categories : Sanitized the plethora of wordpresscategories to which posts belong. At the moment there are just 5 categories : ‘stories’ and ‘web’ (for all posts with low mathcontent) and three categories ‘level1’, ‘level2’ and ‘level3’, loosely… Read more »

noncommutative, web
Jason & David, the Ninja warriors of noncommutative geometry
Posted on by lievenlbSocialMention gives a rather accurate picture of the webbuzz on a specific topic. For this reason I check it irregularly to know what’s going on in noncommutative geometry, at least webwise. Yesterday, I noticed two new kids on the block : Jason and David. Their blogs have (so far ) 44 resp. 27 posts, this… Read more »

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… Read more »

No christmas or newyears family party without heated discussions. Often on quite silly topics. For example, which late 19thcentury 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,… Read more »

This is a belated response to a MathOverflow 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 ConnesMarcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as… Read more »

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… Read more »

Probably the smartest move I’ve made after entering mathschool 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… Read more »

Boy, do I feel stupid for having written close to 500 blogposts hoping (in vain) they might eventually converge into a book project… Gil Kalai is infinitely smarter. Get a fake gmail account, invent a fictitious character and start COMMENTING and provoking responses. That’s how “Gina” appeared on the scene, cut and pasted her comments… Read more »

Creating (or taking) an image and explaining how it depicts your mental picture of a noncommutative space is one thing. Ideally, the image should be strong enough so that other people familiar with it might have a reasonable guess what you attempt to depict. But, is there already enough concordance in our views of noncommutative… Read more »

I really like Matilde Marcolli’s idea to use some of Jackson Pollock’s paintings as metaphors for noncommutative spaces. In her talk she used this painting and refered to it (as did I in my post) as : Jackson Pollock “Untitled N.3”. Before someone writes a post ‘The Pollock noncommutative space hoax’ (similar to my own… Read more »

The general public expects pictures from geometers, even from noncommutative geometers. Hence, it is important for researchers in this topic to make an attempt to convey the mental picture they have of their favourite noncommutative space, … somehow. Two examples : This picture was created by Shahn Majid. It appears on his visions of noncommutative… Read more »

Tim Gowers’ dream of massively collaborative mathematics got me thinking…

In the series “Brave new geometries” we give an introduction to ‘strange’ but exciting new ideas. We start with Grothendieck’s schemerevolution, go on with Soule’s geometry over the field with one element, Mazur’s arithmetic topology, Grothendieck’s anabelian geometry, Connes’ noncommutative geometry etc.

We use Kontsevich’s idea of thin varieties to define complexified varieties over F\_un.

We propose to extend the ConnesConsani definition to noncommuntative F_un varieties.

Some links to posts on Soule’s algebraic geometry over the field with one element.

A quick recap of last time. We are trying to make sense of affine varieties over the elusive field with one element $\mathbb{F}_1 $, which by Grothendieck’s schemephilosophy should determine a functor $\mathbf{nano}(N)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto N(A) $ from finite Abelian groups to sets, typically giving pretty small sets $N(A) $. Using the… Read more »
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