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UID:20251104T070924EST-2615UV3HVE@132.216.98.100
DTSTAMP:20251104T120924Z
DESCRIPTION:On the distribution of poles of rational solutions to the Painl
 eve II hierarchy\n\nThe second Painlevé equation admits a sequence of rati
 onal solutions that can be written as logarithmic derivatives of special e
 valuations of Schur polynomials associated to staircase-shaped integer par
 titions. These polynomials are known in the literature as Vorobev-Yablonsk
 y polynomials\, and their algebraic structure and asymptotic properties ar
 e fairly well understood. In particular\, Bertola and Bothner found that t
 he squares of these polynomials can be written as partition functions ((Ha
 nkel determinants) of a non-hermitian matrix model on a contour and theref
 ore their asymptotic behaviour may be investigated by the Deift-Zhou nonli
 near steepest descent method for Riemann-Hilbert problems. The limiting di
 stribution of the poles of the rational solutions is encoded into the geom
 etry of the steepest descent contours\, and the boundary of the pole accum
 ulation region can be characterized as the locus of points where the assoc
 iated algebraic curve changes its genus. The second Painlevé equation is t
 he first member of the Painlevé II hierarchy of integrable nonlinear ODEs.
  All members of the PII hierarchy admit a sequence of rational solutions\,
  and they are all expressible in terms of special evaluations of Schur pol
 ynomials associated to staircase partitions. The matrix model representati
 on of Bertola and Bothner generalizes naturally to all rational solutions 
 of the PII hierarchy\, and the Deift-Zhou nonlinear steepest descent metho
 d offers an insight into the geometry of their limiting pole structure. In
  this talk I will introduce Schur polynomials associated to staircase part
 itions and explain their interpretation as matrix model partition function
 s. I will show how the Riemann-Hilbert method helps to explain the curious
  limiting shapes of the pole accumulation regions that were observed origi
 nally by Clarkson and Mansfield via numerical methods. The talk is based o
 n a joint work with M. Bertola and T. Bothner.\n
DTSTART:20180306T203000Z
DTEND:20180306T213000Z
LOCATION:Room 4336\, CA\, Pav. André-Aisenstadt\, 2920\, ch. de la Tour
SUMMARY:Ferenc Balogh\, John Abbott College et CRM
URL:/mathstat/channels/event/ferenc-balogh-john-abbott
 -college-et-crm-285494
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