# From Leibniz to Quantum World

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 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, Ramus 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.

The Johann Bernoulli Lecture is organised by the Johann Bernoulli Stichting in cooperation with Studium Generale Groningen.