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 »

  • stories

    according to Groth. IV.22

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    At the Bourbaki Seminar in November 1968 the participants were handed the following (premature) announcement of Bourbaki’s death. The French text can be found at the Canulars Bourbaki, and the English translation below is from Maurice Mashaal’s book Bourbaki, a secret society of mathematicians, page 115. I’ve underlined a couple of riddles in the text…. Read more »

  • stories

    Did Nöbeling discover toposes?

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    Chasing one story, one sometimes tumbles into a different one. For some time I’m trying to debunk the story that Wolfgang Krull was close to inventing the notion of schemes in the early 1930’s. I guess my first encounter with it was through The Rising Sea: Grothendieck on simplicity and generality I by Colin McLarty… Read more »

  • stories

    Where are Grothendieck’s writings?

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    You better subscribe to the French newspaper Liberation if you’re interested in the latest whereabouts of Grothendieck’s ‘gribouillis’. And even then it is hard to turn this info into a consistent tale. A futile attempt… [section_title text=”In the Bibliotheque Nationale de France?”] A year ago it all seemed pretty straightforward. Georges Maltsiniotis gave a talk… Read more »

  • absolute, geometry, number theory

    Topology and the symmetries of roots

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    We know embarrassingly little about the symmetries of the roots of all polynomials with rational coefficients, or if you prefer, the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. In the title picture the roots of polynomials of degree $\leq 4$ with small coefficients are plotted and coloured by degree: blue=4, cyan=3, red=2, green=1. Sums and products of roots… Read more »

  • france, geometry, math, stories

    Grothendieck’s gribouillis (2)

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    We left the story of Grothendieck’s Lasserre notes early 2015, uncertain whether they would ever be made public. Some things have happened since. Georges Maltsiniotis gave a talk at the Gothendieck conference in Montpellier in june 2015 having as title “Grothendieck’s manuscripts in Lasserre”, raising perhaps even more questions. Philippe Douroux, a journalist at the… Read more »

  • stories

    artisanal integers

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    Summer of 2012. Suddenly several “integer-as-a-service-providers” spring from nowhere. They deliver “artisanal integers”. Integers which (they claim) are “hand-crafted and guaranteed to be unique and hella-beautiful”. Are you still with me? Perhaps it helps to have a look at one of the three such services still operational today: Brooklyn Integers, Mission Integers (after San Francisco’s… Read more »

  • absolute, geometry, number theory, stories

    The Log Lady and the Frobenioid of $\mathbb{Z}$

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    “Sometimes ideas, like men, jump up and say ‘hello’. They introduce themselves, these ideas, with words. Are they words? These ideas speak so strangely.” “All that we see in this world is based on someone’s ideas. Some ideas are destructive, some are constructive. Some ideas can arrive in the form of a dream. I can… Read more »

  • absolute, geometry

    Two lecture series on absolute geometry

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    Absolute geometry is the attempt to develop algebraic geometry over the elusive field with one element $\mathbb{F}_1$. The idea being that the set of all prime numbers is just too large for $\mathbf{Spec}(\mathbb{Z})$ to be a terminal object (as it is in the category of schemes). So, one wants to view $\mathbf{Spec}(\mathbb{Z})$ as a geometric… Read more »

  • number theory, rants

    Je (ne) suis (pas) Mochizuki

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    Apologies to Joachim Roncin, the guy who invented the slogan “Je suis Charlie”, for this silly abuse of his logo: I had hoped the G+ post below of end december would have been the last I had to say on this (non)issue: (btw. embedded G+-post below, not visible in feeds) A quick recap : –… Read more »

  • france, math, stories

    Map of the Parisian mathematical scene 1933-39

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    . Michele Audin has written a book on the history of the Julia seminar (hat tip +Chandan Dalawat via Google+). The “Julia Seminar” was organised between 1933 and 1939, on monday afternoons, in the Darboux lecture hall of the Institut Henri Poincare. After good German tradition, the talks were followed by tea, “aimablement servi par… Read more »

  • geometry, stories

    Children have always loved colimits

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    If Chad Orzel is able to teach quantum theory to his dog, surely it must be possible to explain schemes, stacks, toposes and motives to hipsters? Perhaps an idea for a series of posts? It’s early days yet. So far, I’ve only added the tag sga4hipsters (pun intended) and googled around for ‘real-life’ applications of… Read more »

  • geometry, stories

    Can one explain schemes to hipsters?

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    Nature (the journal) asked David Mumford and John Tate (of Fields and Abel fame) to write an obituary for Alexander Grothendieck. Probably, it was their first experience ever to get a paper… rejected! What was their plan? How did they carry it out? What went wrong? And, can we learn from this? the plan Mumford… Read more »

  • geometry, stories, web

    Grothendieck’s gribouillis

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    A math-story well worth following in 2015. What will happen to Grothendieck’s unpublished notes, or as he preferred, his “gribouillis” (scribbles)? Here’s the little I know about this: 1. The Mormoiron scribbles During the 80ties Grothendieck lived in ‘Les Aumettes’ in Mormoiron In 1991, just before he moved to the Pyrenees he burned almost all… Read more »

  • math, number theory

    $\mathbf{Ext}(\mathbb{Q},\mathbb{Z})$ and the solenoid $\widehat{\mathbb{Q}}$

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    Note to self: check Jack Morava’s arXiv notes on a more regular basis! It started with the G+-post below by +David Roberts: Suddenly I realised I hadn’t checked out Morava‘s “short preprints with ambitious ideas, but no proofs” lately. A couple of years ago I had a brief email exchange with him on the Habiro… Read more »

  • featured, stories

    On categories, go and the book $\in$

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    A nice interview with Jacques Roubaud (the guy responsible for Bourbaki’s death announcement) in the courtyard of the ENS. He talks about go, categories, the composition of his book $\in$ and, of course, Grothendieck and Bourbaki. Clearly there are pop-math books like dedicated to $\pi$ or $e$, but I don’t know just one novel having… Read more »

  • math, noncommutative

    A noncommutative moduli space

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    Supernatural numbers also appear in noncommutative geometry via James Glimm’s characterisation of a class of simple $C^*$-algebras, the UHF-algebras. 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 finite-dimensional full matrix algebras $M_{c_1}(\mathbb{C}) \subset M_{c_2}(\mathbb{C}) \subset … \quad… Read more »

  • books, stories

    Oulipo’s use of the Tohoku paper

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    Many identify the ‘Tohoku Mathematical Journal’ with just one paper published in it, affectionately called the Tohoku paper: “Sur quelques points d’algèbre homologique” by Alexander Grothendieck. In this paper, Grothendieck reshaped homological algebra for Abelian categories, extending the setting of Cartan-Eilenberg (their book and the paper both appeared in 1957). While working on the Tohoku… Read more »

  • books, france

    Where is Fogas?

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    A reading suggestion for Grothendieck-stalkers crawling around the Ariège region, near Saint-Girons, in search of ‘another house’ : better bring along the Fogas Chronicles by Julia Stagg.

  • geometry, stories

    Grothendieck’s Café

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    “A story says that in a Paris café around 1955 Grothendieck asked his friends “what is a scheme?”. At the time only an undefined idea of “schéma” was current in Paris, meaning more or less whatever would improve on Weil’s foundations.” (McLarty in The Rising Sea) Finding that particular café in Paris, presumably in the… Read more »

  • geometry, stories

    the birthday of Grothendieck topologies

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    This is the story of the day the notion of ‘neighbourhood’ changed forever (at least in the geometric sense). For ages a neighbourhood of a point was understood to be an open set of the topology containing that point. But on that day, it was demonstrated that the topology of choice of algebraic geometry, the… Read more »

  • stories

    European mathematics in 1927

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    Here’s a map of the (major) mathematical centers in Europe (in 1927), made for the Rockefeller Foundation. Support by the Rockefeller foundation was important for European Mathematics between the two world wars. They supported the erection of the Mathematical Institute in Goettingen between 1926-1929 and creation of the Institut Henri Poincare in Paris at about… Read more »

  • math, noncommutative, representations

    Quiver Grassmannians can be anything

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    A standard Grassmannian $Gr(m,V)$ is the manifold having as its points all possible $m$-dimensional subspaces of a given vectorspace $V$. As an example, $Gr(1,V)$ is the set of lines through the origin in $V$ and therefore is the projective space $\mathbb{P}(V)$. Grassmannians are among the nicest projective varieties, they are smooth and allow a cell… Read more »

  • stories

    le petit village de l’Ariège

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    For me this quest is over. All i did was following breadcrumbs left by others. Fellow-travelers arrived there before. What did they do next? The people from the esoteric site L’Astrée, write literary texts on Grothendieck, mixing strange details (such as the kiosque de la place Pinel, the village of Fougax-et-Barrineuf and even ‘Winnie’ or… Read more »

  • stories

    G-spots : Saint-Girons

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    Roy Lisker (remember him from the Mormoiron post?) has written up his Grothendieck-quest(s), available for just 23$, and with this strange blurb-text: “The author organized a committee to search for him that led to his discovery, in good health and busily at work, in September, 1996. This committee has since become the Grothendieck Biography Project…. Read more »

  • stories

    G-spots : un petit village de l’Ariège

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    We would love to conclude this series by finding the location of the “final” Grothendieck-spot, before his 85th birthday, this thursday. But, the road ahead will be treacherous, with imaginary villages along the way and some other traps planted by the nice people of the Grothendieck Fan Club It is well-known that some members (if… Read more »