BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Andrey Zotov (Steklov Mathematical Institute RAS) DTSTART:20210308T171000Z DTEND:20210308T180000Z DTSTAMP:20260410T120114Z UID:IntegrableSystemsAndGeometry/1 DESCRIPTION:Title: Characteristic Determinant and Manakov Triple for Dou ble Elliptic Integrable System\nby Andrey Zotov (Steklov Mathematical Institute RAS) as part of Elliptic Integrable Systems\n\n\nAbstract\nUsing the intertwining matrix of the IRF-Vertex correspondence we propose a det erminant representation for the generating function of the commuting Hamil tonians of the double elliptic integrable system. More precisely\, it is a ratio of the normally ordered determinants\, which turns into a single de terminant in the classical case. This gives expression for the spectral cu rve and the corresponding L-matrix\, which is obtained explicitly as a wei ghted average of the Ruijsenaars and/or Sklyanin type Lax matrices with th e weights as in the theta function series definition. By construction the L-matrix satisfies the Manakov triple representation instead of the Lax eq uation. Finally\, we discuss a possibility of the Dunkl-Cherednik construc tion type construction and its applications. The talk is mainly based on p aper 2010.08077. Some ideas and results of 2102.06853 are discussed as wel l.\n LOCATION:https://researchseminars.org/talk/IntegrableSystemsAndGeometry/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Simon Ruijsenaars (Leeds University) DTSTART:20210308T181000Z DTEND:20210308T190000Z DTSTAMP:20260410T120114Z UID:IntegrableSystemsAndGeometry/2 DESCRIPTION:Title: Painleve-Calogero correspondence: The elliptic 8-coup ling level\nby Simon Ruijsenaars (Leeds University) as part of Ellipti c Integrable Systems\n\n\nAbstract\nThis seminar is based on joint work wi th M.Noumi and Y.Yamada [1]. The 8-parameter elliptic Sakai difference Pai nleve equation [2] admits a Lax pair formulation. We sketch how a suitable specialization of one of the Lax equations gives rise to the time-indepen dent Schrodinger equation for the $BC_1$ 8-coupling relativistic Calogero- Moser Hamiltonian due to van Diejen[3]. This amounts to a generalization o f previous results concerning the Painleve-Calogero correspondence to the highest level of the two hierarchies. In both settings\, there exists a sy mmetry under the Weyl group of $E_8$ [2\,4]. \n\nReferences: \n[1] M.Noumi \, S.Ruijsenaars and Y.Yamada\, The elliptic Painleve Lax equation vs. van Diejen's 8-coupling elliptic Hamiltonian\, SIGMA 16 (2020)\, 063\, 16 pag es\, 1903.09738. \n[2] H.Sakai\, Rational surfaces associated with affine root systems and geometry of the Painleve equations\, Commun.Math.Phys. 22 0 (2001)\, 165-229. \n[3] J.F.van Diejen\, Integrability of difference Cal ogero-Moser systems\, J.Math.Phys. 35 (1994)\, 2983-3004. \n[4] S.Ruijsena ars\, Hilbert-Schmidt operators vs. Integrable systems of elliptic Caloger o-Moser type IV. The relativistic Heun (van Diejen) case\, SIGMA 11 (2015) \, 004\, 78 pages\, 1404.4392.\n LOCATION:https://researchseminars.org/talk/IntegrableSystemsAndGeometry/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Shlomo Razamat (Technion University) DTSTART:20210308T191000Z DTEND:20210308T200000Z DTSTAMP:20260410T120114Z UID:IntegrableSystemsAndGeometry/4 DESCRIPTION:Title: Three roads to the van Diejen model and beyond\nb y Shlomo Razamat (Technion University) as part of Elliptic Integrable Syst ems\n\n\nAbstract\nFirst\, we will discuss the relation between the E-stri ng theory and the $BC_1$ van Diejen model. In particular we will present t hree different constructions which lead to the same model. These different constructions can be thought of as integrable models formally correspondi ng to $A_{N=1}$\, $C_{N=1}$ and $(A_1)^{N=1}$ root systems. The constructi ons will provide three different types of Kernel functions for the $BC_1$ van Diejen model. Second\, we will argue that the three roads to the van D iejen model generalize to integrable models associated to $A_{N}$\, $C_{N} $ and $(A_1)^{N}$ root systems. We will give details of the construction o f the $A_{N}$ models.\n LOCATION:https://researchseminars.org/talk/IntegrableSystemsAndGeometry/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Gleb Aminov (Stony Brook University) DTSTART:20210309T031000Z DTEND:20210309T040000Z DTSTAMP:20260410T120114Z UID:IntegrableSystemsAndGeometry/6 DESCRIPTION:Title: DELL Systems and the Seiberg-Witten Prepotentials \nby Gleb Aminov (Stony Brook University) as part of Elliptic Integrable S ystems\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/IntegrableSystemsAndGeometry/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Matej Penciak (Northeastern University) DTSTART:20210309T171000Z DTEND:20210309T180000Z DTSTAMP:20260410T120114Z UID:IntegrableSystemsAndGeometry/7 DESCRIPTION:Title: Geometric Action-Angle Coordinates for the spin Ruijs enaars-Schneider system\nby Matej Penciak (Northeastern University) as part of Elliptic Integrable Systems\n\n\nAbstract\nIn this talk we will o ffer a description of a completion of the RS phase space as a moduli space of so-called framed spectral sheaves. An open subset of this moduli space is exactly given by line bundles supported on RS spectral curves\, but th e full space allows for more general sheaves on singular and non-reduced c urves. In this description\, the flows of the RS system are the natural fl ows coming from modifications of sheaves at a point. We begin with a paral lel story for the Calogero-Moser system which was worked out by David Ben- Zvi and Tom Nevins. We then move to the main result for the RS system wher e the identification with the RS phase space goes through the moduli space of multiplicative Higgs bundles. We end with some consequences of these r esults\, and some hopeful speculation about what a similar description of the Dell system would look like.\n LOCATION:https://researchseminars.org/talk/IntegrableSystemsAndGeometry/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Yegor Zenkevich (ITEP Moscow) DTSTART:20210309T181000Z DTEND:20210309T190000Z DTSTAMP:20260410T120114Z UID:IntegrableSystemsAndGeometry/8 DESCRIPTION:Title: Elliptic DIM algebra and elliptic integrable systems< /a>\nby Yegor Zenkevich (ITEP Moscow) as part of Elliptic Integrable Syste ms\n\n\nAbstract\nWe deomnstrate that elliptic Ruijsenaars-Schneider Hamil tonians can be\nunderstood as acting on certain set of symmetric polynomia ls\nE. Moreover\, these polynomials furnish a representation of elliptic\n Ding-Iohara-Miki (ell-DIM) algebra. It has been conjectured that the\npq-d uals of elliptic RS Hamiltonians are given by a trigonometric\ndegeneratio n of quantum Dell Hamiltonians introduced by Koroteev and\nShakirov (KS Ha miltonians). We show that the correct statement is in\nfact more subtle: t he eigenfunctions of the degeneration KS\nHamiltonians are conjugates of t he polynomials E with respect to Schur\nscalar product. \n\nFinally\, we a nalyze the structure of ell-DIM algebra and show that in\na remarkable way it is isomorphic to a direct sum of the ordinary\n(trigonometric) DIM alg ebra and an additional Heisenberg algebra. The\nisomorphism is inspired by the Bogolyubov transformation in the thermo\nfield double formalism. We e xplore the implications of the isomorphism\nfor representations and commut ing subalgebras of ell-DIM algebra.\n\nThis talk is based on the joint wor ks arXiv:2012.15352 with Mohamed\nGhoneim\, Can Kozcaz\, Kerem Kursun\, an d arXiv:2103.02508 with Andrei\nMironov and Alexey Morozov.\n LOCATION:https://researchseminars.org/talk/IntegrableSystemsAndGeometry/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Vyacheslav Spiridonov (JINR Dubna and HSE Moscow) DTSTART:20210309T191000Z DTEND:20210309T200000Z DTSTAMP:20260410T120114Z UID:IntegrableSystemsAndGeometry/9 DESCRIPTION:Title: From Elliptic Hypergeometric Integrals to Complex Hyp ergeometric Functions\nby Vyacheslav Spiridonov (JINR Dubna and HSE Mo scow) as part of Elliptic Integrable Systems\n\n\nAbstract\nElliptic hyper geometric integrals are top transcendental special functions of hypergeome tric type. They have found applications in 4d supersymmetric quantum field s theories (superconformal indices)\, in integrable systems (wave function s in quantum N-body problems) and 2d statistical mechanics (partition func tions). I will describe how these integrals can be degenerated in a chain of limits to complex hypergeometric functions related to the representatio n theory of SL(2\,C).\n LOCATION:https://researchseminars.org/talk/IntegrableSystemsAndGeometry/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrey Smirnov (UNC State University) DTSTART:20210308T011000Z DTEND:20210308T020000Z DTSTAMP:20260410T120114Z UID:IntegrableSystemsAndGeometry/10 DESCRIPTION:Title: Elliptic stable envelope for Hilbert scheme of point s in C^2\nby Andrey Smirnov (UNC State University) as part of Elliptic Integrable Systems\n\n\nAbstract\nIn this talk I discuss the elliptic sta ble envelope classes of torus fixed points in the Hilbert scheme of points in the complex plane.\nI describe the 3D-mirror self-duality of the ellip tic stable envelopes. The K-theoretic limits of these classes provide vari ous special bases in the space of symmetric polynomials\, including well k nown bases of Macdonald or Schur functions.  The mirror symmetry then tr anslates to new symmetries for these functions. In particular\, I outline a proof of conjectures by E.Gorsky and A.Negut on  ``Infinitesimal chang e of stable basis''\, which relate the wall R-matrices of the Hilbert sche me with the Leclerc-Thibon involution for $U_q(\\frak{gl}_b)$.\n LOCATION:https://researchseminars.org/talk/IntegrableSystemsAndGeometry/10 / END:VEVENT BEGIN:VEVENT SUMMARY:Junichi Shiraishi (University of Tokyo) DTSTART:20210308T021000Z DTEND:20210308T030000Z DTSTAMP:20260410T120114Z UID:IntegrableSystemsAndGeometry/11 DESCRIPTION:Title: Branching formula for q-Toda function of type B\ nby Junichi Shiraishi (University of Tokyo) as part of Elliptic Integrable Systems\n\n\nAbstract\nWe present a proof of the explicit formula for the asymptotically free eigenfunctions of the $B_N$ $q$-Toda operator which w as conjectured by Ayumu Hoshino and J.S. \nThis formula can be regarded as a branching formula from the $B_N$ $q$-Toda eigenfunction restricted to t he $A_{N-1}$ $q$-Toda eigenfunctions. \nThe proof is given by a contigulat ion relation of the $A_{N-1}$ Toda eigenfunctions \nand a recursion relati on of the branching coefficients.\n LOCATION:https://researchseminars.org/talk/IntegrableSystemsAndGeometry/11 / END:VEVENT END:VCALENDAR