The dynamics of very simple systems (a particle propagating in a force field) can lead to very complex motions. For example, the trajectories of a particle moving on a billiard table shaped like a stadium are unstable and they end up covering the whole surface of the table. There is a corresponding quantum system: a wave propagating inside a cavity (electrons in a quantum box, light in an optical fiber...). The semi-classical approach can be used to analyze the stationary modes and their localization, from which one can deduce remarkable properties of special mathematical functions. This vast research domain borders with physics and mathematics. It has numerous applications such as, for example, the study of turbulence and of quantum decoherence.
Statistical physics is based on a precise count of the different states and configurations of a system, whence its intimate connection to combinatorics. The quantum Hall effect, the evaporation of dimers on a surface or the melting of a crystal can be analyzed in terms of generalized random walks with constraints, which can be solved by analogy to integrable systems. The relation between critical phenomena on regular two-dimensional (2D) lattices and their “gravitational” version on random lattices provides the key to the solution of numerous problems in statistical mechanics (polymers, hard core particles) and in mathematics (the three color problem, the meander problem). Conversely, the combinatorial approach allows one to go well beyond classical results. For example, the decomposition of graphs into tree graphs allows the study of the statistics of internal distances in random lattices.
Integrable or exactly solvable systems conserve many quantities during their evolution. They enable us to study non-perturbative phenomena in physical systems with strong statistical and quantum fluctuations. Their analysis reveals remarkable algebraic and geometrical structures with applications in both physics and pure mathematics. Conformal field theories are also excellent tools to study critical 2D systems and quantal 1D systems of mesoscopic physics, as well as to characterize certain universal processes of stochastic and fractal growth. The relation between geometry and integrable systems appears also in the study of random matrices. Due to their universality property (the law of large numbers) such models have applications, not only in mathematics, but also in physics (string theory, quantum chromodynamics, quantum chaos, the growth of crystals), in biology (the structure of RNA) and in every day life (economics, the frequency of bus arrivals).
The de Broglie wavelength of a particle, propagating in a region of space where the energy density is very large, in the neighborhood of a black hole for example, can be of the same order of magnitude as the space-time curvature. Classical gravity is then modified by quantum effects. Such phenomena are the subject of string theory, in which particles appear as vibrational modes of extended objects and which offers a unified description of gravity and quantum mechanics. The quantization of string theory in a curved space uses techniques which were developed in matrix models, and the know-how acquired in conformal theories can be applied to the calculation of scattering amplitudes of branes as well as to the decay of unstable modes. In particle physics, string theory has also inspired new and powerful methods to calculate cross-sections in quantum chromodynamics.