
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 »

Sunday january 2nd around 18hr NeBstats 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 Redditmention is to a blog what doping is to a… Read more »

Exactly 7 years ago I wrote my first post. This blog wasn’t called NeB yet and it used pMachine, a then free blogging tool (later transformed into expression engine), rather than WordPress. Over the years NeB survived three hardwareupgrades of ‘the Matrix’ (the webserver hosting it), more themes than I care to remember, and a… 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 »

Next time you visit your mathlibrary, please have a look whether these books are still on the shelves : Michiel Hazewinkel‘s Formal groups and applications, William Fulton’s and Serge Lange’s RiemannRoch algebra and Donald Knutson’s lambdarings and the representation theory of the symmetric group. I wouldn’t be surprised if one or more of these books… Read more »

We have seen that Conway’s big picture helps us to determine all arithmetic subgroups of $PSL_2(\mathbb{R}) $ commensurable with the modular group $PSL_2(\mathbb{Z}) $, including all groups of monstrous moonshine. As there are exactly 171 such moonshine groups, they are determined by a finite subgraph of Conway’s picture and we call the minimal such subgraph… Read more »

A comment to Charles Siegel’s ‘big theorems’series got me checking my stats.

Some quotes of Andre Weil on the Riemann hypothesis.

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

The Monster is the largest of the 26 sporadic simple groups and has order 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 = 2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71. It is not so much the size… Read more »

Next week, our annual summer school Geometric and Algebraic Methods with Applications in Physics will start, once again (ive lost count which edition it is). Because Isar is awol to la douce France, I’ll be responsible (once again) for the webrelated stuff of the meeting. So, here a couple of requests to participants/lecturers : if… Read more »

We are after the geometric trinity corresponding to the trinity of exceptional Galois groups The surfaces on the right have the corresponding group on the left as their group of automorphisms. But, there is a lot more grouptheoretic info hidden in the geometry. Before we sketch the $L_2(11) $ case, let us recall the simpler… Read more »

We saw that the icosahedron can be constructed from the alternating group $A_5 $ by considering the elements of a conjugacy class of order 5 elements as the vertices and edges between two vertices if their product is still in the conjugacy class. This description is so nice that one would like to have a… Read more »

The buckyball is without doubt the hottest mahematical object at the moment (at least in Europe). Recall that the buckyball (middle) is a mixed form of two Platonic solids the Icosahedron on the left and the Dodecahedron on the right. For those of you who don’t know anything about football, it is that other ballgame,… Read more »

There are only a handful of human activities where one goes to extraordinary lengths to keep a dream alive, in spite of overwhelming evidence : religion, theoretical physics, supporting the Belgian football team and … mathematics. In recent years several people spend a lot of energy looking for properties of an elusive object : the… Read more »

I’ve always thought of Alain Connes as the unchallengeable worldchampion opaque mathematical writing, but then again, I was proven wrong. Alain’s writings are crystal clear compared to the monstrosity the AMS released to the world : In search of the Riemann zeros – Strings, fractal membranes and noncommutative spacetimes by Michel L. Lapidus. Here’s a… Read more »

Time to wrap up this series on the BostConnes algebra. Here’s what we have learned so far : the convolution product on double cosets $\begin{bmatrix} 1 & \mathbb{Z} \\ 0 & 1 \end{bmatrix} \backslash \begin{bmatrix} 1 & \mathbb{Q} \\ 0 & \mathbb{Q}_{> 0} \end{bmatrix} / \begin{bmatrix} 1 & \mathbb{Z} \\ 0 & 1 \end{bmatrix} $… Read more »

This semester, I’m running a 3rd year course on Marcus du Sautoy’s The music of the primes. The concept being that students may suggest topics, merely sketched in the book, and then we’ll go a little deeper into them. I’ve been rather critical about the book before, but, rereading it last week (and knowing a… Read more »

Vacation is always a good time to catch up on some reading. Besides, there’s very little else you can do at night in a skiresort… This year, I’ve taken along The Archimedes Codex: Revealing The Secrets Of The World’s Greatest Palimpsest by Reviel Netz and William Noel telling the story of the Archimedes Palimpsest. The… Read more »

(via the Arcadian Functor) At the time of the doing the Perelmanpost someone rightfully commented that “making a voluntary retreat from the math circuit to preserve one’s own wellbeing (either mental, physical, scientific …)” should rather be called doing the Grothendieck as he was the first to pull this stunt. On Facebook a couple of… Read more »

In the series “Noncommutative geometry and the Riemann zeta function” we give an introduction to the BostConnes algebra. We describe its relation to adeles/ideles and to KMSstates leading to the zetafunction as the partition function.

You may not have noticed, but the really hard work was done behind the scenes, resurrecting about 300 old posts (some of them hidden by giving them ‘private’status). Ive only deleted about 10 posts with little or no content and am sorry I’ve selfdestructed about 2030 hectic posts over the years by pressing the ‘delete… Read more »

Via the ncategory cafe (and just now also the Arcadian functor ) I learned that Benjamin Mann of DARPA has constructed a list of 23 challenges for mathematics for this century. DARPA is the “Defense Advanced Research Projects Agency” and is an agency of the United States Department of Defense ‘responsible for the development of… Read more »

In a couple of days I’ll be blogging for 4 years… and I’m in the process of resurrecting about 300 posts from a databasedump made in june. For example here’s my first post ever which is rather naive. This conversion program may last for a couple of weeks and I apologize for all unwanted pingbacks… Read more »
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