Quantum mechanics has led to the emergence of new concepts in various mathematical fields (in analysis: Hilbert spaces formalized by von Neumann; in algebra: representation theory according to Cartan and Weyl). In return, these concepts have led to better formalizations in fundamental physics, as well as important discoveries, such as the standard model of elementary particles (Glashow, Weinberg, Salam). For this EA, the mathematics considered will be based on group theory and the physics targeted will essentially be that of the infinitely small.

In physics, whether at the classical or quantum level, the analysis of the symmetries of a system makes it possible to simplify its study because they generally imply the existence of retained quantities, selection rules, etc. The symmetry groups at play are part of the daily tools of many areas of fundamental physics. Some mathematical subtleties of abstract group theory are strikingly embodied in physics: for example, the difference between the SU(2) and SO(3) groups corresponds to the existence of half-entire spin particles, objects that do not have a classical interpretation. Extensions of orthogonal groups, the Lorentz and Poincaré groups, are interpreted as symmetry groups of relativistic physical systems. It so happens that the unit groups, SU(2) as well as U(1) and SU(3), also appear as "internal" symmetry groups of elementary particles: this discovery led to the formulation of the standard model of particle physics mentioned above. This theory classifies the elementary bricks of matter and describes their interactions, and its many predictions have passed all experimental tests to date.

The mathematical notion of linear representation of a group is central to quantum mechanics, and is a beautiful illustration of the interaction between mathematics and physics that we propose to present: it is a notion that pre-existed in quantum mechanics, but the directions in which it developed have sometimes been very strongly determined by physical considerations (E. Wigner). It is in this spirit that the basics of this theory will be presented (weight diagrams, character representations, tables and Young's diagrams).

The sessions are led alternately by a mathematician teacher and a physics teacher.

In parallel to the teaching, the students prepare a bibliographic project on a subject of their choice, leading to the writing of a report and an oral defense at the end of the period.

**Language of the course**: French or English, depending on the audience

**ECTS credits**: 5

- Profesor: De Suzzoni Anne-Sophie
- Profesor: Munier Stéphane