Professor Andrei Okounkov (Skoltech, Columbia Univ.)
will deliver a series of lectures titled

“Enumerative geometry and geometric representation theory”

Next lecture – Tuesday, December 15, 2020, 18.30

Course whiteboardOneNote
Course announcementsTelegram Messenger
Course sitegoogle.com


    lect #01 / May 13 / video / whiteboard_pdf (3.6 MB)
    an introductory discussion of: indices of supersymmetric operators, their relation to topological K-theory, K-theoretic Field Theories, moduli of vacua/Gibbs states, 3-dimensional mirror symmetry

    lect #02 / May 20 / video / whiteboard_pdf (3.6 MB)
    q-hypergeometric functions and Macdonald polynomials as examples of vertex functions, quantum groups and their categories of modules, R-matrices, reconstruction of a quantum group from R-matrices

    lect #03 / May 27 / video / whiteboard_pdf (3.6 MB)
    more on R-matrices, comultiplication in Yangians, relations in a quantum group, R-matrix as an interface, geometric construction of the R-matrix for Y(sl_2), geometric meaning of the Yang-Baxter equation

    lect #04 / Jun 03 / video / whiteboard_pdf (3.6 MB)
    moduli of vacua in QFTs with extended supersymmetry, their response to variation of external parameters, general idea of “stable envelopes”, equivariant cohomology theories, introduction to equivariant K-theory, equivariant K-theory of the projective space

    lect #05 / Jun 17 / video / whiteboard_pdf (4.3 MB)
    Lecture by I. Krichever: equivariant cobordism

    lect #06 / Jun 24 / video / whiteboard_pdf (3.2 MB)
    Koszul resolutions, localization, first steps in elliptic cohomology, Chern classes, equivariant K-theory and equivariant elliptic cohomology of Hilb(C^2), Thom spaces

    lect #07 / Jul 01 / video / whiteboard_pdf (5.3 MB)
    Modification of the Thom isomorphism in elliptic cohomology, Theta bundles and theta functions, Bott periodicity, pushforwards in elliptic cohomology, change of groups in equivariant elliptic cohomology

    lect #08 / Jul 08 / video / whiteboard_pdf (3.3 MB)
    equivariant elliptic cohomology from the cell decomposition point of view, cofibration sequence, examples

    lect #09 / Jul 15 / video / whiteboard_pdf (3.4 MB)
    Attracting manifolds, strategy for inductive construction of elliptic stable envelopes, stable envelopes as an interpolation problem, interpolation and its relations to cohomology vanishing, cohomology vanishing for line bundles on abelian varieties, Picard and Neron-Severi groups of an abelian variety

    lect #10 / Jul 22 / video / whiteboard_pdf (4.2 MB)
    Attractive line bundles, definition of elliptic stable envelopes, existence and uniqueness of elliptic stable envelopes

    lect #11 / Jul 29 / video / whiteboard_pdf (4.6 MB)
    Elliptic stable envelopes for projective spaces, and for hypertoric varieties, Felder’s elliptic R-matrix

    lect #12 / Aug 05 / video / whiteboard_pdf (3.4 MB)
    Cohomology of the Grassmannian, Schubert classes and interpolation Schur functions, elliptic stable envelopes for Grassmannians, abelianization of stable envelopes

    lect #13 / Aug 12 / video / whiteboard_pdf (3.2 MB)
    triangle lemma, polarization and index of a component of the fixed locus, dynamical Yang-Baxter equation for elliptic stable envelopes

    lect #14 / Aug 19 / video / whiteboard_pdf (4.5 MB)
    duality for stable envelopes, rigidity in elliptic cohomology, coproduct in quantum groups in terms of R-matrices, various factorizations of R-matrices, operators of cup product as vacuum-vacuum matrix elements of geometric R-matrices

    lect #15 / Aug 26 / video / whiteboard_pdf (3.8 MB)
    Nakajima quiver varieties, Hilbert schemes of points and Heisenberg algebra, stable envelopes in K-theory and cohomology, unitarity of R-matrices, classical Yang-Baxter equation, the Lie algebra corresponding to a classical r-matrix, Langrangian correspondences, Steinberg correspondences

    lect #16 / Sep 02 / video / whiteboard_pdf (4.2 MB)
    equivariant symplec resolutions, Lagrangian residues, Lagrangian Steinberg correspondences commute with R-matrices in cohomology, Nakajima-Baranovsky operators and classical r-matrix for instanton moduli spaces, general properies of the Maulik-Okounkov Lie algebras

    lect #17 / Sep 08 / video / whiteboard_pdf (4.2 MB)
    cup product by divisor in cohomology of instanton moduli spaces, its relation to quantum Calogero-Sutherland and Benjamin-Ono integrable systems, full R-matrix for instanton moduli spaces, Yangian of \hat gl(1) and its relation to the Virasoro algebra and W-algebras, Kac determinant from R-matrices, the R-matrix for the Hilbert scheme of points as the reflection operator in Liouville CFT, slices and relations in quantum groups

    lect #18 / Sep 16 / video / whiteboard_pdf (3.2 MB)
    slices in quiver varieties and more general moduli problems, slices for Grassmannians and instanton moduli spaces, slices as quantum group intertwiners, screening operators and Plücker relations, quantum integrable systems from R-matrices, their relation to quantum multiplication for Nakajima varieties

    lect #19 / Sep 22 / video / whiteboard_pdf (5.0 MB)
    classical and quantum integrable systems in enumerative geomety, Plücker relations and Toda equations, Toda equations in the Gromov-Witten theory of P^1, the corresponding quantum integrable system and free fermions, free fermions as the Yangian of \hat gl(1) at \hbar=0, Donaldson-Thomas theory and its relation to GW theory, the full Yangian in the GW/DT theory of local curves in 3-folds

    lect #20 / Sep 29 / video / whiteboard_pdf (11.0 MB)
    Lecture by Melissa Liu: virtual fundamental classes in enumerative geometry

    lect #21 / Oct 06 / video / whiteboard_pdf (3.3 MB)
    exact and approximate self-duality of obstruction theories, symmetrized virtual structure sheaf, the quest for proper moduli spaces with self-dual obstruction theory, quasimaps to GIT quotients, quasimaps to projective spaces and to the Hilbert schemes of points

    lect #22 / Oct 13 / video / whiteboard_pdf (5.1 MB)
    Donaldson-Thomas counts of subschemes in 3-folds, quasimaps to Hilb(C^2) and Pandharipande-Thomas counts for rank 2 bundles over curves, torus fixed points in the Hilbert scheme of curves and it the PT spaces, twisted quasimaps, evaluation maps, relative quasimaps, accordions, nodes and generation formula, the glue matrix

    lect #23 / Oct 20 / video / whiteboard_pdf (4.8 MB)
    diagrammatic notation for different flavors of insertions/boundary conditions in enumerative problems, relative moduli spaces in DT theory, expanded degenerations, degeneration formulas, correspondence between different boundary condition, degeneration and algebraic cobordism, relative counts in GW theory, their correspondence with relative DT counts

    lect #24 / Oct 27 / video / whiteboard_pdf (5.7 MB)
    basic building blocks of quasimap counts, vertex with descendants, its expression as an elliptic stable envelope, relative counts and q-difference equations, q-Gamma functions, vertex with descendants and integral solutions to quantum difference equations, integral solutions and Bethe ansatz, residues in the integral vs. localization formulas for vertices with descendants

    lect #25 / Nov 03 / video / whiteboard_pdf (8.7 MB)
    difference equations in equivariant variables, twisted quasimaps, equivariant localization, formula for the K-theory class of the virtual tangent bundle, edge and vertex contributions in localization formulas, pure edge and q-Gamma functions, q-analog of the Iritani class, the degree of a twisted map, its relation to Kähler line bundles in elliptic cohomology

    lect #26 / Nov 10 / video / whiteboard_pdf (5.3 MB)
    пeometric meaning of the operator in the q-difference equation in equivariant variables, q-difference equations in Kähler variables, quantum Knizhnik-Zamolodchikov equations

    lect #27 / Nov 17 / video / whiteboard_pdf (6.3 MB)
    singularities of the difference equations in equivariant variables, minuscule cocharacters and their geometric meaning, quantum Knizhnik-Zamolodchikov equations for shifts by minuscule cocharacters

    lect #28 / Nov 24 / video / whiteboard_pdf (5.8 MB)
    q-difference equations in Kähler variables, Dubrovin connection for Nakajima varieties, dynamical groupoids, slope R-matrices in equivariant K-theory, Khoroshkin-Tolstoy factorization of R-matrices, slope subalgebras in quantum loop groups, dynamical groupoid associated to slope subalgebras and quantum q-difference equiations for Nakajima varieties

    lect #29 / Dec 01 / video / whiteboard_pdf (5.1 MB)
    Tube=Glue, the glue matrix in terms of the dynamical groupoid, varieties X’ associated to strata in Pic(X) \otimes R, conjectural formula for the wall operators in terms of these X’, what it says for rank r framed sheaves on C^2, capped vertex with descendants, large framing vanishing, Smirnov’s formula for the capped vertex with descendants, the fusion operator J again

    lect #30 / Dec 08 / video / whiteboard_pdf (5.3 MB)
    Capped vertex with descendants as a correspondence between X and the ambient quotient stack, stable and unstable loci in GIT, stratification of the unstable locus, inductive construction of stable envelopes for quotient stacks

    lect #31 / Dec 15 / video / whiteboard_pdf (4.4 MB)
    nonabelian stable envelopes for algebraic symplectic reductions, the relation between abelian and nonabelian stable envelopes, K-theoretic stable envelopes and equivalence between descendant and relative insertions, integral solutions of the quantum difference equations, Bethe equations, K-theoretic stable envelopes as the off-shell Bethe eigenfunction

    lect #32 / Dec 22 / video / whiteboard_pdf (5.8 MB)
    nonabelian stable envelopes for G=\prod GL(V_i), algebraic Bethe Ansatz, vertex with descendents and nonabelian stable envelopes, maps from the formal disk and q-Gamma functions, integral solutions of the quantum difference equations and the monodromy of the vertex functions

    lect #33 / Dec 29 / whiteboard_pdf (8.7 MB)

    lect #32 / Dec 22 / 18:30 – 20:00 (Moscow time)


Let X be the Hilbert scheme of points in C^2, or a more general Nakajima quiver variety, or a more general moduli space of vacua in a 2+1 dimensional supersymmetric gauge theory with 4 times the minimal amount of supersymmetry. In this course, I plan to explain:

  • the construction of certain remarkable correspondences, called stable envelopes, in the equivariant elliptic cohomology of X, and hence in K(X) and H(X);
  • the construction of a quantum group that act in elliptic cohomology, K-theory, and cohomology of X, from these stable envelopes;
  • enumerative K-theory of (quasi)maps from curves to X and its particular building blocks — certain remarkable operators between K(X) and itself, or K(X) and the K-theory of the ambient stack;
  • the formulas for these building blocks obtained from the geometric representation theory of our quantum groups


the main technical references for this course are

  • Andrei Okounkov, “Inductive construction of stable envelopes and applications, I. Actions of tori. Elliptic cohomology and K-theory and Inductive construction of stable envelopes and applications”, [ PDF: arXiv: 2007.09094 ]
  • Andrei Okounkov, “Inductive construction of stable envelopes and applications, II. Nonabelian actions. Integral solutions and monodromy of quantum difference equations”, [ PDF: arXiv: 2010.13217 ]

an overview of the connections between geometric representation theory and enumerative geometry maybe found in

  • Andrei Okounkov, “Enumerative geometry and geometric representation theory”, [ PDF: arXiv: 1701.00713 ]
  • Andrei Okounkov, “On the crossroads of enumerative geometry and geometric representation theory”, [ PDF: arXiv: 1801.09818 ]

an discussion of the fundamentals of Donaldson-Thomas theory, its origins, open questions, and certain recent advances may be found

  • Andrei Okounkov, “Takagi lectures on Donaldson-Thomas theory”, [ PDF: arXiv: 1802.00779 ]

examples, exercises, and proofs for those who want to really understand the subject may be found in

  • Andrei Okounkov, “Lectures on K-theoretic computations in enumerative geometry”, [ PDF: arXiv: 1512.07363 ]


  • English