BEGIN:VCALENDAR
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CALSCALE:GREGORIAN
X-WR-CALNAME:Cascade Lectures in Combinatorics
X-WR-TIMEZONE:Pacific Time (US & Canada)
BEGIN:VEVENT
DTSTAMP:20260722T201151Z
UID:tag:localist.com\,2008:EventInstance_44561499797652
DTSTART:20231104T163000Z
DTEND:20231104T230000Z
DESCRIPTION:The Cascade Lectures in Combinatorics is a series of combinator
 ial workshops held each fall and each spring on a Saturday in the Pacific 
 Northwest Region (or sometimes on Zoom)\, funded by the National Science F
 oundation.  The workshop this fall\, the 5th installment of the Cascade Le
 ctures in Combinatorics\, will be held as an online meeting\, held on Zoom
 \, on Saturday\, November 4\, 2023.  It will include four one hour invited
  talks as well as online social gatherings in the form of coffee breaks an
 d a lunch break. Additional information is on the main CALICO web site at 
 https://pages.uoregon.edu/plhersh/CALICO/ \n\n \n\nTalk titles and abstrac
 ts:\n\n \n\nChris Eur: A Tale of two rings\n\nAbstract: A complex projecti
 ve manifold carries two well-studied rings\, namely\, the cohomology ring 
 and the Grothendieck K-ring of vector bundles. For toric varieties\, these
  have polyhedral descriptions\, as the polytope algebra and the algebra of
  piecewise polynomials. For special toric varieties\, we show an exception
 al isomorphism between these two rings\, different from the classical Hirz
 ebruch-Riemann-Roch theorem\, and discuss its utility in combinatorial con
 texts.\n\nJoint works with Andrew Berget\, Alex Fink\, June Huh\, Matt Lar
 son\, Hunter Spink\, and Dennis Tseng.\n\nSergey Fomin: Incidences and til
 ings\n\nAbstract: We show that various classical theorems of real/complex 
 linear incidence geometry\, such as the theorems of Pappus\, Desargues\, M
 öbius\, and so on\, can be interpreted as special cases of a single "mast
 er theorem" that involves an arbitrary tiling of a closed oriented surface
  by quadrilateral tiles. This yields a general mechanism for producing new
  incidence theorems and generalizing the known ones.\n\nThis is joint work
  with Pavlo Pylyavskyy.\n\nMaria M. Gillespie: Battery-powered tableaux\, 
 Springer theory\, and the Delta conjecture\n\nAbstract: We present new for
 mulas for the t=0 specialization of the polynomials involved in the Delta 
 conjecture. One is a combinatorial Schur expansion in terms of "battery-po
 wered tableaux"\, and its related formulation is simply an adjoint Schur o
 perator applied to a Hall-Littlewood polynomial. This generalizes to give 
 formulas for the Frobenius character of the cohomology ring of the "Delta-
 Springer varieties" defined by Griffin\, Levinson\, and Woo\, which fit in
 to the generalized Springer theory of Borho and MacPherson. We will descri
 be how this generalized theory of partial resolutions of nilpotent varieti
 es leads to our results\, and state some more general conjectural formulas
  towards a Schur expansion for the Delta conjecture at the end.\n\nThis is
  joint work with Sean Griffin.\n\nSheila Sundaram: Stirling representation
 s\, supersolvable matroids and Koszul duality\n\nAbstract: (Joint work wit
 h Ayah Almousa and Vic Reiner)\n\nThe unsigned Stirling numbers c(n\,k) of
  the first kind give the Hilbert function for two algebras associated to t
 he hyperplane arrangement in type A\, the Orlik-Solomon algebra and the gr
 aded Varchenko-Gelfand algebra. Both algebras carry symmetric group action
 s with a rich structure\, and have been well studied by topologists\, alge
 braists and combinatorialists: the first coincides with the Whitney homolo
 gy of the partition lattice\, and the second with a well known decompositi
 on (Thrall's decomposition\, giving the higher Lie characters) of the univ
 ersal enveloping algebra of the free Lie algebra. In each case the graded 
 representations have dimension c(n\,k).\n\nBoth these algebras are example
 s of Koszul algebras\, for which the Priddy resolution defines a group-equ
 ivariant dual Koszul algebra. Now the Hilbert function is given by the Sti
 rling numbers S(n\,k) of the second kind\, and hence the Koszul duality re
 lation defines representations of the symmetric group whose dimensions are
  the numbers S(n\,k). Investigating this observation led to the realisatio
 n that this situation generalises to all supersolvable matroids. The Koszu
 l duality recurrence is shown to have interesting consequences.\n\nFor the
  resulting group representations\, it implies the existence of branching r
 ules which\, in the case of the braid arrangement\, specialise by dimensio
 n to the classical enumerative recurrences satisfied by the Stirling numbe
 rs of both kinds. It also implies representation stability in the sense of
  Church and Farb. The associated Koszul dual representations appear to hav
 e other properties that are more mysterious\; for example\, in the case of
  the braid arrangement\, the Stirling representations of the Koszul dual a
 re sometimes tantalisingly close to being permutation modules. I will ende
 avour to give a flavour of these phenomena in the talk.
LOCATION:
SUMMARY:Cascade Lectures in Combinatorics
URL;VALUE=URI:https://calendar.uoregon.edu/event/cascade_lectures_in_combin
 atorics
CATEGORIES:Lectures & Presentations
CATEGORIES:Lecture
CATEGORIES:Presentation
CATEGORIES:Discussion
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