B-series. From numerical analysis to algebra and differential geometry (Hans Munthe-Kaas, Bergen)

11.02.2016 16:15

B-series, named after John Butcher by Ernst Hairer and Gerhard Wanner in 1974, is an indispensable tool in the numerical analysis of time integration of dynamical systems. The starting point for understanding the importance of this tool is to note that Taylor series expansions of numerical integrators are immensely simplified by indexing the terms over the set of rooted trees, an idea going back to Arthur Cayley in the 1850s, and cleverly applied to numerical integration by Butcher in the 1960s. With the introduction of geometric (structure preserving) integration methods in the last two decades, B-series evolved into a powerful tool for understanding structure preservation of dynamical systems, with important contributions by the Geneva group (Hairer, Vilmart), and other groups around the world.

Recent developments show, however, that the topic of B-series is a universal mathematical structure with diverse applications also in very different areas of mathematical research. We will in this talk give an overview of the recent developments.

Some highlights:
- Recently B-series have been characterised geometrically (McLachlan, Modin, MK, Verdier). The result shows that classical B-series is tightly coupled with the geometry of Euclidean spaces.
- B-series have gradually been ‘married’ with Lie series on manifolds (Lie-Butcher series). This gives a rich algebraic and differential geometric structure which is still giving rise to many new results in the borderland between geometry and algebra. The structure of LB-series is fundamentally linked with the differential geometry of Lie groups.

I aim the talk at a general audience from computational mathematics without particular background in the field, nor in differential geometry or algebra.

Lieu

salle 623, attention horaire inhabituel. Séminaire d'analyse numérique

entrée libre