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DTSTAMP:20260511T025906Z
DESCRIPTION:Everyone is welcome to attend. The rough subject of the worksho
 p is representation theory of double-affine quantum groups\, Cherednik alg
 ebras and Coulomb branches. Talks will start at 10h00 in room PK-5115. The
  afternoon talks will take place in room PK-5675.  \n\n\n	\n\n	Martin Vrabec
  (Université de Montréal\, 10h00—11h00)\n\n	Title: Generalisations of Macdo
 nald–Ruijsenaars operators in external fields and a subalgebra of DAHA\n\n
 	Abstract \n\n	Firstly\, I will present a construction of commuting elements
  in the double affine Hecke algebra (DAHA) of type GL_n whose action on sy
 mmetric polynomials leads to generalisations of Van Diejen's Macdonald–Rui
 jsenaars system with an external Morse potential and other new integrable 
 q-difference operators. Secondly\, as another application of this construc
 tion\, I will discuss a subalgebra A inside the DAHA that flatly deforms t
 he crossed product of the symmetric group with the image of the Drinfeld–J
 imbo quantum group U_q(gl_n) under its q-oscillator representation. The al
 gebra A reduces in the q = 1 limit to the degree zero part of the correspo
 nding rational Cherednik algebra. The degree zero part is a flat deformati
 on of the crossed product of the symmetric group with a quotient of the un
 iversal enveloping algebra of gl_n\, and it is related to generalised Howe
  duality and to the Calogero–Moser integrable system in an external harmon
 ic potential. This talk is based on a joint paper with Misha Feigin and a 
 current work in progress.\n\n\n\n	\n\n	Vasily Krylov (Harvard\, 11h15—12h15)
  –\n\n	Title: Graded traces on quantized Coulomb branches\n\n	Abstract \n\n	H
 iggs and Coulomb branches of quiver gauge theories form two important fami
 lies of Poisson varieties that are expected to be exchanged under so-calle
 d 3D mirror symmetry. The representation theory of quantized Coulomb branc
 hes is deeply connected with the enumerative geometry of Higgs branches. O
 ne important approach to studying modules over quantized Coulomb branches 
 is by analyzing their graded traces. Graded traces generalize the notion o
 f characters and are closely related to the q-characters introduced by Fre
 nkel and Reshetikhin. Any graded trace defines a solution of the D-module 
 of graded traces introduced by Kamnitzer\, McBreen\, and Proudfoot.\n	 \n\n
 	In this talk\, I will discuss techniques that allow one to explicitly comp
 ute characters and graded traces of certain modules over quantized Coulomb
  branches\, using the D-module of graded traces combined with analytic met
 hods. Time permitting\, I will explain how some of these results naturally
  appear on the Higgs side\, leading to an explicit description of the D-mo
 dule of graded traces for a quantized Coulomb branch via the geometry of t
 he Higgs branch. We prove these results for ADE quivers and formulate some
  conjectures in the general case. Talk is based on joint works with Dinkin
 s\, Karpov\, Klyuev\, and Lance.\n\n\n\n	\n\n	Duncan Laurie (University of E
 dinburgh\, 13h30—14h30)\n\n	Title: Representations of quantum toroidal alge
 bras\n\n	Abstract \n\n	Quantum toroidal algebras are the ‘double affine’ obj
 ects within the quantum world. Their principal module category Ô is the na
 tural toroidal analogue of the finite-dimensional modules for quantum affi
 ne algebras.\n\n	 \n\n	After introducing these algebras and discussing their
  structure\, we shall outline some recent results on their representation 
 theory. These include a well-defined tensor product and monoidal structure
  on Ô\, compatible with both Drinfeld polynomials and q-characters\, and a
  meromorphic braiding by R-matrices.\n\n	 \n\n	Time permitting\, I’ll briefl
 y mention work in progress with Théo Pinet exhibiting special subcategorie
 s of Ô as monoidal categorifications of cluster algebras in type A.\n\n\n
 \n	\n\n	Ilya Dumanski (MIT\, 14h30—15h30) –\n\n	Title: Pursuing equivariant s
 heaves on the double affine Grassmannian\n\n	Abstract \n\n	I will explain ho
 w to view equivariant coherent sheaves on the affine Grassmannian as sheav
 es on an affine Grassmannian slice with a certain additional structure. Th
 en I will speculate how one could try to generalize this to non-finite typ
 es\, using the coproduct for Coulomb branches\, which is to be defined. I 
 will explain the relation of this construction to the category of Poisson 
 sheaves\, and discuss the perverse coherent t-structure on this category.
 \n\n\n\n	\n\n	Alex Weekes (Université de Sherbrooke\, 16h00—17h00)\n\n	Title:
  Twisted Yangians and fixed points in the affine Grassmannian \n\n	Abstract
 \n	 \n\n	Yangians are infinite-dimensional quantum groups\, which arise natu
 rally as quantizations of enveloping algebras of current algebras.  Via th
 e quantum duality principle\, they can also be viewed as quantizations of 
 loop groups.  This ultimately leads to a variety of other connections\, in
 cluding to affine Grassmannian slices and to the theory of Coulomb branche
 s.\n	 \n\n	Twisted Yangians are closely related algebras\, which have been s
 tudied essentially the introduction of Yangians themselves\, and which rel
 ate to symmetric pairs.  A natural question is whether they can also be vi
 ewed as quantizations of some geometry related to loop groups.  In this ta
 lk\, I will discuss joint work with Kang Lu and Weiqiang Wang where we add
 ress this question.  We also discuss relations to affine Grassmannian slic
 es\, potential connections to Coulomb branches\, and potential generalizat
 ions such as to affine types.ctions to Coulomb branches\, and potential ge
 neralizations such as to affine types.\n\n
DTSTART:20260317T141500Z
DTEND:20260317T141500Z
SUMMARY:Double affine day 2026
URL:https://www.mcgill.ca/mathstat/channels/event/double-affine-day-2026-37
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