There are many songs in the world about love and loss, heartbreak and heart-ache. There are altogether fewer about algebraic geometry in the style of Alexander Grothendieck. Here is my attempt to fill that gap:
Disclaimer: I hope none of this needs saying, but just in case of misinterpretation:
1. No views expressed therein are attributable to any organisation to which I am affiliated.
2. Most of the views expressed therein are not attributable to me either, but are a deliberate exaggeration, for comic effect, of a common initial reaction to one’s first meeting with Grothendieck-style algebraic geometry.
3. It is not intended as a serious critique of any mathematician or school of mathematics!
When I was a young boy doing maths in class
I thought I knew it all.
Every test that I took, I was sure to pass.
I felt pride, and there never came a fall.
Up at university, I found what life is for:
A world of mathematics, and all mine to explore.
Learning geometry and logic, I was having a ball.
Until I hit a wall…
For I adore Euler and Erdős,
Élie Cartan and Ramanujan
Newton and Noether. But not to sound churlish
There’s one man I cannot understand.
No, I can’t get to grips with Grothendieck,
My palms feel sweaty and my knees go weak.
I’m terrified that never will I master the technique
Of Les Éments de Géométrie Algébrique.
He’s a thoughtful and a thorough theory-builder, sans pareil.
But can anybody help me find the secret, s’il vous plaît
Of this awe-inspiring generality and abstraction?
I have to say it’s driving me to total distraction.
For instance… A Euclidean point is a location in space, and that we can all comprehend.
René Descartes added coordinates for the power and the rigour they lend.
Later came Zariski topology, where a point’s a type of algebraic set
Of dimension nought. Well, that’s not what I thought. But it’s ok. There’s hope for me yet!
But now and contra all prior belief
We hear a point’s a prime ideal
In a locally ringed space, overlaid with a sheaf.
Professor G, is truly this for real?
No, I can’t make head nor tail of Grothendieck
Or Deligne, or Serre, or any of that clique.
I’ll have to learn not to care whenever people speak
Of Les Fondements de la Géométrie Algébrique.
But don’t take me for a geometrical fool.
I can do much more than merely prove the cosine rule.
I’ll calculate exotic spheres in dimension 29
And a variety of varieties, projective and affine.
I’m comfortable with categories (though not if they’re derived)
I’ll tile hyperbolic space in dimension 25
I can compute curvature with the Gauss-Bonnet law
And just love the Leech Lattice in dimension 24.
But algebro-geometric scheming
Leaves me spluttering and screaming.
And in logic too, you may call me absurd
But I wouldn’t know a topos, if trampled by a herd.
I’ve tried Pursuing Stacks but they vanished out of sight,
I’ve fought with étale cohomology with all my might.
And Les Dérivateurs. It’s 2000 pages long.
I reach halfway through line 3, before it all goes badly wrong.
No, I’ll never get to grips with Grothendieck
And I’m frightened that I’m failing as a mathematics geek.
All the same, I can’t deny the lure and the mystique
Of Le Séminaire de Géométrie Algébrique.
– Richard Elwes, 2015