﻿ Skoltech Center for Advanced Studies |

#### | OVERVIEW | PROGRAM | TITLES and ABSTRACTS | PARTICIPANTS | LOCAL |

[ last update: July 12, 2019 / 16:00 ]

### PROGRAM

 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, R-matrices, quantum double 11:00 – 12:00 Pavel Etingof Main examples: quantized enveloping algebras (of finite dimensional, affine, general Kac-Moody algebras). Poisson-Lie 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 D-modules 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: 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
 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. 3-dimensional TQFT 14:30 – 15:30 Pavel Etingof Lusztig’s braid group action, quantum Weyl group. KZ equations, Drinfeld-Kohno theorem, Casimir connection, Toledano-Laredo’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 Casselman-Shalika equivalence. Remarks on perverse sheaves with modular coefficients. Quantum groups 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) 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 Kazhdan-Lusztig-Ginzburg 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 K-theory and equivariant cohomology video 11:00 – 12:30 Nikita Nekrasov 3) Instanton partition functions from localization. K-theoretic and elliptic analogs 4) Y-observables, 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. q-characters 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 q-characters video 11:00 – 12:30 Nikita Nekrasov 5) Nonperturbative Dyson-Schwinger equations I: residue matching, qq-characters in the A case video 14:00 – 15:00 Noah Arbesfeld / TA R-matrices 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 K-theory for quantum loop groups video 11:00 – 12:30 Nikita Nekrasov 6) Nonperturbative Dyson-Schwinger equations II: crossed instantons, compactness theorem, qq-characters for general quivers video 14:00 – 15:00 Petr Pushkar / TA Baxter Operator and the Transfer Matrix from Enumerative K-theory video 15:30 – 16:30 Pavel Etingof Dynamical R-matrices 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 Dyson-Schwinger equations III: limits to q-characters, limit shapes, Baxter equations, and Bethe ansatz video 14:00 – 15:15 15:30 – 16:30 Nikita Nekrasov 8) Nonperturbative Dyson-Schwinger equations IV: applications to BPS/CFT correspondence, BPZ and KZ equations from gauge theory video