Rahbar Virk

I work(ed) primarily in representation theory, with a bias for topological and geometric methods.

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Publications

• Equivariant motives and geometric representation theory
  with W. Soergel and M. Wendt
• On Euler-Poincare characteristics,
  Comptes Rendus 353, Issue 5 (2015), 449-453.
• Some geometric facets of the Langlands correspondence for real groups,
  Bull. LMS 47 (2) (2015), 225-232.
• A remark on braid group actions on coherent sheaves,
  Osaka J. of Math. 51 (2014), 719-728.
• Affine and degenerate affine BMW algebras: actions on tensor space,
  with Z. Daugherty and A. Ram, Selecta Math. 19 (2013), 611-653.
• Affine and degenerate affine BMW algebras: the center,
  with Z. Daugherty and A. Ram, Osaka J. of Math. 51 (2014), 257-283.
• Derived equivalences and sl2 categorifications for Uq(gln),
  J. of Algebra 346 (2011), 82-100.

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Miscellaneous articles
(most of these are available on the arXiv, but will probably never appear in an official publication)

• Motivic splitting principle
• Chow groups and equivariant geometry
• Tate classes, equivariant geometry and purity
  This is a complement to `Chow groups and equivariant geometry' above.
• Graded tensoring and crystals
• On Hall-Littlewood polynomials
• On the geometric Hecke algebra,
  Enveloping Algebras and Geometric Representation theory,
  Oberwolfach Reports 9, Issue 1 (2012).
• A note on Hecke patterns in Category O
• A remark on some bases in the Hecke algebra
• Derived equivalences and category O,
  PhD thesis, UW-Madison, 2011.

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Incomplete/scratch work

• Motivic Leray-Hirsch
  This is closely related to the motivic splitting principle.
• Discussion around the motivic equivariant derived category
  There is an inaccuracy in this discussion - to obtain duality one needs to restrict to constructible motives. Note that Beilinson motives satisfy etale descent (Theorem 14.3.4 here). Further, the exact same constructions, with a minor modification, also yield a motivic derived category for algebraic stacks (replace the forgetful functor by pullback to an atlas; replace induction by pushforward). This is all of course with rational coefficients. Similar constructions can be done away from "interesting torsion primes", but by definition that doesn't capture any interesting torsion information.
• The Hodge conjecture is trivial in equivariant geometry
  Some silliness related to the note`Chow groups and equivariant geometry'.
• Basic observation: alternate proof
• Sheaf theoretic Kunneth
• Summands of Bott-Samelson motives
• Cohomology of homogeneous varieties is Tate
• Equivariant nearby cycles [unstable]
• Misc. discussion around smooth base change and induction equivalence
• Discussion around smooth morphisms and the equivariant derived category
  This is related to the equivariant part of `On Euler-Poincare characteristics' above.
• Discussion around the cohomology of the complement of a normal crossings divisor
• Notes on motivic models for category O
• Some geometric facets of the Langlands correspondence for real groups
  This is an ugly version of its namesake above. The primary difference is that appropriate changes (technical and linguistic) are made so that the arguments work both in the setting of mixed Hodge modules and mixed l-adic sheaves. There are some interesting questions (related to `Frobenius semisimplicity') that arise in the l-adic setting. There is no discussion of convolution, the Hecke algebra, or formality in this version.
• Projective functors and free pro-unipotent sheaves
• Computing extensions between sheaves (a la Beilinson-Ginzburg-Soergel)

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Notes from conferences/workshops

• Proudfoot Talbot (Oregon, 2012)
• Geometric Langlands (Freiburg, 2012)
• Enveloping algebras and geometric representation theory (Oberwolfach, 2012)
• Algebraic Lie theory (Newton Institute, 2009)

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Notes from various talks

• Rook placements and Jordan forms (Martha at CU-Boulder, 2013)
• Satake isomorphisms and affine Beilinson-Bernstein localization (Masoud at Freiburg, 2012)
• F to C ala [BBD] (Soergel at Freiburg, 2012)
• Cotangent complexes of moduli spaces and symplectic structures (Z. Zhang at Freiburg, 2012)
• Geometric Langlands for GL_1 (Oliver at Freiburg, 2012)
• Geometric Langlands and Physics (E. Scheidegger at Freiburg, 2012)
• Generic vanishing fails for singular varieties in characteristic p (S. Kovacs at CSU, 2012)
• Kazhdan-Lusztig cells and Calegoro-Moser space (Rouquier at UC-Riverside, 2012)
• Geometric realizations of quantum groups (Yiqiang Li at NCSU, 2012)
• The Hodge theorem as a derived self intersection (Andrei at Buchweitz's birthday conference, 2012)
• Index theory for elliptic operators (E. Van Erp at CU-Boulder, 2012)
• Tensor triangulated geometry (P. Balmer at ANU, 2009)

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Notes for/from courses

• Topology
• Algebra
• Algebraic number theory
• Complex analysis

Some notes that I wrote as a student (graduate and undergraduate) can be found here.

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Last updated: 24 August, 2019