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## Integrable Systems

Integrable systems are physical systems in which the equations of motion can (in

principle) be solved exactly. They exist in classical and quantum versions ; and

can have finite number of degrees of freedom, or infinite (the latter being the

case of field theories — in the quantum case, integrable quantum field theories

are almost entirely restricted to two dimensions). The 20th century has witnessed

considerable progress in the field of integrable systems, including the

development of modern methods such as the classical and quantum inverse

scattering methods.

Integrable systems have many applications : to statistical mechanics (exactly

solvable lattice models in two dimensions), to condensed matter (quasi-1D

systems), to pure mathematics and more recently to string theory.

In the recent years, the activity of the group has been as follows. O. Babelon

has been working on the separated variables and their quantization. F. Smirnov’s

work concerns the connection between integrable quantum field theories and

conformal field theories, and the hidden free fermionic structure of quantum

integrable models. P. Zinn-Justin and J.-B. Zuber are working on relations

between two-dimensional loop models and combinatorics, in particular alternating

sign matrices and plane partitions. P. Zinn-Justin also works on applications of

quantum integrable models to combinatorial problems in algebraic geometry.