| 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 | |
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Problems for week 1 for the minicourse by P. Etingof will be from the book “Tensor categories” that can be found here: |
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| 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 |
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| 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 | |
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Problems from A. Okounkov’s lectures, part 1: pdf_file Problems for N. Nekrasov lectures: pdf_file Problems for P. Etingof lecture: pdf_file |
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| 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 |
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| 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 |