BEGIN:VCALENDAR VERSION:2.0 PRODID:-//CERN//INDICO//EN BEGIN:VEVENT SUMMARY:Rational interpolation/approximation and integrability DTSTART;VALUE=DATE-TIME:20230623T074500Z DTEND;VALUE=DATE-TIME:20230623T081500Z DTSTAMP;VALUE=DATE-TIME:20260410T054325Z UID:indico-contribution-6-2@cern.ch DESCRIPTION:Speakers: Prof. DOLIWA\, Adam (University of Warmia and Mazury )\nIt is well known that the rational (or Padé) approximants are closely related with the discrete-time Toda equation. The structural connection wi th orthogonal polynomials provides a link between the theory of integrable systems and various classical results of applied mathematics and numerica l analysis. Inspired by recent advances on symmetry and integrability of d ifference equations I would like to disuss several generalizations of the relation (some of them already known):\n\n* the role of Hirota's discrete KP system\;\n* non-commutative versions of the classical results\;\n* appr oximation as a confluent limit of the interpolation.\n\n[1] A. Doliwa\, A. Siemaszko\, Integrability and geometry of the Wynn recurrence\, Numer. Al gorithms doi: 10.1007/s11075-022-01344-5\n\n[2] A. Doliwa\, A. Siemaszko\, Hermite-Padé approximation and integrability\, arXiv:2201.06829\n\n[3] A . Doliwa\, Non-commutative Hermite-Padé approximation and integrability\, Lett. Math. Phys. 112 68 (2022) doi: 10.1007/s11005-022-01560-z\n\n[4] A. Doliwa\, Non-autonomous multidimensional Toda system and multiple interpo lation problem\, J. Phys. A: Math. Theor. 55 (2022) 505202 (17 pp.) doi: 1 0.1088/1751-8121/acad4d\n\nhttp://indico.fuw.edu.pl/contributionDisplay.py ?contribId=2&sessionId=6&confId=67 LOCATION:Faculty of Physics\, University of Warsaw Lecture hall: 0.06 URL:http://indico.fuw.edu.pl/contributionDisplay.py?contribId=2&sessionId= 6&confId=67 END:VEVENT BEGIN:VEVENT 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:20260410T054325Z UID:indico-contribution-6-62@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 END:VEVENT BEGIN:VEVENT SUMMARY:TCD maps DTSTART;VALUE=DATE-TIME:20230623T081500Z DTEND;VALUE=DATE-TIME:20230623T084500Z DTSTAMP;VALUE=DATE-TIME:20260410T054325Z UID:indico-contribution-6-56@cern.ch DESCRIPTION:Speakers: Mr. AFFOLTER\, Niklas (TU Berlin)\nA TCD map is a ma p from a triple crossing diagrams to projective space\, satisfying an inci dence requirement. We introduce dynamics on TCD maps based on Menelaus the orem and show multi-dimensionally consistency with Desargues theorem. To each TCD map we associate a hierarchy of dimer models\, which provides loc al and global invariants. Conveniently\, TCD maps include as special cases a large number of known maps\, including Q-nets\, Line complexes\, Darbou x maps\, Desargues maps\, dSKP lattices\, t-embeddings\, T-graphs\, the pe ntagram map\, integrable cross-ratio systems and others. Additionally\, we show how to relate the resistor subvariety of the dimer model to dBKP red uctions and the Ising subvariety to dCKP reductions.\n\nhttp://indico.fuw. edu.pl/contributionDisplay.py?contribId=56&sessionId=6&confId=67 LOCATION:Faculty of Physics\, University of Warsaw Lecture hall: 0.06 URL:http://indico.fuw.edu.pl/contributionDisplay.py?contribId=56&sessionId =6&confId=67 END:VEVENT END:VCALENDAR