New geometric and numeric tools for the analysis of differential equations

The Banff International Research Station will host the "New geometric and numeric tools for the analysis of differential equations" workshop from August 13 to 15, 2010.

Experts Develop New Geometric Modeling Tools A mathematics workshop, to be held at the Banff International Research Station, for mathematical innovation and discovery, brings together researchers in the Geometry of differential equations and in Numerical Analysis. The complex motions of a medical robot or the moving components of a vehicle are described by differential equations often involving polynomial expressions in the quantities of interest and in design parameters. From a geometrical viewpoint, the states of such mechanisms are represented by points in a high dimensional space (e.g. 100-dimensional space, versus the 3-dimensional space we are familiar with). These kind of models typically involve parameters which are only known approximately. Hence to analyse and solve such models we must make sure that the algorithms used are stable with respect to small perturbations in the data. This can be achieved by combining ideas of Algebraic Geometry (the study of polynomial equations and their solutions) with Numerical Analysis. Charles Wampler (General Motors Research) and Andrew Sommese are speakers at the workshop, and pioneers in the unification of Numerical Analysis with the traditionally exact area of Geometry. Indeed it was applied problems in automotive design that originally inspired Sommese to start working on this unification. Topological ideas including continously deforming the equations into forms in which they could be easily solved; and representing higher dimensional solution sets by certain generic points on the solutions, were key in the generalization and creation of a new subject, representing the unification of Numerical Analysis and Algebraic Geometry: Numerical Algebraic Geometry. Because of the computational size of even simple problems, it is important to thoroughly understand qualitative, in other words geometric, features of the model. In this task the analysis of the algebraic structure of the problem using symbolic computation is essential. For example analysing and then exploiting the symmetry of a problem may easily transform a computationally impossible problem to a tractable one. For example, participant Elizabeth Mansfield, has worked with the UK Meteorological Office, on the problem of modeling storm systems, using symmetry to reduce the complexity of the systems. Although the geometry of the problem is the unifying theme of the workshop it is clear that in any real applications extensive symbolic and numeric computation is required. This implies that a lot of care is needed in the implementation of relevant algorithms and indeed a number of participants are deeply involved with developing new software which is helpful and useful in the analysis of the differential systems.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnologia (CONACYT).

BIRS Scientific Director, Nassif Ghoussoub
E-mail: birs-director[@]birs.ca
http://www.birs.ca/~nassif