Emma Previato

Complex Algebraic Geometry Applied to Integrable Dynamics: Concrete examples and open problems

Lecture I: Elliptic and Hyperelliptic Theta Functions: Applications to integrable dynamics. The lecture will cover definition and significance of elliptic and hyperelliptic curves, their moduli, and special functions, together with Klein's generalization of the Weierstrass sigma function and its relationship with Sato's tau function. As an application, solutions to examples of integrable hierarchies such as non-linear wave equations and the Toda Lattice will be constructed.

References:
David Mumford, Tata Lectures on Theta I and II, Birkh\"auser, 1984.
Morikazu Toda, Theory of Nonlinear Lattices (2 ed.), Berlin: Springer (1989)

Lecture II: Vector bundles over curves. This lecture will cover definitions of vector bundles over curves and their moduli, together with applications to algebraically completely integrable Hamiltonian systems such as the geodesic and monopole equations.

References:
Robert C. Gunning, Lectures on Vector Bundles over Riemann Surfaces. Princeton University Press, 1967.
Nigel Hitchin, On the construction of monopoles. Communications in Mathematical Physics 89 (2) (1983) 145-190.

Lecture III: This lecture will bring together integrable fibrations and classical theorems of projective geometry, to conclude with applications to random-matrix theory, ordinary differential equations with no movable singularities (except poles), and open questions.

References:
Olivier Babelon, Denis Bernard and Michel Talon, Introduction to Classical Integrable Systems, Cambridge Monographs on Mathematical Physics, 2003.

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