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

Arriving in Banff, Alberta Friday, August 13 and departing Sunday August 15, 2010

Organizers

Elizabeth Mansfield (University of Kent)

(University of Western Ontario)

(University of Notre Dame)

Jukka Tuomela (University of Joensuu)

Objectives

The aim of the workshop is to foster interaction between mathematicians, engineers, and scientists working with algorithmic geometrical methods for differential equations at the intersection of the three areas: geometrical approximation techniques; symbolic invariant calculus; discrete variational and discrete exterior calculus.

A key motivation underlying the objectives of the workshop is that problems from applications are now big enough and the inputs have enough numerical error that methods are needed that take full advantage of the geometry with the numerics always in sight. Symbolic tools should be used to the extent they can be, but in the end we need to develop tools that address problems that actually arise.

The expected outcomes include significant progress on creation and development of new algorithmic tools for the geometric analysis of differential equations and their approximations.

Specific goals of the workshop include sharing progress made, open problems, and technical set-ups, with a view to developing or applying tools that utilize the best ideas in the three areas. In particular the specialists of each subtopic will learn of the techniques and software being developed in other areas which will help them to tackle overlapping facets of the common problems.

The relevant technical background comes from numerical algebraic geometry, algorithmic algebraic geometry, differential algebra, symbolic invariant calculus, moving frames, and invariantization methods. The growing common realization that nontrivial computational advances in the above sub-areas, requires a combination of geometrical and numerical techniques, has already prompted initial contacts between people working in the above sub-areas.

While many conferences and workshops have considered geometric analysis of differential equations, and discrete exterior and variational calculus, none have considered them all with an underlying theme of approximation using numerical algebraic geometry.

The unique combination of themes exposes many new open problems and has a high potential for important breakthroughs. As examples we can cite combining Lie group integrators with numerical applications of moving frames; combining discrete variational methods with discrete exterior calculus in the context of Finite Element Methods; using Numerical Algebraic Geometry to characterize singular orbits of group actions and special solutions of PDE; determining hidden constraints and compatibility conditions using numerical approaches.

A special feature of the workshop will be three scheduled group discussions on the open problems, resulting from the overlaps between the three areas of the workshop. In addition a small number of educational survey talks will be followed by brainstorming.