Integrable systems student seminar

Fall 2013

Organizers: Gus Schrader and Alexander Shapiro

Time and place: Thursday 2pm, 891 Evans

Mailing list: If you want to be added, let me know, preferably by email (guss@math.b....)

Plan: We're planning to start with Nigel Hitchin's lecture notes in the book "Integrable systems: twistors, loops groups and Riemann surfaces", and his original paper "Stable bundles and integrable systems."

We then plan on covering the material in Michael Semenov-Tyan-Shansky's notes "Quantum and classical integrable systems."

Here's a rough plan of the topics for individial talks:

For Hitchin's lectures:

Algebraic geometry refresher: vector bundles on Riemann surfaces
Spectral curves and Lax equations
Hitchin systems

For Semenov-Tyan-Shanksy's lectures:

Construction of integrable systems: the linear case
Examples: open Toda lattice, classical Gaudin model
Construction of integrable systems: the quadratic case
Quantization in the linear case and the open Toda lattice
Affine Lie algebras and the quantization of the Gaudin model
Quantization in the quadratic case and quasitriangular Hopf algebras

Suggestions for further references and topics to cover are welcome!

Additional references:

David Mumford, Tata Lectures on Theta II
Pol Vanhaecke, Integrable systems in the realm of algebraic geometry
Boris Dubrovin, Integrable systems and Riemann surfaces

Notes:

Intoduction, October 5
Hamilton-Jacobi theory and Jacobians, October 17
Algebraic construction of integrable systems: Toda chains and Gaudin model, October 24

Calendar of participant talks:

September
5	Gus Schrader	Introduction.
12	Piotr Achinger	Geometric prerequisites for Hitchin integrable systems.
19	Qiao Zhou	Algebraic integrable systems.
26	Alexander Shapiro	Hitchin systems.
October
3	Piotr Achinger	The Hitchin fibration.
10	Nicolai Reshetikhin	KZ equations and Hitchin systems.
17	Gus Schrader	The Hamilton Jacobi method and Jacobians
24	Gus Schrader	Algebraic construction of classical integrable systems: Toda chains and the Gaudin model.

September 5, Gus Schrader: Introduction.

I'll introduce the notion of a classical integrable system, and talk a litle about where the seminar might be heading. Notes.

September 12, Piotr Achinger: Geometric prerequisites for Hitchin integrable systems.

This talk will cover algebraic geometry prerequisites necessary for the construction of the Hitchin integrable system. We will review the theory of algebraic curves and line and vector bundles on them.

September 19, Qiao Zhou: Algebraic integrable systems.

We are going to introduce the notion of an algebraic integrable system. We will discuss the spectral curve, the Lax pair of equations, and the dynamics of the system on the Jacobian of the spectral curve.

September 26, Alexander Shapiro: Hitchin systems.

I will explain how an integrable system arises in a natural way on the moduli space of stable vector bundles (of fixed rank and degree) over a Riemann surface

October 3, Piotr Achinger: The Hitchin fibration.

We will study properties of the Hitchin map associated to a moduli space of stable bundles on a curve. We will identify a general fiber of this map with an open subset of the Jacobian of the corresponding spectral curve and construct a "relative compactification" of the Hitchin map.

October 10, Nicolai Reshetikhin: KZ equations and Hitchin systems.

I will talk about classical and quantum Knizhnik-Zamolodchikov equations and their relation to Hitchin systems.

October 17, Gus Schrader: The Hamilton-Jacobi method and Jacobians.

I will explain the Hamilton-Jacobi method of constructing angle coordinates on the Liouville tori of an integrable system. In these coordinates, the time evolution of the system is simply linear motion with constant velocity. We will apply this method to the example of the Gaudin spin chain introduced last week. In this case, we will identify our change-of-coordinate map with the Abel map associated to the system's spectral curves. Notes.

October 24, Gus Schrader: Algebraic construction of classical integrable systems: Toda chains and the Gaudin model.

I'll return to the examples of the open Toda chain and the Gaudin model, and explain how to construct (and solve) them Lie-theoretically. Notes.