10:00 — 10:50 | 11:30 — 12:20 | 12:35 — 13:25 | 15:30 — 16:20 | 17:00 — 17:50 | after | place | |

Dec 13 | Khoroshkin | Serganova | Olshanski | Arkhipov | E.Feigin | IUM | |

Dec 14 | Finkelberg | Shiraishi | Belavin | Mukhin | Smirnov | Welcome party | IUM |

Dec 15 | Toledano | Odesski | Shoikhet | Mirkovic | Ostrik | IUM | |

Dec 16 | Stroppel | Vasserot | Shan | Varagnolo | Cherednik | Steklov MI | |

Dec 17 | Frenkel B.Feigin | Gaitsgory | Braverman | Bezrukavnikov | Beilinson | Banquet | Steklov MI |

Dec 18 | Kapustin | Nekrasov | Okounkov | Nakajima | Enriquez | Steklov MI | |

Dec 19 | Baranovsky | Veselov | Kaledin | Miwa | IUM |

**S.Arkhipov**, Demazure Descent Data and Braid group actions on categories.

Abstract: we recall the classical notion of Demazure operators acting on the K-theory of a G-variety X, G being a reductive algebraic group.

Then we propose a categorification of the algebra generated by Demazure operators and introduce the notion of Demazure Descent Data (DDD) on a category. We define the descent category for a DDD on a triangulated category C.

We explain how DDD arises naturally from a monoidal action of the tensor category of quasicoherent sheaves on B\ G/B on a category. A natural example of such picture is provided by the derived category of quasicoherent sheaves on X/B for a scheme X with an action of the reductive group G. The descent category in this case is the derived category of quasicoherent sheaves on X/G.

Next we replace the category of quasicoherent sheaves by DG-modules over the algebra of differential forms on X. We explain how an analog of the construction above gives rise to a braid group action of a category.

**V.Baranovsky**, Quantization of Lagrangians in algebraic geometry.

Abstract: let X be a smooth algebraic variety with an algebraic symplectic form and Y a smooth Lagrangian subvariety. For a fixed deformation quantization A of X and a line bundle L on Y we could ask whether L deforms to a module over A. We answer this question under a mild technical assumption. Joint work with D. Kaledin, V. Ginzburg and J. Pecharich

** R.Bezrukavnikov**, Generic quantum geometric Langlands in positive characteristic (after R. Travkin)

Abstract: I will talk about the work of Roman Travkin in which he established "generic" quantum geometric Langlands duality over fields of positive characteristic for the group GL(n). Time permitting I will present some speculations on the relation of this work to the nonabelian Hodge theory of C. Simpson and higher Teichmuller theory of V.Fock and A. Goncharov.

** A.Beilinson**, Relative continuous K-theory and cyclic homology

Abstract: Back in the 80s, Goodwillie has shown that for characteric zero rings the formal completions of K-theory and of classical cyclic homology coincide. I will discuss a continuous p-adic version of the Goodwillie theorem. The work is motivated by results of Bloch-Esnault-Kerz.

** A.Belavin**, Frobenious manifold structure for Douglas string equation and the correlation numbers for Minimal Liouville gravity

Abstract: I am going to present a reviw of some results of the joint works with my colleagues about so-called (p,q) Minimal Liouville gravity. I will argue that the generating function of the correlators in genus zero in Minimal Liouville gravity (MLG) is nothing but logarithm of the Sato tau-function for dispersionless Gefand-Dikii hierarchy with the special initial condition given by Douglas string equation. The correlators of Minimal Liouville gravity are not equal to the expansion coefficients of log of the tau-function in respect to KdV times as in Matrix models. Instead the correlators of MLG are the expansion coefficients of Log of the tau-function in respect to the new variables connected with KdV variables by a special noliniear "resonance" transformation. These correlators of MLG satisfy to the necessary conformal and fusion rules as it should be M(p/q) conformal minimal models. I will use the connection between Minimal Liouville gravity and Frobenious manifolds to get an explicit and useful expression for log Sato tau-function corresponding to Douglas string equation in dispersionless limit.

** A.Braverman**, Kazhdan-Lusztig conjecture for affine Lie algebras
via Uhlenbeck spaces

Abstract: we present a new geometric approach to Kazhdan-Lusztig type conjectures for finite and affine
Lie algebras (the actual results are new only in the case of affine Lie algebras and critical level).
One nice property of this approach is that allows to treat the cases of representation of affine Lie algebras with positive, negative and critical level in a uniform way.

This is a joint work in progress with M.Finkelberg and H.Nakajima.

** I.Cherednik**, Rogers-Ramanujan identities via Nil-DAHA

Abstract: almost by design, Nil-DAHA provides Dunkl operators
and other tools (algebraic and analytic) in the Q-Toda
theory. As Boris Feigin and the speaker demonstrated
recently, Nil-DAHA has important connections with the
coset algebras and can be used to build the theory of
Rogers-Ramanujan identities of modular type associated
with root systems. The Rogers-Ramanujan sums we obtain
quantize the constant Y-systems (of type RxA_{n} for any
reduced root systems R). This involves physics, dilogarithm
identities and a lot of interesting arithmetic, though
the talk will be mainly focused on the core construction.

** B.Enriquez**, A universal version of the KZB flat bundle and extensions of the Grothendieck-Teichmuller Lie algebra.

Abstract: we construct a bundle with flat connection over the moduli space of marked curves of genus one; this is a universal version of the KZB connection on the same moduli space. We study the holonomy representation of this connection. The building blocks of this representation are analytic objects, satisfying genus one analogues of the associator identities. The corresponding scheme is a torsor under an extension of the Grothendieck-Teichmuller Lie algebra, which we make explicit. We discuss the analogous constructions in genus greater that one. (Partly joint w. D. Calaque and P. Etingof)

** E.Feigin**, PBW and toric degenerations

Abstract: we will describe the PBW degeneration of the irreducible representations of simple Lie algebras and of the corresponding flag varieties. We will also explain how to degenerate the representations and flag varieties further to get toric structures. The connection with monomial bases will be clarified.

** M.Finkelberg**, Drinfeld compactification of Calogero-Moser space

Abstract: this compactification was discovered 15 years ago by G.Wilson. Its striking similarities to Uhlenbeck or zastava spaces for simple Lie algebras suggest that it plays the role of the zastava space for the Heisenberg Lie algebra. This is a joint project with A.Ionov.

** D.Gaitsgory**, Local geometric Langlands for the semi-infinite Whittaker category

Abstract: I'll describe work-in-progress by my graduate student Sam Raskin.
Local geometric Langlands in its most general form is a conjecture that certain two 2-categories (one associated with the group G, and another with its Langlands dual) are equivalent. However, the very formulation of this conjecture relies on a series (of largely conjectural) statements that certain pairs of *factorization categories* are equivalent. The most fundamental among them is a description of that the category of Whittaker D-modules on the semi-infinite flag space G(K)/N(K)T(O) in Langlands dual terms. In the talk I'll explain this description, as well as the role that it plays in the (usual) global unramfied geometric Langlands.

** D.Kaledin**, Cyclic homology of a different kind

Abstract: periodic cyclic homology of an associative algebra is defined by
taking the total complex of a certain bicomplex. The bicomplex happens to
be infinite, so that there are two ways to totalize it. One is widely
assumed to be "incorrect" since at least over Q, it gives identically
zero. As it turns out, however, in all the other cases — in char p, over
Z_{p}, over Z — this gives not zero but an interesting and meaningful
homology theory. This is what I am going to discuss, in the particular
case of algebras in char p.
I should mention that the idea was first suggested by Kontsevich in 2005,
but it was not taken seriously at the time. Recently, a similar phenomenon
was observed in the work of Beilinson and Bhatt on B_{DR} and B_{cris}.
This gave an impetus to reexamine and realize Kontsevich's suggestion; if
time permits, I will also explain the relation to Beilinson and Bhatt's
work.

** A.Kapustin**, Topological Quantum Field Theory of 2-form gauge fields

Abstract: I define and study Topological Quantum Field Theories (TQFTs) which describe the low-energy limit of gapped phases of gauge theories. These TQFTs generalize the Dijkgraaf-Witten TQFT and can be described on a lattice using discrete 1-form and 2-form gauge fields. The gauge group is replaced by a gauge 2-group (a 2-category with a single object and invertible 1-morphisms and 2-morphisms). It is proposed that 2-group TQFTs are associated with new types of symmetry-protected gapped phases of matter.

** A.Khoroshkin**, Highest weight categories and Macdonald polynomials

Abstract: the goal of the talk is to explain an approach to the problem of categorification of Macdonald polynomials based on derived categories of modules over Lie algebra of currents. First, I recall the definition of Macdonald polynomials as the orthogonalisation of the linear monomial basis in the ring of symmetric functions with respect to the certain given pairing depending on two parameters. I will give the relationship of the latter pairing with the Grothendieck ring of the category of modules over the Lie algebra of currents. Second, I will explain the orthogonalisation procedure in derived categories and give a hint on categorification problem. Third, I discuss when it is possible to avoid the derived setting and get different applications for the category of modules.

In particular, we will prove the BGG reciprocity for the category of modules over the Lie algebra g tensored with C[x] with g-semisimple.

** I.Mirkovich**, Loop Grassmannians and quivers

Abstract: the goal of the project is to organize the algebraic geometry of loop Grassmannians according to the combinatorics of quivers. Most of this is a work with Joel Kamnitzer and Allen Knutson.

** E.Mukhin**, Lower bounds in real Schubert Calculus

Abstract: we discuss the numbers of real solutions to osculating instances of Schubert problems in Grassmannians. We use the relation of those problems to the Gaudin model to establish a lower bound on these numbers. (This is a report on a joint work with V. Tarasov)

** H.Nakajima**, Instantons moduli spaces and W-algebras

Abstract: consider the equivariant intersection cohomology group of the Uhlenbeck space with the structure group G (partial compactification of the G-instanton moduli spaces over R^{4} with framing). We endow it with a structure of an integral form of the W-algebra, at least when G is of type ADE. We have the infinite dimensional R-matrix and the Yang-Baxter equation quite naturally,
which give us a new view point to the W-algebra.

** H.Nekrasov**, The index of M-theory I

Abstract: motivated by M-theory physics we will discuss a conjectural curve-counting theory in Calabi-Yau 5-folds and its relation to K-theoretic Donaldson-Thomas invariants of smooth 3-folds. Specifically, if a Calabi-Yau 5-fold Z admits an automorphism $q$ with 3-dimensinal fixed locus X, then the trace of q on K-theoretic curve counts in Z is written in terms of DT invariants of X with boxcounting parameter q.

**A.Odesski**, A simple construction of integrable Whitham type hierarchies

Abstract: a simple construction of Whitham type hierarchies in all genera is suggested. Potentials of these hierarchies are written in terms of integrals of hypergeometric type.

** A.Okounkov**, The index of M-theory II

Abstract: motivated by M-theory physics we will discuss a conjectural curve-counting theory in Calabi-Yau 5-folds and its relation to K-theoretic Donaldson-Thomas invariants of smooth 3-folds. Specifically, if a Calabi-Yau 5-fold Z admits an automorphism $q$ with 3-dimensinal fixed locus X, then the trace of q on K-theoretic curve counts in Z is written in terms of DT invariants of X with boxcounting parameter q.

** G.Olshanski**, Algebraic approach to infinite-dimensional Markov dynamics

Abstract: I will explain how to construct a model of Markov dynamics with infinitely many interacting particles. The model originated from the problem of harmonic analysis on the infinite symmetric group. The construction is essentially algebraic (no advanced probabilistic technique is used), and the algebra of symmetric functions serves as one of principal technical tools.

** V.Ostrik**, Level-rank duality via fusion product

Abstract: I will report on my joint work with Michael Sun. We apply the theory of fusion product in order to give a simple proof of affine branching rules for the restriction of integrable highest weight modules over sl(MN) at level 1 to sl(M) plus sl(N).

** V.Serganova**, Representations of the Lie superalgebra P(n) and Brauer algebras with signs

Abstract: The "strange" Lie superalgebra P(n) is the algebra of endomorphisms of an (n|n)-dimensional vector space V equipped with a non-degenerate odd symmetric form. The centralizer of the P(n)-action in the k-th tensor power of V is given by a certain analogue of the Brauer algebra. We discuss some properties of this algebra in application to representation theory of P(n) and P(∞). We also construct a universal tensor category such that for all n the categories of P(n) modules can be obtained as quotients of this category. In some sense this category is an analogue of the Deligne categories GL(t) and SO(t).

** P.Shan**, Cyclotomic rational Cherednik algebras and categorifications (I)

Abstract: This will be part one of a joint talk with M. Varagnolo. We will present a proof of Varagnolo-Vasserot's conjecture on the equivalence between the category O of rational Cherednik algebras and a parabolic category O of affine Lie algberas and some applications of this result (joint work with R. Rouquier and E. Vasserot). In this talk, we will present the conjecture, discuss its relationship with Schur-Weyl duality and explain one of the main ingredients of our proof - an extended version of Rouquier's theory on unicity of highest weight covers.

** B.Shoikhet**, Deligne conjecture for higher-monoidal abelian categories

Abstract: We discuss and outline a proof of some generalization of "Deligne conjecture" for n-fold monoidal abelian categories. We illustrate it in two examples: the category of bimodules over an associative algebra (n=1), and the category of tetramodules over a Hopf algebra (n=2). As well, we discuss some approach to formality phenomena in deformation theory, natural from this point of view.

** F.Smirnov**, Reflection relations and fermionic basis

Abstract: I shall explain what are the reflection relations and why they are important for study of perturbations of CFT. Then I shall introduce the fermionic basis which solves the reflection relations.

** C.Stroppel**, Quantum symmetric pairs and categorification

** V.Toledano Laredo**, From Yangians to quantum loop algebras via abelian difference equations

Abstract: the finite-dimensional representations of the Yangian Y_{h}(g) and quantum loop algebra U_{q}(Lg) of a complex, semisimple Lie algebra have long been known to share many similar features. Assuming that q is not a root of unity, I will explain how to construct an equivalence of categories between finite-dimensional representations of U_{q}(Lg) and
an explicit subcategory of finite-dimensional representations of Y_{h}(g). This equivalence is governed by the monodromy of an additive, abelian difference equation, and can be upgraded to a meromorphic tensor equivalence.
This is joint work with Sachin Gautam, and is based on: arXiv:1310.7318 and arXiv:1012.3687

** B.Tsygan**, Index theory and algebraic K theory

Abstract: the Atiyah-Singer index theorem expresses the index of an elliptic differential operator as the integral over the cotangent bundle of a cohomology class depending only on the principal symbol. We will present a conjectural generalization of this theorem. It expresses the virtual space which is the difference of the kernel and the cokernel as the integral over the cotangent bundle of a cohomology class of the cotangent bundle with coefficients in a presheaf of spectra. In other words, the morphism associating to an elliptic complex its Euler characteristic can be extended to a morphism from the algebraic K theory spectrum of the category of elliptic complexes to the algebraic K theory spectrum K(C) of complex numbers, and we give a conjectural formula for it. This formula involves integration over the cotangent bundle of a K(C)-valued cohomology class. A more general statement gives a similar formula for the Euler characteristic of an elliptic pair. There is also a conjectural generalization along the same line of the index theorem for Toeplitz operators. Together, these conjectures unite many known results, such as the classical Atiyah-Singer, various formulas for the determinant line of a family of elliptic operators, higher index theorems for commuting operators due to Carey-Pincus and Kaad-Nest, Beilinson's microlocal formula for the determinant line of the cohomology of a constructible sheaf, etc.

** M.Varagnolo**, Cyclotomic rational Cherednik algebras and categorifications (II)

Abstract: This will be the second part of a joint talk with P. Shan. We will present a proof of Varagnolo-Vasserot's conjecture on the equivalence between the category O of rational Cherednik algebras and a parabolic category O of affine Lie algebras and some applications of this result (joint work with R. Rouquier and E. Vasserot). In this talk, we will speak about categorifications and we explain another ingredient of the proof, that is reduction to codimension one. We will also present two corollaries of the result which involve the category O of cyclotomic rational Cherednik algebra: this category is Koszul and the multiplicities of simples in standard modules are given by parabolic KL-polynomials (the first was conjectured by Chuang and Miyachi and the second by Rouquier).

** E.Vasserot**, AGT conjecture and q-vertex operators

Abstract : I will talk about a work in progress in order to compute some particular vertex operators which occurs in the AGT conjecture.

** A.Veselov**, Dunkl operators at infinity and Calogero-Moser systems

Abstract: we define the Dunkl and Dunkl-Heckman operators in infinite number of variables and use them to construct the quantum integrals of the Calogero-Moser-Sutherland problems at infinity. As a corollary we have a simple proof of integrability of the deformed quantum CMS systems related to classical Lie superalgebras. The talk is based on a joint work with A.N. Sergeev.