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DTSTAMP:20260511T025901Z
DESCRIPTION:one-day workshop on March 18th at LACIM (5th floor of UQaM's PK
  building). \n\nEveryone is welcome to attend. The rough subject of the wo
 rkshop is representation theory of double-affine quantum groups\, Cheredni
 k algebras 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\nMartin
  Vrabec (Université de Montréal\, 10h00—11h00)\n\nTitle: Generalisations o
 f Macdonald–Ruijsenaars operators in external fields and a subalgebra of D
 AHA\n\nAbstract \n\nFirstly\, I will present a construction of commuting e
 lements in the double affine Hecke algebra (DAHA) of type GL_n whose actio
 n on symmetric polynomials leads to generalisations of Van Diejen's Macdon
 ald–Ruijsenaars system with an external Morse potential and other new inte
 grable q-difference operators. Secondly\, as another application of this c
 onstruction\, I will discuss a subalgebra A inside the DAHA that flatly de
 forms the crossed product of the symmetric group with the image of the Dri
 nfeld–Jimbo quantum group U_q(gl_n) under its q-oscillator representation.
  The algebra A reduces in the q = 1 limit to the degree zero part of the c
 orresponding rational Cherednik algebra. The degree zero part is a flat de
 formation of the crossed product of the symmetric group with a quotient of
  the universal enveloping algebra of gl_n\, and it is related to generalis
 ed Howe duality and to the Calogero–Moser integrable system in an external
  harmonic potential. This talk is based on a joint paper with Misha Feigin
  and a current work in progress.\n\n\n\nVasily Krylov (Harvard\, 11h00—12h
 00) –\n\nTitle: Graded traces on quantized Coulomb branches\n\nAbstract \n
 \nHiggs and Coulomb branches of quiver gauge theories form two important f
 amilies of Poisson varieties that are expected to be exchanged under so-ca
 lled 3D mirror symmetry. The representation theory of quantized Coulomb br
 anches is deeply connected with the enumerative geometry of Higgs branches
 . One important approach to studying modules over quantized Coulomb branch
 es is by analyzing their graded traces. Graded traces generalize the notio
 n of characters and are closely related to the q-characters introduced by 
 Frenkel and Reshetikhin. Any graded trace defines a solution of the D-modu
 le of graded traces introduced by Kamnitzer\, McBreen\, and Proudfoot.\n\n
  \n\nIn this talk\, I will discuss techniques that allow one to explicitly
  compute characters and graded traces of certain modules over quantized Co
 ulomb branches\, using the D-module of graded traces combined with analyti
 c methods. Time permitting\, I will explain how some of these results natu
 rally appear on the Higgs side\, leading to an explicit description of the
  D-module of graded traces for a quantized Coulomb branch via the geometry
  of the Higgs branch. We prove these results for ADE quivers and formulate
  some conjectures in the general case. Talk is based on joint works with D
 inkins\, Karpov\, Klyuev\, and Lance.\n\n\n\nDuncan Laurie (University of 
 Edinburgh\, 13h30—14h30)\n\nTitle: Representations of quantum toroidal alg
 ebras\n\nAbstract \n\nQuantum toroidal algebras are the ‘double affine’ ob
 jects within the quantum world. Their principal module category Ô is the n
 atural toroidal analogue of the finite-dimensional modules for quantum aff
 ine algebras.\n\n \n\nAfter introducing these algebras and discussing thei
 r structure\, we shall outline some recent results on their representation
  theory. These include a well-defined tensor product and monoidal structur
 e on Ô\, compatible with both Drinfeld polynomials and q-characters\, and 
 a meromorphic braiding by R-matrices.\n\n \n\nTime permitting\, I’ll brief
 ly mention work in progress with Théo Pinet exhibiting special subcategori
 es of Ô as monoidal categorifications of cluster algebras in type A.\n\n\n
 \nIlya Dumanski (MIT\, 14h30—15h30) –\n\nTitle: Pursuing equivariant sheav
 es on the double affine Grassmannian\n\nAbstract \n\nI will explain how to
  view equivariant coherent sheaves on the affine Grassmannian as sheaves o
 n an affine Grassmannian slice with a certain additional structure. Then I
  will speculate how one could try to generalize this to non-finite types\,
  using the coproduct for Coulomb branches\, which is to be defined. I will
  explain the relation of this construction to the category of Poisson shea
 ves\, and discuss the perverse coherent t-structure on this category.\n\n
 \n\nAlex Weekes (Université de Sherbrooke\, 16h00—17h00)\n\nTitle: Twisted
  Yangians and fixed points in the affine Grassmannian \n\nAbstract\n\n \n
 \nYangians are infinite-dimensional quantum groups\, which arise naturally
  as quantizations of enveloping algebras of current algebras.  Via the qua
 ntum duality principle\, they can also be viewed as quantizations of loop 
 groups.  This ultimately leads to a variety of other connections\, includi
 ng to affine Grassmannian slices and to the theory of Coulomb branches.\n
 \n \n\nTwisted Yangians are closely related algebras\, which have been stu
 died essentially the introduction of Yangians themselves\, and which relat
 e to symmetric pairs.  A natural question is whether they can also be view
 ed as quantizations of some geometry related to loop groups.  In this talk
 \, I will discuss joint work with Kang Lu and Weiqiang Wang where we addre
 ss this question.  We also discuss relations to affine Grassmannian slices
 \, potential connections to Coulomb branches\, and potential generalizatio
 ns such as to affine types.\n
DTSTART:20260310T190000Z
DTEND:20260310T190000Z
SUMMARY:Double affine day 2026
URL:https://www.mcgill.ca/mathstat/channels/event/double-affine-day-2026-37
 1868
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