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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Numerical Generatingfunctionology: Counting with T
oeplitz Determinants\, Hayman-Admissibility\, and
the Wiener-Hopf-Factorization - Folkmar Bornemann
(Technische Universität München)
DTSTART;TZID=Europe/London:20191211T113000
DTEND;TZID=Europe/London:20191211T123000
UID:TALK135577AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/135577
DESCRIPTION:Counting related to representation theory and symm
etric functions can be framed as generating functi
ons given by Toeplitz determinants. Prime examples
are counting all permutations with no long increa
sing subsequence or lattice paths in last passage
percolation. Intricate scaling limits of those gen
erating functions have been used\, e.g.\, in the s
eminal work by Baik/Deift/Johansson\, to obtain as
ymptotic formulae in terms of random matrix theory
. In this talk\, we address the question whether g
enerating functions can be used to numerically ext
ract the counts in a mesoscopic regime where combi
natorial methods are already infeasible and the ra
ndom matrix asymptotics is still too inaccurate. T
he stable computation of the counts by means of co
mplex analysis is possible\, indeed\, and can be e
xplained by the theory of Hayman admissibility. As
a bonus track from complex analysis\, the numeric
al evaluation of the Toeplitz determinant itself h
as to be stabilized by a variant of the Borodin-Ok
ounkov formula based on the Wiener-Hopf factorizat
ion. This way\, we obtain\, e.g.\, exact 1135-digi
t counts in permutations of order 500 or\, by taki
ng Hayman&rsquo\;s famous generalization of Stirli
ng&rsquo\;s formula at face value\, a blazingly fa
st\, surprisingly robust and accurate numerical as
ymptotics.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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