The following are remarks made during a panel discussion held during the workshop on the future prospects, problems, and opportunities in the applications of group theory to numerical methods and discrete systems.
A basic issue is to relate the continuous symmetries in physics with the discrete symmetries in numerics. In quantum mechanics, the operator differential equations are difficult to solve, and so one wishes to approximate by applying finite elements or linear interpolation. There is only one discretization that preserves the Heisenberg algebra. Generalizing this to other groups and resolving the operator ordering problems remains a significant challenge.
The use of contact and higher order symmetries for differential equations and integrability is well developed. A significant challenge how to properly develop the methods of contact and higher order symmetries for difference equations.
The goal of an opportunistic numerical analyst is to borrow from any area of mathematics. The specific purpose is to unify the understanding of differential equations through both analysis and numerics. The challenge is to compute while maintaining as much structure in the problem as possible. No-go theorems that prevent one from discretizing while trying to maintain too much structure must be kept in mind. The need is for a synthesis of analysis and geometry with computation. Further work will focus on on maintaining inequalities, maximum principles, asymptotics, stability, and so on.
A particular challenge is the study of discrete problems in physics and biology. Symmetries are very important for their analysis and solution. The Hamiltonian structure is not completely determined, and certain ambiguities in the formulation can be resolved by use of symmetry.
Use of symbolic algebra in numerics is a great opportunity. The Lie methods for determining symmetries of partial differential equations have been one of the great successes of computer algebra and serve as a benchmark for evaluating systems and algorithms. Generalizations lead into the study of overdetermined systems of partial differential equations and variational methods, in which some progress has been made but challenges are still there. The Cartan and moving frame methods for studying equivalence problems are beginning to be addressed by the computer algebra community. The goal is to develop algebraic tools to combine geometry and analysis. Continuous symmetries on discrete lattices and geometric integration is an important new direction of research.
A key challenge is to incorporate infinite-dimensional groups into numerical algorithms for dynamical systems on manifolds. Cartan's classification of the primitive transitive groups leads, in order of the extent of analysis, to the classes of general diffeomorphisms, symplectic maps, volume-preserving maps, contact transformations, conformal symplectic maps and conformal volume-preserving maps. Non-primitive groups have not been classified and only a couple of papers have studied their discretizations. The classification and discretization of semigroups and symmetric spaces is even more incomplete. Finally, Hopf algebras arise in the algebraic description of Runge-Kutta methods. Other integration methods have hardly been studied from this point of view.
There is a need to properly evaluate the success of the series on meetings in symmetries and integrability of differential equations (SIDE) and numerical methods (ICDEA) Combining the two communities has led to new collaborations and new directions of research.