Quantum geometry and integrable systems

next Friday 25th of March, Carslaw 175 at 2:30 PM.

We will take the speaker to lunch at the Grandstand with folks from the Algebra seminar, leaving from Carslaw level 2 at 1 PM.

Title: Quantum Geometry and Integrable Systems

Abstract: In physics, an exact solution is a concise and complete mathematical description of the problem without using an approximation. Of course, not every physical problem can be solved exactly but, remarkably, it looks as though some of the most fundamental problems which determine our being are indeed exactly solvable (eg, the Kepler problem of planetary motion and the hydrogen atom in quantum mechanics). Intensive development of the theory of exactly solvable or "integrable" systems with a large or even infinite number of degrees of freedom started in the 1970s after the discovery of the Yang-Baxter equations. On the math side this led to spectacular advances in the theory of infinite dimensional algebras and quantum groups.

In this lecture I will explain that this algebraic theory naturally originates from consideration of geometry of three-dimensional lattices.

In this approach the Yang-Baxter equations become connected to incidence theorems of elementary geometry, whereas integrable models of statistical mechanics and quantum field theory become models describing quantum fluctuations of lattice geometry. The classical geometry arises as a stationary configuration in the quasi-classical limit. The lecture is intended for a general physical and mathematical audience, no special knowledge is assumed.