Workshop on Geometry and Symmetry in Numerical Computation

Colorado State University, Fort Collins CO

Investigators

Abstract

Some of the most exciting developments in modern computational science have resulted from exploiting ideas in areas of mathematics not traditionally associated with numerical computation. Conversely, numerical techniques have been applied to solve computational problems arising in "non-traditional" fields. A good example is the fruitful interaction between computational mathematics and algebraic geometry. Singularity theory, which builds on ideas of algebraic geometry, has been embraced by computational scientists to compute paths of critical points in multi-parameter systems of differential and partial differential equations. Symmetries and group actions are used to create numerical methods for specific types of problems with significantly improved accuracy and stability properties. Techniques in algebraic geometry are also very useful for finding solutions of differential equations on manifolds, and are currently being applied to develop algorithms to compute decompositions of higher order tensors. On the other hand, numerical techniques for continuation, homotopy and symmetry provide the basis for methods in numerical algebraic geometry that are used to compute solution components of systems of polynomial equations. The ability to carry out the numerical decomposition of polynomial systems has yielded applications in mechanical engineering including the understanding and design of mechanisms that transmit, control, or constrain relative motion, robotics, control theory (pole placement), integer programming, and statistics. The development of hybrid exact/approximate methods for finding solutions of polynomial equations gives rise to issues of errors and stability that confront numerical analysts in many other contexts. The Workshop on geometry and symmetry in numerical computation will bring together experts from computational mathematics and algebraic geometry in order to explore and develop the potential in this rich interdisciplinary area. The program has been planned specifically to introduce and attract students and young investigators to this area. Each session will begin with an introductory lecture followed by four talks by leading experts. The introductory speakers will prepare a short "guide" describing some basic language and results that the audience can use during the invited talks. The one hour lectures themselves will be aimed towards an audience of advanced graduate students and researchers from different areas of mathematics. We expect the Workshop to break down disciplinary barriers and encourage cross-fertilization between researchers from algebraic geometry and numerical analysis. The lectures and discussion sections will encourage students and young researchers to become involved in the intersection of algebraic geometry and numerical analysis, and will provide stimulation and support for those already engaged in this activity. Potential outcomes range from improved methods to compute large complex physical systems governed by systems of partial differential equations, to advances in computational methods for general relativity, to new geometric methods for the analysis of large data sets, and to more efficient numerical methods for robotics and control.