"Summer" 2020

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

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

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

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

Lecture by I. Krichever: equivariant cobordism

Koszul resolutions, localization, first steps in elliptic cohomology, Chern classes, equivariant K-theory and equivariant elliptic cohomology of Hilb(C^2), Thom spaces

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

equivariant elliptic cohomology from the cell decomposition point of view, cofibration sequence, examples

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

Attractive line bundles, definition of elliptic stable envelopes, existence and uniqueness of elliptic stable envelopes

Elliptic stable envelopes for projective spaces, and for hypertoric varieties, Felder's elliptic R-matrix

Cohomology of the Grassmannian, Schubert classes and interpolation Schur functions, elliptic stable envelopes for Grassmannians, abelianization of stable envelopes

triangle lemma, polarization and index of a component of the fixed locus, dynamical Yang-Baxter equation for elliptic stable envelopes

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

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

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

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,

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

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

Lecture by Melissa Liu: virtual fundamental classes in enumerative geometry

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

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

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

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

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 . For a theoretical discussion of localization in DT theory, see e.g. Section 6 of my Takagi lectures. For a tutorial on computer-aided localization, see the talk by Henry Liu

Geometric meaning of the operator in the q-difference equation in equivariant variables, q-difference equations in Kähler variables, quantum Knizhnik-Zamolodchikov equations

Singularities of the difference equations in equivariant variables, minuscule cocharacters and their geometric meaning, quantum Knizhnik-Zamolodchikov equations for shifts by minuscule cocharacters

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, fusion operators J for slope subalgebras, dynamical groupoid associated to slope subalgebras and quantum q-difference equiations for Nakajima varieties.

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

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

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

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

video defective

Monodromy of vertex functions and elliptic stable envelopes, K-theory limit and the monodromy of the quantum differential equations, categorification thereof and relation with quantization in characteristic p >> 0, a-solutions and z-solutions, 3d mirror symmetry, duality interface, categorification of the duality interface, q-difference equations and multiplicative quantizations in roots of unity