Mathematics

Integrable systems are special  systems which can be solved exactly in some sense. They arise in a variety of settings, ranging from Hamiltonian systems, nonlinear wave equations   and probability. This  course covers the origins of the subject as well as modern topics in  nonlinear waves  and  integrable probability.

1.  The Korteweg de Vries equation (KdV) 

• Scattering and inverse scattering of the Schrodinger equation on the line and solution of KdV equation;
• as a Fredholm determinant. Tau-function and Fredholm determinants;
• Periodic problem  of KdV  and finite gap solutions on Riemann surfaces;
• Modulation theory of nonlinear waves  and Whitham equations.

2. The Toda lattice: an example of a non linear integrable system of particles

• The Lax matrix and the Adler-Kostant-Syme scheme of integration;
• Factorisation theorem  and asymptotic behaviour of solutions;
• Gibbs measure initial data and random matrices.

3. Random matrices 

• Hermitian random matrices. Partition function as an example of tau-function of the Toda lattice;
• Partitions and Schur function expansion of tau functions;
• Topological expansion, enumeration of maps and Hurwitz numbers.

Exam

The evaluation in this course will consist of homework assignments, and a final exam, in the form of a seminar.

References

• O. Babelon, D. Bernard and M. Talon, Introduction to Classical Integrable Systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2003. 
• M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series 149, Cambridge University Press, 1992. 
• J. Harnad and F. Balogh, Tau functions and their applications.  Cambridge University Press Online publication date: January 2021 Online ISBN: 9781108610902 DOI: https://doi.org/10.1017/9781108610902
• G. Segal "Integrable systems and inverse scattering”     in the book "Integrable Systems"; Oxford Sci. Publ. 1999 
• J. Moser "Various aspects of integrable Hamiltonian systems."    Progr. Math. 8, Birkhauser, Boston, Mass.,1980, pp. 233-289
• Greg W. Anderson, Alice Guionnet, Ofer Zeitouni  An Introduction to Random Matrices, Cambridge University Press Print publication year: 2009/ https://doi.org/10.1017/CBO9780511801334

Rooms

For Febraury

• Always A-132
• 21/02 A-004
• 28/02 A-138

From March 19

• Morning A-134
• Afternoon A-005