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 object over something ‘deeper’, the… Read more →

# Je (ne) suis (pas) Mochizuki

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 : – in august 2012, Shinichi Mochizuki… Read more →

# Map of the Parisian mathematical scene 1933-39

. 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 Mmes Dubreil et Chevalley”. A… Read more →

# Children have always loved colimits

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 sheaves, cohomology, and worse. Sooner… Read more →

# Can one explain schemes to hipsters?

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 and Tate set themselves an… Read more →

# Grothendieck’s gribouillis

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 What will happen to Grothendieck’s unpublished notes, or as he preferred, his “gribouillis” (scribbles)?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 of his personal notes in… Read more →

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

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 topology on the roots of… Read more →

# On categories, go and the book $\in$

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 as its title a single… Read more →

# A noncommutative moduli space

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 $M_{c_1}(\mathbb{C}) \subset M_{c_2}(\mathbb{C}) \subset … \quad \subset A$ Such embedding are… Read more →

# Oulipo’s use of the Tohoku paper

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 paper in Kansas, Grothendieck did… Read more →

# the birthday of Grothendieck topologies

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 Zariski topology, needed a drastic… Read more →

# Quiver Grassmannians can be anything

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 decomposition. A quiver $Q$ is… Read more →

# le petit village de l’Ariège

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 ‘Fred le Belge, notre indic… Read more →

# G-spots : Saint-Girons

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. All of this is recorded… Read more →

# G-spots : un petit village de l’Ariège

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 not all) of the GFC… Read more →