Spectral Geometry is the field of mathematics that aims to relate the geometry of an object to its vibrational spectrum. The field had its first birth in the years 1910, when the precursors of quantum mechanics sought to calculate the spectrum of atoms from geometric considerations on the planetary model. The question then evolved into the study of the spectrum of Schrödinger operators, in connection with symplectic geometry in the phase space of classical mechanics.

The second birth of the field dates back to the years 1960 with the index theorem, which gives relationships between certain " topological indices " (e.g. the Euler characteristic of a topological space) and the bottom of the spectrum of an elliptic operator (such as the Laplace operator). This field is currently experiencing intense activity on the physics side, with the discovery of the role of the notion of" index " in the description of topological materials.

Among the major issues in Spectral Geometry are :

• Quantum chaos : the study of the spectrum of a Schrödinger operator when the corresponding Hamiltonian system in classical mechanics is chaotic ;
• Inverse problems : what can we guess about the geometry of an object from the measurement of its vibration spectrum ?
• The link between spectrum and topology, via various avatars of the index theorem ;
• The spectrum of disordered systems or random geometric objects ;
• The link between geometry and wave control : what are the best places to " direct " a wave ?

The lecture will focus on the mathematical aspects of these questions, but in some years the seminar will be an opportunity to hear physicists present their work in relation to the lecture.