Geometric integration: Numerical
methods for differential equations on manifolds, e.g. symplectic and
isospectral flows and the solution of Lie-type
equations by the method of Magnus, Fer and
Cayley series and their generalizations;
Highly oscillatory ordinary
differential equations and their numerical solution by asymptotic-numerical expansions, applications e.g. in electronic engineering. Extensions to delay-differential, differential-algebraic and partial DEs with highly oscillatory forcing terms;
Fast (i.e. O(n log n)) methods for the computation of orthogonal expansions, e.g. in orthogonal polynomials;
Spectral theory of highly oscillatory Fredholm operators, e.g. the Fox--Li equation in both its operator-theoretic and computational aspects;
Highly oscillatory integrals in one or
more dimensions, their asymptotics and quadrature;
Approximation of the matrix exponential
from a Lie algebra to a Lie group, mainly by linear-algebraic
techniques and by exploiting the Lie-algebraic structure;
Functional-differential equations with
proportional delay. The structure of attractors of the discretized
pantograph equation. Connections between the pantograph equation and orthogonal polynomials on the unit circle;
The theory of orthogonal
polynomials on the real line and on the complex unit circle -- in particular, lately, explicit representations of OPUCs;
Isospectral flows with Poisson structure, e.g. the Bloch--Iserles system, their features, integrability and Lie-algebraic representation;
Approximation of smooth functions in one or more dimensions by the means of Birkhoff expansions: both its theoretical properties and computation by highly oscillatory quadrature, polynomial subtraction and the hyperbolic cross..