Past

In this talk I'll explore the meaning of the Hankel determinant formula for the general solutions of the Painleve' equations both from the analytic and the geometric point of view. I'll start with the simple example of PII and I'll show how the generating function for the Hankel determinant satisfies two Riccati equations. These linearize into the Jimbo-Miwa-Ueno isomonodromic deformation problem. Indeed this occurs for all the Painleve' equations PII,..,PVI and it is due to the link between their solutions and the infinite Toda lattice equation. I'll then explore the geometric meaning of the Hankel determinants by looking at the (suitably defined) spectral curve of the Toda lattice equation.

Chakravarty and Ablowitz (PRL vol.76 p.857, 1996). showed that a fifth-order equation arizing from the Bianchi IX system can be described asmonodromy evolving (non-preserving) deformations. In my talk, we will show that general Halphen-type systems, which comes from generic DH-IX systems, can be represented as monodromy evolving deformations.

  In this talk we will discuss some recent developments in the study of nonlinear partial differential equations of mixed type, including the mixed parabolic-hyperbolic type and mixed elliptic-hyperbolic type. Examples include nonlinear degenerate diffusion-convection equations and transonic flow equations in fluid mechanics, as well as nonlinear equations of mixed type in a fluid mechanical formulation for isometric embedding problems in differential geometry. Further ideas, trends, and open problems in this direction will be also addressed.  

We present a continuous-time equilibrium-based model for large economic agent, where she trades with market makers at their utility indifference prices. The presentation is based on a joint project with Peter Bank.

There will be three discussion meetings based on aspects of the

programme open to all internal project members. Others interested in

attending should contact Carlos Mora-Corral.

Team meetings, held roughly every four weeks, are open to anyone who is

interested. OxMOS post docs and Dphil students will give updates on the

research.