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TITLES and ABSTRACTS of the LECTURES:

Alexander Braverman /

Perimeter Institute, University of Toronto, Skoltech

**Introduction to local geometric Langlands correspondence and its applications** / July 1-5 / Plan (subject to changes):

**Lecture 1**: Digression on classical local Langlands. The idea of geometric version. Digression on D-modules**Lecture 2**: The stack of de Rham local systems on the formal punctured disc. Local geometric class field theory (after Laumon)**Lecture 3**: The notion of strong group action on a category. Formulation of classical and quantum local geometric Langlands. Some examples**Lecture 4**: More examples. Some applications: derived geometric Satake and categorification of Kazhdan-Lusztig-Ginzburg geometric realization of the affine Hecke algebra (after R. Bezrukavnikov)**Lecture 5**: More examples. Possibility of a discussion of a connection to 3d N=4 quantum field theories

Roman Bezrukavnikov /

MIT

**Lecture 1**: More on geometric Satake equivalence (complementary to Sasha’s presentation). Geometric Casselman-Shalika equivalence. Remarks on perverse sheaves with modular coefficients /

July 3**Lecture 2**: Quantum group at root of unity, Kazhdan-Lusztig equivalence, realization of a principal block for quantum group at a root of unity via the affine Grassmannian. Remarks on the modular analogue (“Finkelberg-Mirkovic” conjecture) /

July 3**Lecture 3**: Koszul duality in representation theory, (conjectural) relation to derived Satake and the above equivalences and coherent realizations for the categories of constructible sheaves on the affine flag variety /

July 4**Lecture 4**: Some ideas of symplectic duality and Koszul duality for categories O, examples: cotangent to partial flag manifolds, possibly also hypertoric varieties /

July 4**Lecture 5**: Symplectic duality and loop spaces: examples and conjectures /

July 5

The **problems for week 1** for the minicourse by R. Bezrukavnikov: **pdf_file**

Pavel Etingof /

MIT

**Lecture 1**: Rigid tensor categories, fiber functors, Hopf algebras, braided categories, Drinfeld center, R-matrices, quantum double /

July 1, 1 hour**Lecture 2**: Main examples: quantized enveloping algebras (of finite dimensional, affine, general Kac-Moody algebras). Poisson-Lie groups, Lie bialgebras, Manin triples, their quantization /

July 1, 1 hour**Lecture 3**: Representations of quantized enveloping algebras (category O, finite dimensional representations). Knot invariants. Roots of unity. 3-dimensional TQFT /

July 2, 1 hour**Lecture 4**: Lusztig’s braid group action, quantum Weyl group. KZ equations, Drinfeld-Kohno theorem, Casimir connection, Toledano-Laredo’s monodromy theorem /

July 2, 1 hour**Lecture 5**: Yangians. Loop realizations of Yangians and quantum affine algebras /

July 8, 1 hours**Lecture 6**: Finite dimensional representations of Yangians and quantum affine algebras. q-characters /

July 9, 1 hours**Lecture 7**: Intertwining operators, fusion operator, ABRR equation. KZ and quantum KZ equations for matrix elements of intertwiners in the affine case / July 10, 1 hour**Lecture 8**: Dynamical R-matrices and Weyl groups. Difference Casimir connections /

July 11, 1 hour

The **problems for week 1** for the minicourse “Quantum groups” by P. Etingof will be from the book “Tensor categories” that can be found here: **www-math.mit.edu/~etingof/egnobookfinal.pdf**

( errata are **here** )

**Exercises**

5.2.5, 5.2.7, 5.3.3, 5.3.17(ii), 5.5.9, 5.5.10, 5.6.3, 5.8.2, 5.8.3, 5.8.4, 5.8.6, 5.14.2, 6.5.10, 7.14.4, 7.15.3, 8.3.11, 8.3.13, 8.3.25, 8.3.27, 8.4.3, 8.4.6, 8.4.7, 8.4.10, 8.4.11, 8.5.5, 8.5.7, 8.9.6.

Some exercises require looking through the subsection they are in.

Students are encouraged to look at the exercises and choose the ones to solve, as many as they can

Nikita Nekrasov /

Simons Center for Geometry and Physics, Skoltech

**BPS/CFT correspondence and qq-characters** / July 8-12 :

**Lecture 1**: Supersymmetric field theory as intersection theory /

July 8**Lecture 2**: Instantons: moduli spaces, compactifications, symmetries, fixed points /

July 8**Lecture 3**: Instanton partition functions from localization. K-theoretic and elliptic analogs /

July 9**Lecture 4**: Y-observables, poles, and residues. Comparison to matrix models /

July 9**Lecture 5**: Nonperturbative Dyson-Schwinger equations I: residue matching, qq-characters in the A case /

July 10**Lecture 6**: Nonperturbative Dyson-Schwinger equations II: crossed instantons, compactness theorem, qq-characters for general quivers /

July 11**Lecture 7**: Nonperturbative Dyson-Schwinger equations III: limits to q-characters, limit shapes, Baxter equations, and Bethe ansatz /

July 12**Lecture 8**: Nonperturbative Dyson-Schwinger equations IV: applications to BPS/CFT correspondence, BPZ and KZ equations from gauge theory /

July 12

Andrei Okounkov /

Columbia University, HSE University, Skoltech

**Quiver varieties and representations of quantum group** / June 8-12

This will be an introductory review of the following topics:

**Lecture 1.**Construction and basic examples of Nakajima quiver varieties, their basic properties**Lecture 2.**Introduction to equivariant K-theory and equivariant cohomology**Lecture 3.**Basic ideas in represenation theory of quantum groups and their geometric realization. Stable envelopes and geometric construction of

the braiding for Yangians. Geometric meaning of q-characters**Lecture 4.**Same in equivariant K-theory for quantum loop groups**Lecture 5.**Spaces of maps to Nakajima varieties and first introduction to enumerative geometry