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SUMMARY:Discrete integrable systems and orthogonal polynomials from contin
 ued fractions in function fields
DTSTART;VALUE=DATE-TIME:20230623T071500Z
DTEND;VALUE=DATE-TIME:20230623T074500Z
DTSTAMP;VALUE=DATE-TIME:20260923T003619Z
UID:indico-contribution-62@cern.ch
DESCRIPTION:Speakers: Prof. HONE\, Andrew (University of Kent)\nIt has bee
 n known for some years that there are deep connections between continued f
 ractions and integrable systems: one of the earliest examples appears in M
 oser's work on solutions of the finite Kac-van Moerbeke (or Volterra) latt
 ice\, but there are many other examples e.g. in recurrence coefficients fo
 r orthogonal polynomials arising in random matrix theory\, which satisfy (
 discrete and continuous) Painleve equations. In this talk we describe our 
 recent work on continued fractions of Jacobi type (J-fractions) for a cert
 ain family of functions on hyperelliptic curves\, based on a construction 
 of van der Poorten related to Somos-4 sequences (corresponding to genus g=
 1). We explain how to interpret van der Poorten's result for all genera g\
 , in terms of a family of discrete integrable systems. This not only leads
  to an elementary derivation of Hankel determinant formulae for Somos-4 fo
 und by Chang\, Hu & Xin\, but also provides a natural construction of high
 er genus analogues of Chebyshev polynomials\, and produces genus g solutio
 ns of the infinite Toda lattice. The J-fractions naturally arise from even
  models of hyperelliptic curves\, but recent work with John Roberts and Po
 l Vanhaecke also reveals another family of discrete integrable systems ass
 ociated with continued fraction of Stieltjes-type (S-fractions) and odd mo
 dels of the same curves\, which yield solutions of the infinite Volterra l
 attice. In particular\, we find that the g=2 S-fraction corresponds to an 
 integrable map with 2 degrees of freedom\, discovered in a recent classifi
 cation of 4D maps with Lagrangian structure by Gubbiotti\, Joshi\, Viallet
  & Tran. Other 4D integrable maps found by the latter authors turn out to 
 be connected with the modified Volterra lattice: our additional joint work
  with Federico Zullo has revealed that they arise from the same genus 2 S-
 fraction\, but as BTs in the sense of Sklyanin.\n\nhttp://indico.fuw.edu.p
 l/contributionDisplay.py?contribId=62&sessionId=6&confId=67
LOCATION:Faculty of Physics\, University of Warsaw Lecture hall: 0.06
URL:http://indico.fuw.edu.pl/contributionDisplay.py?contribId=62&sessionId
 =6&confId=67
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