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CATEGORIES:Applied and Computational Analysis
SUMMARY:Attractive-repulsive equilibrium problems via orth
ogonal polynomials - Sheehan Olver (Imperial)
DTSTART;TZID=Europe/London:20230504T150000
DTEND;TZID=Europe/London:20230504T160000
UID:TALK198043AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/198043
DESCRIPTION:When particles interact\, say by attracting or rep
ulsing\, they tend to form nice distributions as t
he number of particles become large. Examples incl
ude both physical (electrons in a potential well)
and biological (flocking birds\, bacteria). Naive
simulation via differential equations proves insuf
ficient\, with computational cost becoming prohibi
tively expensive in more than one dimensions. Inst
ead\, we will introduce techniques based on a meas
ure minimisation reformulation using weighted orth
ogonal polynomials\, where by incorporating the co
rrect singularities of the distributions we can ra
pidly and accurately compute many such distributio
ns in arbitrary dimensions. This leads to high acc
uracy confirmation of open conjectures on gap form
ation (imagine a flock of birds with no density in
the middle). \n\nThese techniques involve underst
anding the relationship between orthogonal polynom
ials and singular integral (Hilbert\, Riesz\, and
log kernel) transforms\, which have wide reaching
consequences. We further explore connections to or
thogonal polynomials and random matrix theory\, th
e numerical solution of partial differential equat
ions using boundary integral reformulation\, and f
ractional differential equations.
LOCATION:Centre for Mathematical Sciences\, MR14
CONTACT:Matthew Colbrook
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