School of Mathematics

Cluster algebras were introduced by Fomin and Zelevinsky in 2000, and since then appear in various contexts. A large class of cluster algebras can be constructed using triangulated borded surfaces with marked points. In the talk, I will discuss the construction and combinatorial properties of these algebras as well as their applications to classification of cluster algebras of finite mutation type.

The Gromov--Witten invariants of a space X count, roughly speaking, the number of holomorphic curves that meet various cycles in X. They have important applications in algebraic geometry, symplectic topology, and theoretical physics. Let X be the canonical line bundle over the projective plane P^2. I will describe joint work with Iritani in which we show that generating functions for Gromov--Witten invariants of X are modular forms for the group Gamma_0(3). There are tantalizing hints of a connection with integrable systems.

In dynamical systems with discrete space, the distinction between regular and irregular motions must be re-considered. I articulate the main questions using examples, with emphasis on arithmetical phenomena.

In dynamical systems with discrete space, the distinction between regular and irregular motions must be re-considered. I articulate the main questions using examples, with emphasis on arithmetical phenomena.

Algebra, Geometry and Integrable Systems Colloquium.

Cubic surfaces in complex projective space can be classified by a map to the quotient of the unit ball in C^4 by an arithmetic group. This map is called the period map. The arithmetic group reflects and refines classical the classical relations between cubic surfaces and the Weyl group of E_6. We will explain the construction of this map, explain some of its geometry, and then talk about what is known and what we would like to know about values of this map. This is joint work with Allcock and Carlson.

We construct new noncommutative deformations of toric varieties by combining methods from toric geometry, isospectral deformation theory and noncommutative geometry in braided monoidal categories. We apply these techniques to the construction of a certain class of noncommutative instantons and discuss the interrelationships between their description in terms of deformed ADHM data, torsion-free modules and a noncommutative twistor correspondence.

This talk will have three parts: in the first, I will describe the beautiful classical theory of isothermic surfaces in the 3-sphere due to Christoffel, Darboux, Bianchi and others. Then I will indicate how the 3-sphere may be replaced by any symmetric R-space (a conjugacy class of real parabolic subalgebras with abelian nilradicals) with essentially no loss of integrable structure. Finally, I shall show how dynamics of the simplest examples (curves in the real projective space) provide a geometric interpretation of the KdV equation, its relation to the mKdV equation via the Miura transform and the Baecklund transformations of KdV discovered by Walhquist-Estabrook.

I shall present a solution to a long standing problem: what is the natural geometry of moduli spaces of hyperbolic monopoles. The answer, it turns out, is a completely new type of geometry, generalising hyperkaehler and hypercomplex geometries. Remarkably, even the underlying linear algebraic structure appears not to have been studied before. In this talk, I'll present the geometry "from bottom up", starting with the linear-algebraic version and then describing integrability conditions required for manifolds with such geometry. No knowledge of monopoles is necessary. This work is joint with L. Schwachhoefer.