Symplectic singularities and symplectic duality in examples, Spring 2025.

Time: 12–1:15 pm, MF, at Science Center (1 Oxford St), 507.

Office hours: 1:30-2:30, MF, at Science Center (1 Oxford St), 535.

Problem Sets:

  • PSet 1.
  • PSet 2.
  • PSet 3.
  • PSet 4.
  • PSet 5.

    Schedule:

  • 1/27: Symplectic singularities -- definitions, motivations and first examples, notes.
  • 1/31: Nilpotent cone, Springer resolution, Slodowy varieties -- basic properties, notes.
  • 2/3: Poisson and symplectic structures on our varieties, Hamiltonian actions and Hamiltonian reduction, notes.
  • 2/7: Symplectic structure on Slodowy varieties, Gan and Ginzburg's approach, notes.
  • 2/10: Finish Gan and Ginzburg's approach, notes.
  • 2/14: Cancelled.
  • 2/17: Holiday.
  • 2/21: Kleinian singularities as Slodowy slices, notes.
  • 2/24: Resolutions of Kleinian singularities as Slodowy varieties, first examples of dual varieties, and first predictions of symplectic duality, notes.
  • 2/28: Duality for Slodowy varieties, notes.
  • 3/3: Cotangent bundles of partial flag varieties and functions on covers of nilpotent orbits, notes.
  • 3/7: The Spectrum of functions on nilpotent covers is conical, universal deformation of cotangent bundles of partial flag varieties, notes.
  • 3/10: Parabolic Slodowy varieties, Gan-Ginzburg's approach, duality in this setting, Hikita-Nakajima conjecture, notes.
  • 3/14: Quantizations of parabolic Slodowy varieties, category O, irreducible objects there and quantized Hikita-Nakajima conjecture, notes.
  • 3/17: Holiday.
  • 3/21: Holiday.
  • 3/24: Category O for symplectic resolutions I: definitions, Cartan subquotients notes.
  • 3/28: Category O for symplectic resolutions II: more on Cartan subquotients, their classical versions and Verma modules, notes.
  • 3/31: Category O for symplectic resolutions III: main properties, irreducible objects, sheaf versions of Cartan subquotients, notes.
  • 4/4: Quantized Hikita-Nakajima conjecture, Affine Grassmannians -- basic properties, notes.
  • 4/7: More about Affine Grassmannians, notes.
  • 4/11: Slices in affine Grassmannians: basic properties, notes.
  • 4/14: More on slices in affine Grassmannians, generalized slices, examples and quantizations, notes.
  • 4/18: Moduli space realizations of generalized slices, notes.
  • 4/21: Resolutions, torus fixed points, notes.
  • 4/25: Multiplication morphisms between generalized slices, Darboux coordinates and Verma modules, notes.
  • 4/28: Type A transversal slices are isomorphic to parabolic Slodowy varieties, sketch of the proof of the quantized Hikita-Nakajima conjecture for parabolic type A Slodowy varieties, notes.