New geometric and numeric tools for the analysis of differential equations (10w2134)


Elizabeth Mansfield (University of Kent)

(University of Western Ontario)

(University of Notre Dame)

Jukka Tuomela (University of Joensuu)


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

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).