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SUMMARY:Why B-series\, rooted trees\, and free algebras? - 1 - Hans Munthe
 -Kaas (Universitetet i Bergen)
DTSTART:20190708T090000Z
DTEND:20190708T100000Z
UID:TALK126961@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 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 pr
 ecise mathematical picture underlying B-series requires a combination of d
 ifferent perspectives and tools from geometry (connections)\; analysis (ge
 neralisations of Taylor expansions)\, algebra (pre-/post-Lie and Hopf alge
 bras) and combinatorics (free algebras on rooted trees). This summarises a
 lso the scope of these lectures. &nbsp\; In the first lecture we will outl
 ine the geometric foundations of B-series\, and their cousins Lie-Butcher 
 series. The latter is adapted to studying differential equations on manifo
 lds. The theory of connections and parallel transport will be explained. I
 n 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 ter
 ms in Taylor series and generalisations thereof. The final lecture will di
 scuss various applications of the theory in the numerical analysis of inte
 gration schemes.
LOCATION:Seminar Room 1\, Newton Institute
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