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 »

  • games, number theory

    Life on Gaussian primes

    Posted on by

    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 »

  • stories, web

    Grothendieck’s gallery No. 154

    Posted on by

    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 »

  • absolute, geometry, number theory

    How to dismantle scheme theory?

    Posted on by

    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 »

  • groups, number theory, representations

    Moonshine for everyone

    Posted on by

    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 »

  • games, geometry

    The geometry of football

    Posted on by

    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 »

  • geometry, stories

    The subway singularity

    Posted on by

    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 »

  • geometry

    Forgetting can’t be that hard, can it?

    Posted on by

    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 »

  • geometry, personal, rants, representations

    Stirring a cup of coffee

    Posted on by

    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 »

  • geometry, stories

    Where are Grothendieck’s writings? (2)

    Posted on by

    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 »

  • books, geometry

    how much to spend on (cat)books?

    Posted on by

    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 »

  • books, geometry, noncommutative, representations

    let’s spend 3K on (math)books

    Posted on by

    Santa gave me 3000 Euros to spend on books. One downside: I have to give him my wish-list before monday. So, I’d better get started. Clearly, any further suggestions you might have will be much appreciated, either in the comments below or more directly via email. Today I’ll focus on my own interests: algebraic geometry,… Read more »

  • books, games

    human-, computer- and fairy-chess

    Posted on by

    It was fun following the second game last night in real time. Carlsen got a winning endgame with two bishops against a rook, but blundered with 62. Bg4?? (winning was Kf7), resulting in stalemate. There was this hilarious message around move 60: “The computer has just announced that white mates in 31 moves. Of course,… Read more »

  • books, geometry, personal, stories

    NaNoWriMo (3)

    Posted on by

    In 2001, Eugenia Cheng gave an interesting after-dinner talk Mathematics and Lego: the untold story. In it she compared math research to fooling around with lego. A quote: “Lego: the universal toy. Enjoyed by people of all ages all over the place. The idea is simple and brilliant. Start with some basic blocks that can… Read more »

  • books, web

    Ulysses and LaTeX

    Posted on by

    If you’re a mathematician chances are that your text-editor of choice will be TeXShop, the perfect environment for writing papers. Even when writing a massive textbook, most of us stick to this or a similar LaTeX-frontend. The order of chapters in such a book is usually self-evident, and it is enough to use one TeX-file… Read more »

  • books, geometry, personal, stories

    NaNoWriMo (2)

    Posted on by

    Two more days to go in the NaNoWriMo 2016 challenge. Alas, it was clear from the outset that I would fail, bad. I didn’t have a sound battle plan. Hell, I didn’t even have a a clue which book to write… But then, I may treat myself to a SloWriMo over the Christmas break. For… Read more »

  • geometry, math

    Grothendieck topologies as functors to Top

    Posted on by

    Either this is horribly wrong, or it must be well-known. So I guess I’m asking for either a rebuttal or a reference. Take a ‘smallish’ category $\mathbf{C}$. By this I mean that for every object $C$ the collection of all maps ending in $C$ must be a set. On this set, let’s call it $y(C)$… Read more »

  • geometry, stories

    le lemme de la Gare du Nord

    Posted on by

    Theorems have the tendency to pop into existence when you least expect them: taking a bath, during your sleep, dreaming away during a dull lecture, waiting for an airplane, bicycling, whatever. One of the most famous (and useful) lemmas was dreamed up in the Parisian Gare du Nord station, during a conversation between Saunders Mac… Read more »

  • geometry

    Car crashes in scheme theory

    Posted on by

    What do you get when two cars crash head on at full speed? A heap of twisted metal. What do you get when two tiny cars crash head on at full speed? A smaller heap of twisted metal. In the limit, what do you get when two point cars crash head on at full speed?… Read more »

  • books, france, math, personal, stories

    NaNoWriMo (1)

    Posted on by

    Some weeks ago I did register to be a participant of NaNoWriMo 2016. It’s a belated new-year’s resolution. When PS (pseudonymous sister), always eager to fill a 10 second silence at family dinners, asked (PS) And Lieven, what are your resolutions for 2016? she didn’t really expect an answer (for decades my generic reply has… Read more »

  • stories

    Where’s Bourbaki’s tomb?

    Posted on by

    In according to Groth IV.22 we tried to solve one of the riddles contained in Roubaud’s announcement of Bourbaki’s death. Today, we’ll try our hands on the next one: where was Bourbaki buried? The death announcement gives this fairly opaque clue: “The burial will take place in the cemetery for Random Functions (metro stations Markov… Read more »

  • math

    from chocolate bars to constructivism

    Posted on by

    A fun way to teach first year students the different methods of proof is to play a game with chocolate bars, Chomp. The players take turns to choose one chocolate block and “eat it”, together with all other blocks that are below it and to its right. There is a catch: the top left block… Read more »

  • personal

    Gitte exhibits in Ghent

    Posted on by

    Gitte Le Bruyn, the artist previously known as PD1 on this blog, exhibits some of her work in Ghent. The exhibition VGC Visual Art Gitte Le Bruyn is hosted by the Van Crombrugghe’s Genootschap, Huidevetterskaai 39, 9000 Gent, Belgium. Gitte shows hundreds of stills from her animation projects. She has a laborious working method: for… Read more »

  • stories

    Hasse = “le P. Adique, de l’Ordre des Diophantiens”

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    The Bourbaki wedding invitation is probably the most effective branding- and marketing-campaign in the history of mathematics. It contains this, seemingly opaque, paragraph: [quote name=”Bourbaki wedding card”]The trivial isomorphism will be given to them by P. Adic, of the Diophantine Order, at the Principal Cohomology of the Universal Variety, the 3 Cartember, year VI, at… Read more »

  • geometry

    Toposes alive and kicking at IHES

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    After 50 years, vivid interest in topos theory seems to have returned to one of the most prestigious research institutes, the IHES. Last november, there was the meeting Topos a l’IHES. At the meeting, Celine Loozen filmed a documentary which is supposed to have as its title “Unifying Worlds”. Its very classy trailer is now… Read more »

  • absolute, geometry, number theory

    The group algebra of all algebraic numbers

    Posted on by

    Some weeks ago, Robert Kucharczyk and Peter Scholze found a topological realisation of the ‘hopeless’ part of the absolute Galois group $\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. That is, they constructed a compact connected space $M_{cyc}$ such that etale covers of it correspond to Galois extensions of the cyclotomic field $\mathbb{Q}_{cyc}$. This gives, at least in theory, a handle on… Read more »