curso "Abelian varieties in algebraic integrable systems"

Contents: 6 lectures, each one 55 min., given in Spanish with slides in English.   


Description of the first 3 lectures.


Lecture 1. Algebraic curves, Abelian varieties, and algebraic integrable systems.


- A first example:  The classical Euler top on so(3). Equations of motion, elliptic curve, linearization.

- Abelian varieties: definitions, basic properties, examples. Jacobi varieties (algebraic and analytical description).

- Algebraic integrable systems. Examples. Historical remarks. 

- Exercises.


Lecture 2. Algebraic integrable systems and the Painlevé property.


- More properties of Abelian varieties: divisors, linear systems, embedding to projective space. 

- The essence of the Kovalevskaya--Painlevé (KP) analysis: Laurent series solutions, Kowalewski exponents, the theorems of

   Adler & van Moerbeke and of Yoshida, and their generalizations.

- Geometric interpretation of the KP analysis.

- Examples: the Euler top again, the Clebsch integrable case of the Kirchoff equations. 


Lecture 3.  Lax representation and spectral curves.


- What really is a Lax representation of an integrable system. Examples. 

- The eigenvector map to the Jacobian of the spectral curve S. Linearization of the flow on the Jacobian.

- The problem of separation of variables. 

- Spectral curves with an involution and Prym subvarieties of the Jacobian. 

- Relation between the Jacobian of S and the complex invariant manifold of the system.  


Yuri Fedorov, Universidad Politécnica de Cataluña
Sala de Lectura, U.D. Geometría y Topología
this course is not about algebraic geometric aspects of Abelian varieties or recent advances in this area, but rather on their applications to some finite-dimensional integrable systems of classical mechanics and mathematical physics. It is intended to be self-contained, although some knowledge of algebraic curves and elliptic functions will be required. During the course various examples will be presented, and they will be considered from different points of view.
Thursday, 3 November 2016