Life on Gaussian primes

At the moment I’m re-reading Siobhan Roberts’ biography of John Horton Conway, Genius at play – the curious mind of John Horton Conway. In fact, I’m also re-reading Alexander Masters’ biography of Simon Norton, The genius in my basement – the biography of a happy man. [full_width_image] [/full_width_image] If you’re in for a suggestion, try… Read more »

Grothendieck’s gallery No. 154

Since mid May the Montpellier part of Grothendieck’s gribouillis are online and for everyone available at the Archives Grothendieck. The story is well-known. End of June 1990, Grothendieck phoned Jean Malgoire warning him to come asap if he wanted to safeguard the better part of G’s mathematical archive, for he was making a bonfire… A… Read more »

How to dismantle scheme theory?

In several of his talks on #IUTeich, Mochizuki argues that usual scheme theory over $\mathbb{Z}$ is not suited to tackle problems such as the ABC-conjecture. The idea appears to be that ABC involves both the additive and multiplicative nature of integers, making rings into ‘2-dimensional objects’ (and clearly we use both ‘dimensions’ in the theory… Read more »

Moonshine for everyone

Today, Samuel Dehority, Xavier Gonzalez, Neekon Vafa and Roger Van Peski arXived their paper Moonshine for all finite groups. Originally, Moonshine was thought to be connected to the Monster group. McKay and Thompson observed that the first coefficients of the normalized elliptic modular invariant \[ J(\tau) = q^{-1} + 196884 q + 21493760 q^2 +… Read more »

The geometry of football

Soon, we will be teaching computational geometry courses to football commentators. If a player is going to be substituted we’ll hear sentences like: “no surprise he’s being replaced, his Voronoi cell has been shrinking since the beginning of the second half!” David Sumpter, the author of Soccermatics: Mathematical Adventures in the Beautiful Game, wrote a… Read more »

The subway singularity

The Boston subway is a complex system, spreading out from a focus at Park Street. On March 3rd, the Boylston shuttle went into service, tying together the seven principal lines, on four different levels. A day later, train 86 went missing on the Cambridge-Dorchester line. The Harvard algebraist R. Tupelo suggested the train might have… Read more »

Forgetting can’t be that hard, can it?

Geometers will tell you there are two ways to introduce affine schemes. You can use structure sheaves. That is, compute all prime ideals of your ring and turn them into a space. Then, put a sheaf of rings on this space by localisation. You’ll get your ring back taking global sections. Or, you might try… Read more »

Stirring a cup of coffee

Please allow for a couple of end-of-semester bluesy ramblings. I just finished grading the final test of the last of five courses I lectured this semester. Most of them went, I believe, rather well. As always, it was fun to teach an introductory group theory course to second year physics students. Personally, I did enjoy… Read more »

Where are Grothendieck’s writings? (2)

A couple of days ago, there was yet another article by Philippe Douroux on Grothendieck’s Lasserre writings “Inestimables mathématiques, avez-vous donc un prix?” in the French newspaper Liberation. Not that there is much news to report. I’ve posted on this before: Grothendieck’s gribouillis, Grothendieck’s gribouillis (2), and more recently Where are Grothendieck’s writings? In that… Read more »

how much to spend on (cat)books?

My favourite tags on MathOverflow are big-lists, big-picture, soft-question, reference-request and the like. Often, answers to such tagged questions contain sound reading advice, style: “road-map to important result/theory X”. Two more K to go, so let’s spend some more money. [section_title text=”Category theory”] [full_width_image] [/full_width_image] One of the problems with my master course on algebraic… Read more »

• web

wp-latex’ sweet revenge : wp+MathJax-> ePub

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In the early days of math-blogging, one was happy to get LaTeXRender working. Some years later, the majority of math-blogs were using the, more user-friendly, wp-latex plugin to turn LaTeX-code into png-images. Today, everyone uses MathJax which works with modern CSS and web fonts instead of equation images, so equations scale with surrounding text at…

• absolute

master seminar ncg 2011

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Note to students following this year’s ‘seminar noncommutative geometry’ : the seminar starts friday september 30th at 13h in room G 0.16. However, if you have another good reason to be in Ghent on thursday september 22nd, consider attending the inaugural lecture of Koen Thas at 17h in auditorium Emmy Noether, campus De Sterre, Krijgslaan… Read more »

• stories

the Bourbaki code : offline

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If you’ve downloaded recently the little booklet containing the collection of my posts on the Bourbaki code, either in pdf- or epub-format, cherish it. I have taken all Bourbaki-code posts offline (that is, changed their visibility from ‘Public’ to ‘Private’). Here’s why. Though all speculations and the few ‘discoveries’ in these posts are entirely my… Read more »

• web

From WordPress to ePublishing (1)

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Perhaps, the tips and tricks I did receive to turn a selection of wordpress-posts into a proper ePub-file may be of use to others, so I will describe the procedure here in some detail. It makes a difference whether or not some of the posts contain TeX. This time, I’ll sketch the process for non-LaTeX… Read more »

• web

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There were some great comments by Peter before this post was taken offline. So, here they are, once again.

• stories

What’s Pippa got to do with the Bourbaki wedding?

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Last time we’ve seen that on June 3rd 1939, the very day of the Bourbaki wedding, Malraux’ movie ‘L’espoir’ had its first (private) viewing, and we mused whether Weil’s wedding card was a coded invitation to that event. But, there’s another plausible explanation why the Bourbaki wedding might have been scheduled for June 3rd :… Read more »

• absolute, web

eBook ‘geometry and the absolute point’ v0.1

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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 »

• stories

the birthday of the primes=knots analogy

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Last time we discovered that the mental picture to view prime numbers as knots in $S^3$ was first dreamed up by David Mumford. Today, we’ll focus on where and when this happened. 3. When did Mazur write his unpublished preprint? According to his own website, Barry Mazur did write the paper Remarks on the Alexander… Read more »

• stories

Brigitte Bardot, miniskirts and homological algebra

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The papers by Liliane Beaulieu on the history of the Bourbaki-group are genuine treasure troves of good stories. Though I’m mostly interested in the pre-war period, some tidbits are just too good not to use somewhere, sometime, such as here on a lazy friday afternoon … In her paper Bourbaki’s art of memory she briefly… Read more »

• stories

Art and the absolute point (3)

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Previously, we have recalled comparisons between approaches to define a geometry over the absolute point and art-historical 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 »

• noncommutative

Penrose tilings and noncommutative geometry

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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 »

• stories

If Bourbaki=WikiLeaks then Weil=Assange

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In an interview with readers of the Guardian, December 3rd 2010, Julian Assange made a somewhat surprising comparison between WikiLeaks and Bourbaki, sorry, The Bourbaki (sic) : “I originally tried hard for the organisation to have no face, because I wanted egos to play no part in our activities. This followed the tradition of the… Read more »

• stories

Who dreamed up the primes=knots analogy?

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One of the more surprising analogies around is that prime numbers can be viewed as knots in the 3-sphere $S^3$. The motivation behind it is that the (etale) fundamental group of $\pmb{spec}(\mathbb{Z}/(p))$ is equal to (the completion) of the fundamental group of a circle $S^1$ and that the embedding $\pmb{spec}(\mathbb{Z}/(p)) \subset \pmb{spec}(\mathbb{Z})$ embeds this circle… Read more »

• stories

What happened on the Bourbaki wedding day?

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Early on in this series we deciphered part of the Bourbaki wedding invitation The wedding was planned on “le 3 Cartembre, an VI” or, for non-Bourbakistas, June 3rd 1939. But, why did they choose that particular day? Because the wedding-invitation-joke was concocted sometime between mid april and mid may 1939, the most probable explanation clearly… Read more »

• rants, stories

what have quivers done to students?

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A few years ago a student entered my office asking suggestions for his master thesis. “I’m open to any topic as long as it has nothing to do with those silly quivers!” At that time not the best of opening-lines to address me and, inevitably, the most disastrous teacher-student-conversation-ever followed (also on my part, i’m… Read more »

• absolute, stories

Art and the absolute point (2)

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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… Read more »

• absolute, stories

Art and the absolute point

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In his paper Cyclotomy and analytic geometry over $\mathbb{F}_1$ Yuri I. Manin sketches and compares four approaches to the definition of a geometry over $\mathbb{F}_1$, the elusive field with one element. He writes : “Preparing a colloquium talk in Paris, I have succumbed to the temptation to associate them with some dominant trends in the… Read more »

• noncommutative, web

Noncommutative algebra and geometry master-degree

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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… Read more »

• web

mathblogging and poll-results

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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,… Read more »

• noncommutative

On the Reality of Noncommutative Space

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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… Read more »

• web

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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… Read more »

• stories

the Reddit (after)effect

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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… Read more »

• games

How to win transfinite Nimbers?

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Last time we introduced the game of transfinite Nimbers and asked a winning move for the transfinite game with stones a at position $~(2,2)$, b at $~(4,\omega)$, c at $~(\omega+2,\omega+3)$ and d at position $~(\omega+4,\omega+1)$. Above is the unique winning move : we remove stone d and by the rectangle-rule add… Read more »

• games

n-dimensional and transfinite Nimbers

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Today, we will expand the game of Nimbers to higher dimensions and do some transfinite Nimber hacking. In our identification between $\mathbb{F}_{16}^*$ and 15-th roots of unity, the number 8 corresponds to $\mu^6$, whence $\sqrt{8}=\mu^3=14$. So, if we add a stone at the diagonal position (14,14) to the Nimbers-position of last time… Read more »

• games

How to play Nimbers?

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Nimbers is a 2-person game, winnable only if you understand the arithmetic of the finite fields $\mathbb{F}_{2^{2^n}}$ associated to Fermat 2-powers. It is played on a rectangular array (say a portion of a Go-board, for practical purposes) having a finite number of stones at distinct intersections. Here’s a typical position The players alternate making… Read more »