Presented by:
Kurusch Ebrahimi-Fard
Date:
Tuesday 9th July 2019 - 11:30 to 12:30
Venue:
INI Seminar Room 1
Event:
Abstract:
"We regard Butcher’s work on the classification of
numerical integration methods as an impressive example that concrete
problem-oriented work can lead to far-reaching conceptual results”. This quote
by Alain Connes summarises nicely the mathematical depth and scope of the
theory of Butcher's B-series.
The aim of this joined lecture is to answer the question
posed in the title by drawing a line from B-series to those far-reaching
conceptional results they originated. Unfolding the precise mathematical
picture underlying B-series requires a combination of different perspectives
and tools from geometry (connections); analysis (generalisations of Taylor
expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free
algebras on rooted trees). This summarises also the scope of these lectures.
In the first lecture we will outline the geometric
foundations of B-series, and their cousins Lie-Butcher series. The latter is
adapted to studying differential equations on manifolds. The theory of
connections and parallel transport will be explained. In the second and third
lectures we discuss the algebraic and combinatorial structures arising from the
study of invariant connections. Rooted trees play a particular role here as
they provide optimal index sets for the terms in Taylor series and
generalisations thereof. The final lecture will discuss various applications of
the theory in the numerical analysis of integration schemes.