My main research is on various aspects of numerical ordinary differential equations but I am also interested in highly oscillatory phenomena, functional equations, approximation theory, orthogonal polynomials and expansions, numerical partial differential equations and integral equations.
In the last few years I have been working more specifically on
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;
Design and stability of PDEs of evolution with an emphasis on operatorial splitting methods, in particular the symmetric Zassenhaus splitting and its applications to the Schrödinger equation in the semiclassical regime and to other PDEs of quantum mechanics.
Structured differentiation matrices, e.g. skew-symmetric matrices approximating the first derivative operator;
Magnus and Fer expansions for Sturm–Liouville spectral problems;
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, in particular expansions 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 theory, asymptotic expansion and quadrature methods;
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 Lie–Poisson structure, e.g. double-bracket and Bloch–Iserles systems, 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.
Orthogonal polynomials with respect to complex-valued measures.
The above list is incomplete and, perhaps most importantly, it is (almost by definition) out of date. Yet, it provides a general impression on the happenings in my frontal cortex.
Some of this research is done on my own, much of it with colleagues and friends: one of great joys of doing mathematics is working with inspiring and brilliant people!