Cached June 14, 2010, from http://www.rug.nl/informatica/nieuws/actueel/bernoulli2004-2005

Johann Bernoulli Lecture 2004-2005

Date:  March 22, 2005 

 The next Johann Bernoulli Lecture will take place on

 March 22 2005, 19.30, Akademiegebouw Groningen

 The speaker will be:

 Jean-Pierre RAMIS, Université Paul Sabatier, Toulouse

The title and summary of his lecture are

 FROM LEIBNIZ TO QUANTUM WORLD:
SYMMETRIES, PRINCIPLE OF SUFFICIENT REASON AND AMBIGUITY IN THE SENSE OF GALOIS


Classical ideas on symmetry mix in fact two related but quite different things:

1. The notion of group operating on a set.

2. The notion of group of ambiguity in the sense of the last letter of Evariste Galois.

In the first case, one knows a priori very well the group (formally or intuitively). In the second case the group is more mysterious but contains a lot of precious and powerful information.

G. D. Birkhoff and H. Weyl remarked that the symmetry of ambiguity (in Galois sense) is in fact very similar to some fundamental symmetries of Physical Theories (like classical mechanics, relativity, classical and quantum electrodynamics...): "Galois theory is relativity applied to finite sets". G. D. Birkhoff formulated these ideas in a general principle valid in mathematics and physics and extending the Principle of Sufficient Reason of Leibniz.

Today ambiguity groups appear frequently in mathematics and physics and are extremely powerful tools. After some elementary examples introducing the idea of symmetry group, I will give an idea of some recent applications to the old and classical problem of integrability of dynamical systems (top, three bodies problem, lunar problem...) and to quantum physics (the Standard Model and the Renormalization in Quantum Fields Theory). Surprisingly the recent work of A. Connes and his collaborators about ambiguity groups in quantum fields theory seems strongly related to very deep questions in arithmetic and we return to E. Galois last letter.

Last modified: February 11, 2005 13:40

From a flyer for the same lecture--

___________________________________________
Symmetry is a concept which functions in almost
any science. In this lecture the focus is on its
importance in mathematics and physics. Imagine a
square. You can turn it 90° and find the same
square again. We say that the square has rotational
symmetry. It also has reflectional symmetry: you can
reflect the square, e.g. in one of the diagonals, and
find the same square. Two of these actions
combined one after another also produce the square
again. There are 24 [sic] operations which will produce
the same square again. They build what is called a
symmetry group.

A next step is the group of ambiguity, a concept
written down by the visionary 20 year old Evariste
Galois in the night before he was shot in a duel.
G.D. Birkhoff and H. Weyl remarked that the
symmetry of ambiguity (in the way Galois defined
it) is very similar to some fundamental symmetries
of Physical Theories (like relativity and electrodynamics).
Birkhoff formulated these ideas in a
general principle valid in mathematics and physics
and extending the Principle of Sufficient Reason of
Leibniz.

Ambiguity groups appear frequently in present day
mathematics and physics and are extremely powerful.
After some elementary examples introducing the
idea of symmetry group, Ramis will give an idea of
some recent applications to the theory of dynamical
systems (three bodies problem, lunar problem) and
to quantum physics (the Standard Model and the
Renormalization in Quantum Fields Theory).

Surprisingly it appears that ambiguity groups in
quantum fields theory seem strongly related to deep
questions in arithmetic and this will bring us back to
the letter that Galois wrote on the night before he
was shot.
__________________________________________
Jean-Pierre Ramis is professor of mathematics at the
Université Paul Sabatier (Toulouse 3). As a
researcher he is connected to the Laboratoire Emile
Picard, where he studies, among other subjects, the
Galois theory of differential equations.