Schedule

All lectures will take place in room 146 of the HEDCO Education building.
Coffee and pastries will be available every morning in the lecture room, starting at 8:30.

Monday, August 5th -- Schubert calculus in finite dimensions
  • 9:00 (Allen Knutson) Overview

  • 9:45 (Balazs Elek) Schubert varieties on Grassmannians and flag manifolds. The Borel presentation. P-orbits on G/Q as Morse-Bott strata. Kleiman transversality. References: [Coskun-Vakil], [Akyildiz]

  • 11:00 (Omar Ortiz) Equivariant cohomology. Kirwan injectivity for Hamiltonian actions on compact symplectic manifolds. GKM manifolds. References: [Kostant-Kumar-1], [Kostant-Kumar-2], [Harada-Henriques-Holm].

  • 1:30 (William Slofstra) Equivariant K-homology and K-cohomology. Double Grothendieck polynomials. Stanley-Reisner schemes (especially of shellable balls) and their K-classes. References: [Miller-Sturmfels], [Knutson-Miller]

  • 2:30 First problem set

  • 4:30 Discuss problem set
    • Topics for problem set

    Schubert calculus on Grassmannians, via mosaics and puzzles. References: [Purbhoo], [Knutson-Tao]

    (Double) Schubert polynomials, matrix Schubert varieties, pipe dreams. References: [Miller-Sturmfels]

    Problem Set 1
    Tuesday, August 6th -- affine Schubert calculus
  • 9:00 (Allen Knutson) Triage

  • 9:45 (Philsang Yoo) The Hilbert manifold Map(S1,K) of smooth based loops into a compact group, and its Kahler structure. Morse-Bott theory of the energy functional. Presentation of the homology groups. References: [Pressley-Segal], [Kumar]

  • 11:00 (Daniel Rowe) The lattice model of the affine Grassmannian for U(k). The affine flag manifold and its finite-dimensional and finite-codimensional Schubert varieties. Affine Bruhat order. References: [Anderson-Kogan], [Snider]

  • 1:30 (Laura Escobar) k-double Schur functions. References: [Shimozono], [Lam-Shimozono-2]

  • 2:30 Second problem set

  • 4:30 Discuss problem set
    • Topics for problem set

    Stanley symmetric functions, from reduced words, and from stabilizing Schubert polynomials.

    Affine Stanley symmetric functions, from reduced words, and from

    affine Grassmannian → Omega K → LK → LK/T ← affine flag manifold.

    References: [Ross-Yong], [Lam]

    Problem Set 2
    Wednesday, August 7th -- quantum Schubert calculus
  • 9:00 (Allen Knutson) Triage

  • 9:45 (Khoa Nguyen) Basics of quantum cohomology. Reference: [Fulton-Pandharipande]

  • 11:00 (Clelia Pech) Quantum cohomology of Grassmannians. References: [Buch-Kresch-Tamvakis] [Chaput-Perrin]

  • 1:37 hike

  • 6:00 picnic in University Park
    • Thursday, August 8th -- more quantum Schubert calculus
  • 9:00 (Allen Knutson) Triage

  • 9:45 (Kaisa Taipale) Quantum Schubert polynomials. References: [Fomin-Gelfand-Postnikov], [Ciocan-Fontanine - Fulton]

  • 11:00 (Elizabeth Beazley) The equivariant quantum Chevalley formula. Reference: [Mihalcea]

  • 1:30 (Alex Ellis) Peterson's theorem, via the Toda lattice. References: [Lam-Shimozono-1], [Lam-Shimozono-3]

  • 2:30 Third problem set

  • 4:30 Discuss problem set
    • Topics for problem set

    Quantum equivariant cohomology of CP1, via double quantum Schubert polynomials.

    Quantum equivariant cohomology of CPk, via 2-step puzzles.
    Friday, August 9th
  • 9:00 (Allen Knutson) Triage

  • 9:45 TBA

  • 11:00 (Anna Bertiger) Application: the unitary Horn problem. References: [Agnihotri-Woodward], [Belkale]

  • 1:30 (Allen Knutson) The positroid stratification of the Grassmannian. Reference: [Knutson-Lam-Speyer]

  • 2:45 (Allen Knutson) Positroid varieties, and the affine Grassmannian, and quantum cohomology