Monday / July 1, 2019  Skoltech / Blue Bld. / 3 Nobel str., aud. 408  
10:00 – 11:00  Pavel Etingof  Rigid tensor categories, fiber functors, Hopf algebras, braided categories, Drinfeld center, Rmatrices, quantum double 
11:00 – 12:00  Pavel Etingof  Main examples: quantized enveloping algebras (of finite dimensional, affine, general KacMoody algebras). PoissonLie groups, Lie bialgebras, Manin triples, their quantization 
13:30 – 15:30  Alexander Braverman  Digression on classical local Langlands. The idea of geometric version. Digression on Dmodules 
16:00 – 18:00  problems session  
Problems for week 1 for the minicourse by P. Etingof will be from the book “Tensor categories” that can be found here: wwwmath.mit.edu/~etingof/egnobookfinal.pdf 
Tuesday / July 2, 2019  Skoltech / Blue Bld. / 3 Nobel str., aud. 408  
10:00 – 12:00  Alexander Braverman  The stack of de Rham local systems on the formal punctured disc. Local geometric class field theory (after Laumon) 
13:30 – 14:30  Pavel Etingof  Representations of quantized enveloping algebras (category O, finite dimensional representations). Knot invariants. Roots of unity. 3dimensional TQFT 
14:30 – 15:30  Pavel Etingof  Lusztig’s braid group action, quantum Weyl group. KZ equations, DrinfeldKohno theorem, Casimir connection, ToledanoLaredo’s monodromy theorem 
16:00 – 18:00  problems session 
Wednesday / July 3, 2019  Skoltech / Blue Bld. / 3 Nobel str., aud. 408  
10:00 – 12:00  Roman Bezrukavnikov  More on geometric Satake equivalence (complementary to Braverman’s presentation). Geometric CasselmanShalika equivalence. Remarks on perverse sheaves with modular coefficients. Quantum groups at root of unity, KazhdanLusztig equivalence, realization of a principal block for quantum group at a root of unity via the affine Grassmannian. Remarks on the modular analogue (“FinkelbergMirkovic” conjecture) 
13:30 – 15:30  Alexander Braverman  The notion of strong group action on a category. Formulation of classical and quantum local geometric Langlands. Some examples 
16:00 – 18:00  problems session  
Problems for the minicourse by R. Bezrukavnikov: pdf_file 
Thursday / July 4, 2019  Skoltech / Blue Bld. / 3 Nobel str., aud. 408  
10:00 – 12:00  Roman Bezrukavnikov  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. Some ideas of symplectic duality and Koszul duality for categories O, examples: cotangent to partial flag manifolds, possibly also hypertoric varieties video 
13:30 – 15:30  Alexander Braverman  More examples. Some applications: derived geometric Satake and categorification of KazhdanLusztigGinzburg geometric realization of the affine Hecke algebra video 
16:00 – 18:00  problems session  
Problems for R. Bezrukavnikov and A. Braverman lectures: pdf_file 
Friday / July 5, 2019  Skoltech / Blue Bld. / 3 Nobel str., aud. 408  
10:00 – 12:00  Alexander Braverman  More examples. Possibility of a discussion of a connection to 3d N=4 quantum field theories 
13:30 – 15:30  Roman Bezrukavnikov  Symplectic duality and loop spaces: examples and conjectures video 
16:00 – 18:00  Pablo Boixeda Alvarez / problems session video 

The problems for R. Bezrukavnikov’s lecture 3: pdf_file 
Saturday / July 6, 2019  
Sunday / July 7, 2019  
Monday / July 8, 2019  Skoltech / Blue Bld. / 3 Nobel str., aud. 408  
09:45 – 10:45  Andrei Okounkov  Construction and basic examples of Nakajima quiver varieties, their basic properties video 
11:00 – 12:30  Nikita Nekrasov  1) Supersymmetric field theory as intersection theory 2) Instantons: moduli spaces, compactifications, symmetries, fixed points video 
14:00 – 15:00  Saebyeok Jeong / TA  video 
15:30 – 16:30  Pavel Etingof  Yangians. Loop realizations of Yangians and quantum affine algebras video 
17:00  problems  
Problems from A. Okounkov’s lectures, part 1: pdf_file Problems for N. Nekrasov lectures: pdf_file Problems for P. Etingof lecture: pdf_file 
Tuesday / July 9, 2019  Skoltech / Blue Bld. / 3 Nobel str., aud. 408  
09:45 – 10:45  Andrei Okounkov  Introduction to equivariant Ktheory and equivariant cohomology video 
11:00 – 12:30  Nikita Nekrasov  3) Instanton partition functions from localization. Ktheoretic and elliptic analogs 4) Yobservables, poles, and residues. Comparison to matrix models video 
14:00 – 15:00  Alisa Knizel / TA  Asymptotic computations with discrete loop equations 
15:30 – 16:30  Pavel Etingof  Finite dimensional representations of Yangians and quantum affine algebras. qcharacters video 
17:00  problems 
Wednesday / July 10, 2019  Skoltech / Blue Bld. / 3 Nobel str., aud. 408  
09:45 – 10:45  Andrei Okounkov  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 qcharacters video 
11:00 – 12:30  Nikita Nekrasov  5) Nonperturbative DysonSchwinger equations I: residue matching, qqcharacters in the A case video 
14:00 – 15:00  Noah Arbesfeld / TA  Rmatrices for Hilbert schemes of points on surfaces video 
15:30 – 16:30  Pavel Etingof  Intertwining operators, fusion operator, ABRR equation. KZ and quantum KZ equations for matrix elements of intertwiners in the affine case video 
17:00  problems  
Problems from A. Okounkov’s lectures, part 1 + part 2: pdf_file 
Thursday / July 11, 2019  Skoltech / Blue Bld. / 3 Nobel str., aud. 408  
09:45 – 10:45  Andrei Okounkov  Same in equivariant Ktheory for quantum loop groups video 
11:00 – 12:30  Nikita Nekrasov  6) Nonperturbative DysonSchwinger equations II: crossed instantons, compactness theorem, qqcharacters for general quivers video 
14:00 – 15:00  Petr Pushkar / TA  Baxter Operator and the Transfer Matrix from Enumerative Ktheory video 
15:30 – 16:30  Pavel Etingof  Dynamical Rmatrices and Weyl groups. Difference Casimir connections video 
17:00  problems video 1 video 2 

Problems for N. Nekrasov lectures (update): pdf_file 
Friday / July 12, 2019  Skoltech / Blue Bld. / 3 Nobel str., aud. 408  
09:45 – 10:45  Andrei Okounkov  Spaces of maps to Nakajima varieties and first introduction to enumerative geometry video 
11:00 – 12:30  Nikita Nekrasov  7) Nonperturbative DysonSchwinger equations III: limits to qcharacters, limit shapes, Baxter equations, and Bethe ansatz video 
14:00 – 15:15 15:30 – 16:30 
Nikita Nekrasov  8) Nonperturbative DysonSchwinger equations IV: applications to BPS/CFT correspondence, BPZ and KZ equations from gauge theory video 