Algebro-geometric solutions of KP and KdV by means of the Baker Akhiezer functions

Matveev-Its formula: u =2 \partial^2_x \log \theta

Real smooth solutions of KdV and M-curves

Explcit solution of Jacobi inverse problems on hyperelliptic curves by means of theta functions.

Baker-Akhiezer functions, definition and theorems on sufficient conditions for uniqueness

Theory:

Theorems on the zeros of theta functions and solution of the Jacobi inverse problem

Exercise: Integration on the 4g-gon, meromorphic differentials on the hyperelliptic surfaces, Jacobi inversione in genus 1.

Theory: Abel's theorem, theory of divisors, Riemann-Roch theorem, special divisors on hyperelliptic surfaces

Exercise:Riemann's bilinear relations, Riemann matrix of B cycles, functions of the torus as ratio of theta functions, or logarithmic derivatives of theta functions, Weierstrass P as second logarithmic derivative of theta_1, zeroes of P'

Theory: Residue theorem on compact surfaces, total residue vanishes, compactification of hyperelliptic surfaces by adding one or two points at infinity,

Riemann-Hurwitz formula.

Exercises:location of zeroes of 1 dimensional theta functions of half-periods, isomorpism between elliptic curve and 1-dimensional torus