Colloquia
Greg Reid
CAM COLLOQUIUM
“The Generalization of Numerical Algebraic Geometry to Partial Differential Equations”
Greg Reid
Applied Mathematics Department
University of Western Ontario - Canada
Monday - April 14, 2008- 4:00PM in 127 Hayes-Healy Center
*TEA AT 3:30PM in the Math Lounge - 257 Hurley*
Numerical Algebraic Geometry is a new field that was created and pioneered by Andrew Sommese and Charles Wampler. This area gave the first stable numerical methods for finding and characterizing all the irreducible solution components of solutions of polynomial systems.
This area is both beautiful and is of fundamental importance, since systems in applications have approximate coefficients. Moreover the traditional symbolic exact tools (e.g. Grobner Bases), are inefficient, and unstable when applied to approximate systems. My collaborator Wenyuan Wu and I have been generalizing this method to systems of polynomially nonlinear PDE. Like Numerical Algebraic Geometry the fundamental objects are witness points, generic points cut out by intersection of solution components, with random linear varieties.
The talk will be largely introductory in nature - there will be pictures - and will focus on the progress and challenges in this area. Recent progress on developing a witness point construction more tailored to the PDE case, and determination of singular solution components will be mentioned as will results for the semi-discretization of partial differential equations.
