Are
there hidden relations between mathematical and physical constants such
as
or are these numerical relations mere accidents? A couple of years
ago, Pierre Cartier proposed in his paper A mad day’s work : from Grothendieck to Connes and
Kontsevich : the evolution of concepts of space and symmetry that
there are many reasons to believe in a cosmic Galois group acting on the
fundamental constants of physical theories and responsible for relations
such as the one above.
The Euler-Zagier numbers are infinite
sums over
and there are polynomial relations with rational coefficients between
these such as the product relation
It is
conjectured that all polynomial relations among Euler-Zagier numbers are
consequences of these product relations and similar explicitly known
formulas. A consequence of this conjecture would be that
Drinfeld
introduced the Grothendieck-Teichmuller group-scheme over
whose Lie algebra
algebra on infinitely many generators which correspond in a natural way
to the numbers
group itself plays the role of the Galois group for the Euler-Zagier
numbers as it is conjectured to act by automorphisms on the graded
of the numbers
such that
The Grothendieck-Teichmuller
group also appears mysteriously in non-commutative geometry. For
example, the set of all Kontsevich deformation quantizations has a
symmetry group which Kontsevich conjectures to be isomorphic to the
Grothendieck-Teichmuller group. See section 4 of his paper Operads and motives in
deformation quantzation for more details.
It also appears
in the renormalization results of Alain Connes and Dirk Kreimer. A very
readable introduction to this is given by Alain Connes himself in Symmetries Galoisiennes
et renormalisation. Perhaps the latest news on Cartier’s dream of a
cosmic Galois group is the paper by Alain Connes and Matilde Marcolli posted
last month on the arXiv : Renormalization and
motivic Galois theory. A good web-page on all of this, including
references, can be found here.