Aaron Siegel has now dramatically extended our previous list and calculated the $\alpha_p$ for all primes $p \leq 181$. He mails :

The sad story of Alice Silverberg‘s cat when she came too close to tori-cryptography…

After more than 30 years we make a small addition to Hendrik Lenstra’s list of Conway’s mystery elements $\alpha_p$:

Of course, they’re not. It’s an Arjen Lenstra joke on the abbreviation of the tori key-compression system ECSTR.

John Conway defined the simplest possible addition and multiplication on the class of all ordinal numbers, turning it magically into an algebraically closed field…

What has Hilbert’s Satz 90 to do with finding more efficient ways to exchange keys?

Alain Connes asked for a concrete realisation of the algebraic closure $\overline{\mathbb{F}_2}$ of the finite field on two elements. Well, here it is:

According to Wikipedia the Russell paradox was discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen.

The fractal nature of the devil’s staircase has a large group of (self)symmetries.

There’s this classic Hardy story on his visit to Ramanujan in a taxicab numbered 1729:

Did Diffie and Hellman discover the key-exchange principle, or was this one of the hidden secrets of GCHQ?

Who was the first mathematician to give a slide show talk? I don’t have the definite answer to this question, but would like to offer a strong candidate : Hermann Minkowski.

Note to self: check Jack Morava’s arXiv notes on a more regular basis!

Four months ago, Leila Schneps started a crowd-funding project to translate part 3 of Scharlau’s biography of Grothendieck. So far, she raised 3350 of the required 6000 dollars from 45 donations.

In two months I’ll be teaching ‘Logic and Set Theory’ for the first time, a first year, first semester course on Foundations. And no (though I considered it for a nanosecond) I will not trow HoTT at them…

At times it is far more rewarding to enter into an exchange on G+ than to try to write yet another blog post here…

Supernatural numbers also appear in noncommutative geometry via James Glimm’s characterisation of a class of simple $C^*$-algebras, the UHF-algebras.

Below, a great G+-post by Allen Knutson, pointing to a talk given by Voevodsky in which he explains why some errors (by himself and others) convinced him that mathematics needed a new foundation.

Bourbaki’s death announcement mentions ‘Respectively their father, brother, son, grandson, great-grandson, and grand-cousin’ giving us 5 generations, among which 4 generations of Bourbakis.

Here’s the punchline : a large chunk of the Connes-Marcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as an exploration of the noncommutative boundary to the Langlands program (at least for $GL_1 $ and $GL_2 $ over the rationals $\mathbb{Q} $).

This time we will describe the points of the arithmetic site with Steinitz’ supernatural numbers and adele-classes.

In Bourbaki est mort, CQFD, Pierre Cartier asserts: “Yes, Bourbaki is dead. He was killed by May 68!”

The Leech lattice was, according to wikipedia, ‘originally discovered by Ernst Witt in 1940, but he did not publish his discovery’ and it ‘was later re-discovered in 1965 by John Leech’. However, there is very little evidence to support this claim.

Now that Alain Connes’ talk at the IHES is online, giving hints to prove some of the statements of the arithmetic site, we can continue our story.

From left to right: Jacques Dixmier, Jean Dieudonné, Pierre Samuel, André Weil, Jean Delsarte and Laurent Schwartz on some terrace during the 1951 ‘Ecumenical’ Bourbaki summer-conference in Pelvoux. Where, exactly, was this?

Last week, the IHES shared Pierre Cartier “Les mathématiques de Grothendieck (un survol)” (talk in English though). I don’t know whether this was part of the evening activities of the conference for Maxim Kontsevich 50th birthday (one Groth-activity/conference is quickly becoming the standard…).

For the better part of the 30ties, Ernst Witt (1) did hang out with the rest of the ‘Noetherknaben’, the group of young mathematicians around Emmy Noether (3) in Gottingen.

Roubaud’s motif (pardon the expression) for writing the announcement of Bourbaki’s death in 1968 can be read between the lines in his book Mathematics, a novel from which all quotes below are taken.

The announcement of Bourbaki’s death ends with the remarkable line:

“For God is the Alexandrov compactification of the universe.” Groth. IV.22

What has this to do with Simone Weil attending Bourbaki congresses?

“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)

Perhaps the fact that Bourbaki died on November 11th, 1968 (exactly 50 years after the end of WW1) is an allusion to the mandatory retirement age for members of Bourbaki, as suggested by the Canulars Bourbaki.

Even though the Bourbaki group dissolved itself in the late 90ties, the (premature) death of N. Bourbaki was already announced in november 1968…

“Sammy Eilenberg was suggesting (as a joke) that it should be called “entonnoir” : each step (“page” as one says now) is smaller than the preceding one, and what comes out at the end is delicious.” (Serre on AlgTop)

At the 1951 Pelvoux-congress a schism between the mountaineers and couch potatoes threatened Bourbaki.

“I sat in the seminaire Cartan, as a stupified witness to his discussions with Serre, loaded with ‘Spectral Sequences’ (brr!) and drawings (called ‘diagrams’) full of arrows covering the blackboard. It was the heroic age of the theory of ‘sheaves’ and ‘carapaces’ and of a whole arsenal whose sense entirely escaped me.” (Grothendieck, ReS, p.19)

“The adjective *tropical* was coined by French mathematicians in honor of their Brazilian colleague Imre Simon. There is no deeper meaning in the adjective *tropical*. It simply stands for the French view of Brazil.”
(From Speyer-Sturmfels)

A very nice talk by Maxim Kontsevich, intended for a general audience, on formal languages, free groups and algebraic power series.

A movie by Catherine Aira and Yves Le Pestipon “Alexande Grothendieck, sur les routes d’un genie” is touring mathematical departments and conferences in the south(-west) of France.

This is the story of the day our notion of ‘neighbourhood’ changed forever (at least in the geometric sense).

Jacques Herbrand was considered one of the greatest younger logicians and number-theorists when he fell to his death on july 27th 1931, only 23 years old, while mountain-climbing in the Massif des Ecrins.

Note to fellow library users: I’ve taken out SGA4, Mac Lane-Moerdijk and the Elephants. There’s an arithmetic site I need to check out.

After more than 70 years, credit is finally given to a fine, inspiring and courageous Basque algebraic geometer.

An (at times) hilarious interview on the dangers of category theory and Alain’s conversion to topoi. Here, an attempt to provide subtitles for 2 minutes (50.36-52.27):

Welcome to my new blog – **a mad day’s work**.