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SUMMARY:Attractive-repulsive equilibrium problems via orthogonal polynomia
 ls - Sheehan Olver (Imperial)
DTSTART:20230504T140000Z
DTEND:20230504T150000Z
UID:TALK198043@talks.cam.ac.uk
CONTACT:Matthew Colbrook
DESCRIPTION:When particles interact\, say by attracting or repulsing\, the
 y tend to form nice distributions as the number of particles become large.
  Examples include both physical (electrons in a potential well) and biolog
 ical (flocking birds\, bacteria). Naive simulation via differential equati
 ons proves insufficient\, with computational cost becoming prohibitively e
 xpensive in more than one dimensions. Instead\, we will introduce techniqu
 es based on a measure minimisation reformulation using weighted orthogonal
  polynomials\, where by incorporating the correct singularities of the dis
 tributions we can rapidly and accurately compute many such distributions i
 n arbitrary dimensions. This leads to high accuracy confirmation of open c
 onjectures on gap formation (imagine a flock of birds with no density in t
 he middle). \n\nThese techniques involve understanding the relationship be
 tween orthogonal polynomials and singular integral (Hilbert\, Riesz\, and 
 log kernel) transforms\, which have wide reaching consequences. We further
  explore connections to orthogonal polynomials and random matrix theory\, 
 the numerical solution of partial differential equations using boundary in
 tegral reformulation\, and fractional differential equations. 
LOCATION:Centre for Mathematical Sciences\, MR14
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