A long goodbye to mathematics (a secret hello to physics?) /

A sad day's post

Grothendieck did not derive his inspiration from physics and its mathematical problems. Not that his mind was incapable of grasping this area—he had thought about it secretly before 1967—but the moral principles that he adhered to relegate physics to the outer darkness, especially after Hiroshima. It is surprising that some of Grothendieck’s most fertile ideas regarding the nature of space and symmetries have become naturally wed to the new directions in modern physics. It is this unexpected marriage—and its occasionally comical aspects—that I would like to talk about here. “A mad day’s work”, as you know, is the subtitle given to The Marriage of Figaro by Beaumarchais. From a certain distance there is less cause for astonishment; the concepts of space and symmetry are so fundamental that they are necessarily central to any serious scientific reflection. Mathematicians as influential as Bernhard Riemann or Hermann Weyl, to name only a few, have undertaken to analyze these concepts on the dual levels of mathematics and physics...

Grothendieck’s broken dream was to develop a theory of motives, which would in particular unify Galois theory and topology. At the moment we have only odd bits of this theory, but I would like to conclude with a magnificent, quite unexpected development, in which physics and mathematics come together again...

Drinfeld has introduced a group GRT1 called the (graded) Grothendieck{Teichmuller group. It is a scheme of groups over the field Q, and it therefore has a Lie algebra, denoted grt1. To describe this Lie algebra would require me to give precise information on the Knizhnik/Zamolodchikov equations, which play a fundamental role in the theory of conformal elds. It is conjectured that the Lie algebra grt1 is a free Lie algebra with generators ψ₃, ψ₅, ψ₇ ... corresponding in a natural way to the numbers ζ (3), ζ (5), ζ (7) ... Moreover, GRT1 plays the role of the Galois group for transcendental numbers of the form ζ (k₁... k_r), since it acts (conjecturally) on the algebra A by automorphisms.  

At almost the same time, at the institute, Connes and Kontsevich had just discovered a natural occurrence of the group GRT1 in fundamental problems of physics:

1. Connes and Kreimer [3] discovered how to make the Lie algebra grt1 (and other similar Lie algebras) act on the algebra corresponding to Feynman diagrams. It represents a new type of symmetry, not acting on any particular model of eld theory, but sweeping away a whole class of possible Lagrangians.

2. Kontsevich [9] has recently solved the problem of quantization by deformation for Poisson manifolds. The set of possible quantizations has a symmetry group, and Kontsevich conjectures that it is isomorphic to GRT .

In both problems the numbers ζ (k₁... k_r) arise as the values of certain integrals...

A MAD DAY'S WORK: FROM GROTHENDIECK TO CONNES AND KONTSEVICH, THE EVOLUTION OF CONCEPTS OF SPACE AND SYMMETRY

PIERRE CARTIER

Article electronically published on July 12, 2001

(Version française du texte de Cartier disponible ici)

Last words to Alexander Grothendieck himself

... séduit par le prestige soudain de la physique atomique, c’est pourtant pour des études de physique que je me suis d’abord inscrit à l’ Université de Montpellier, avec l’idée de m’initier aux mystères de la structure de la matière et de la nature de l’énergie. Mais j’ai vite compris que si je voulais m’initier à des mystères, ce n’était pas en suivant les cours de la Fac que j’y arriverais, mais en travaillant par mes propres moyens, seul, avec ou sans livres. Comme je n’avais pas le flair, ni l’appareillage, pour apprendre la physique de cette façon-là ; j’ai renvoyé la chose à des temps plus propices, Je me suis alors mis à faire des maths, tout en suivant "de loin" quelques cours, dont aucun ne pouvait me satisfaire, ni m’apporter rien au delà de ce que je pouvais trouver dans les manuels courants.

Récoltes et semailles , p. 495

Le petit enfant découvre le monde comme il respire - le flux et le reflux de sa respiration lui font accueillir le monde en son être délicat, et le font se projeter dans le monde qui l’accueille. L’adulte aussi découvre, en ces rares instants où il a oublié ses peurs et son savoir, quand il regarde les choses ou lui-même avec des yeux grands ouverts, avides de connaître, des yeux neufs - des yeux d’enfant.

Récoltes et semailles , p. 128 (version Yves Pocchiola?)

//last edit 6 june 2016