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SUMMARY:Why B-series\, rooted trees\, and free algebras? - 2 - Kurusch Ebr
 ahimi-Fard (Norwegian University of Science and Technology)
DTSTART:20190709T103000Z
DTEND:20190709T113000Z
UID:TALK127030@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:"We regard Butcher&rsquo\;s work on the classification of nume
 rical integration methods as an impressive example that concrete problem-o
 riented work can lead to far-reaching conceptual results&rdquo\;. This quo
 te by Alain Connes summarises nicely the mathematical depth and scope of t
 he theory of Butcher&#39\;s B-series.  The aim of this joined lecture is t
 o answer the question posed in the title by drawing a line from B-series t
 o those far-reaching conceptional results they originated. Unfolding the p
 recise mathematical picture underlying B-series requires a combination of 
 different perspectives and tools from geometry (connections)\; analysis (g
 eneralisations of Taylor expansions)\, algebra (pre-/post-Lie and Hopf alg
 ebras) and combinatorics (free algebras on rooted trees). This summarises 
 also the scope of these lectures.  &nbsp\;  In the first lecture we will o
 utline the geometric foundations of B-series\, and their cousins Lie-Butch
 er series. The latter is adapted to studying differential equations on man
 ifolds. The theory of connections and parallel transport will be explained
 . In the second and third lectures we discuss the algebraic and combinator
 ial structures arising from the study of invariant connections. Rooted tre
 es 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 i
 ntegration schemes.
LOCATION:Seminar Room 1\, Newton Institute
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