250100 VO Topics in Integrable Models (2010W)

Integrable Models is a very broad topic that can be divided into two major
sub-topics: Classical Integrable Models, the starting point of which is
nonlinear partial differential equations that typically have soliton
solutions, and Quantum Integrable Models, the starting point of which is
exactly solvable models in statistical mechanics and in quantum field
theory.

In this course, we discuss only Classical Integrable Models which by itself
is a very broad subject with many complementary approaches to it, and we
choose to concentrate on an algebraic approach (also known as Sato's Theory)
which plays a central role in modern mathematical physics, particularly
mathematical aspects of string theory.

The course has three parts:

1. The Lax formulation of integrable models: Integrable nonlinear PDE's are
understood as consistency conditions of systems of linear PDE's (we will see
what this means exactly in due course). This part requires 4 to 5 sets of
2-hour lectures.

2. The Fermionic formulation of integrable models: Solutions of integrable
nonlinear PDE's are re-written in terms of expectation values (to be
defined) of fermion (Clifford) operators (also to be defined). This part
requires 4 to 5 sets of 2-hour lectures.

In the third part, we can discuss either

3A. The Geometric formulation of integrable models: Solutions of integrable
nonlinear PDE's are points on a Grassmannian (to be defined) embedded in a
suitable projective space (also to be defined), or
3B. Applications of the Fermionic formulation to algebraic combinatorial
aspects of problems in modern mathematical physics.

Parts 1, 2 and 3A would be based on the textbook

Solitons, by Miwa, Jimbo and Date, Cambridge U Press, 2000

Part 3B would be based on recent works by various authors, including
Nekrasov, Okounkov, Orlov and collaborators.

Prerequisites of the course are basic undergraduate courses, particularly
Calculus, Complex Analysis and Differential Equations.