**Content:**

Part 1

1.1Short review of the classical theory of finite-dimensional integrable systems

1.2 Be-Hamiltonian Geometry and Lax pair

1.3 The Toda system

1.4 The Korteweg de Vries equation: direct and inverse scattering on the line with decreasing initial data

1.5 Long time asymptotic for the solution of the KdV equation with decreasing initial data and Deift-Zhou steepest descent method

1.6 The Cauchy problem for the KdV equation with periodic initial data and action-angle variables.

Part 2 Riemann Surfaces

2.1 Deﬁnition, examples, and topological properties

2.2 Holomorphic and meromorphic functions and differentials on a Riemann surface.

2.3 Jacobi variety and Abel theorem

2.4 Riemann Roch theorem and applications

2.5 Theta function and Riemann vanishing theorem

2.6 Baker-Akhiezer function and periodic solution of integrable nonlinear PDEs