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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Why B-series\, rooted trees\, and free algebras? -
2 - Kurusch Ebrahimi-Fard (Norwegian University o
f Science and Technology)
DTSTART;TZID=Europe/London:20190709T113000
DTEND;TZID=Europe/London:20190709T123000
UID:TALK127030AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/127030
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 lect
ure is to answer the question posed in the title b
y drawing a line from B-series to those far-reachi
ng conceptional results they originated. Unfolding
the precise mathematical picture underlying B-ser
ies requires a combination of different perspectiv
es and tools from geometry (connections)\; analysi
s (generalisations of Taylor expansions)\, algebra
(pre-/post-Lie and Hopf algebras) and combinatori
cs (free algebras on rooted trees). This summarise
s also the scope of these lectures. \; In t
he first lecture we will outline the geometric fou
ndations of B-series\, and their cousins Lie-Butch
er series. The latter is adapted to studying diffe
rential equations on manifolds. The theory of conn
ections and parallel transport will be explained.
In the second and third lectures we discuss the al
gebraic and combinatorial structures arising from
the study of invariant connections. Rooted trees p
lay a particular role here as they provide optimal
index sets for the terms in Taylor series and gen
eralisations thereof. The final lecture will discu
ss various applications of the theory in the numer
ical analysis of integration schemes.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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