MSI Weekly Bulletin - Week starting Monday 1 October, 2007

Bilinear transformations possess a unique structure that allows for the construction of a class of fractal homeomorphisms. In this talk I will demonstrate the construction of the bilinear transformation between two irregular quadrilaterals. I will then introduce the fractal homeomorphism between pairs of bilinear IFSs sharing the same code structure.

Plurisubharmonic functions are not nearly as well known as analytic functions, but they are ubiquitous in modern complex analysis and complex geometry. They were first explicitly defined in the 1940s, but they had already appeared in attempts to geometrically describe domains of holomorphy at the very beginning of several complex variables in the first decade of the 20th century. Since the 1960s, one of their most important roles has been as weights in a priori estimates for solving the Cauchy-Riemann equation. They are intimately related to the complex Monge-Ampere equation, the second fundamental differential equation of complex analysis. There is also a potential-theoretic aspect to plurisubharmonic functions, which is the subject of pluripotential theory. We will give an overview of the theory of plurisubharmonic functions and briefly describe some very recent applications to algebraic geometry and number theory.