# Classical and quantum integrable systems

The broad area of research are classical and quantum integrable systems, connected to the analysis of linear and nonlinear ODEs in the complex plane, and to Kac-Moody algebras.

For example, a main themes of research are the ODE/IM correspondence and the asymptotic analysis of Painlevé equation.

- The ODE/IM correspondence: we study the spectral problems of some linear operators with values in a Kac-Moody algebra, and analytic coefficients, to provide solutions of the Bethe Ansatz equations of an integrable quantum field theory.
- Painleve equations are nonlinear ODEs whose solutions extends meromorphically to the whole complex plane (hence they are called " integrable"): we study the pole distributions (in the complex plane) of special solutions of Painleve equations and the existenceof smooth real solutions.