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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Why B-series\, rooted trees\, and free algebras? -
2 - Hans Munthe-Kaas (Universitetet i Bergen)
DTSTART;TZID=Europe/London:20190712T150000
DTEND;TZID=Europe/London:20190712T160000
UID:TALK127231AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/127231
DESCRIPTION:"We regard Butcher&rsquo\;s work on the classifica
tion of numerical integration methods as an impres
sive example that concrete problem-oriented work c
an lead to far-reaching conceptual results&rdquo\;
. This quote by Alain Connes summarises nicely the
mathematical depth and scope of the theory of But
cher'\;s B-series. The aim of this joined lectu
re is to answer the question posed in the title by
drawing a line from B-series to those far-reachin
g conceptional results they originated. Unfolding
the precise mathematical picture underlying B-seri
es requires a combination of different perspective
s and tools from geometry (connections)\; analysis
(generalisations of Taylor expansions)\, algebra
(pre-/post-Lie and Hopf algebras) and combinatoric
s (free algebras on rooted trees). This summarises
also the scope of these lectures. \; In the
first lecture we will outline the geometric founda
tions of B-series\, and their cousins Lie-Butcher
series. The latter is adapted to studying differen
tial equations on manifolds. The theory of connect
ions and parallel transport will be explained. In
the second and third lectures we discuss the algeb
raic and combinatorial structures arising from the
study of invariant connections. Rooted trees play
a particular role here as they provide optimal in
dex sets for the terms in Taylor series and genera
lisations thereof. The final lecture will discuss
various applications of the theory in the numerica
l analysis of integration schemes.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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