Meeting Time: Friday, 3:00-5:00 p.m. | Location: 2-361

Contact: Pavel Etingof and Victor Kac

Date and Time | Speaker | |
---|---|---|

September 8, 3-5 p.m. | Ivan Losev (Northeastern) |
Modular categories O for rational Cherednik algebras. The talk deals with representations of rational Cherednik algebras of type A over fields of characteristic p>>0. There are several reasons to be interested in this kind of representations. An "ideological" reason is that passing from complex numbers to fields of characteristic p>>0 "affinizes" the representation theory, as evidenced by the representation theory of semisimple Lie algebras. The representation theory of rational Cherednik algebras over C is already of "affine type" so passing to characteristic p>>0 should result in a representation theory of "double affine type". Such kinds of representation theory presently and not understood but it seems that they should appear in several other contexts. A "practical" reason to be interested in characteristic p representations of type A rational Cherednik algebras is that structural results in this area should lead to (re)proving some difficult results in the combinatorics of Macdonald polynomials, such as Haiman's n! theorem. In my talk I will concentrate on analogs of categories O in characteristic p>>0. By definition, they consist of finite dimensional graded modules. It is more or less classical result that this category is highest weight in a suitable sense (roughly speaking, highest weight means certain upper triangularity properties). I will define filtrations on these categories (standardly stratified structures) and relate the associated graded categories to more classical categories O from characteristic 0. Time permitting I will explain a relation of wall-crossing functors to these standardly stratified structures. No preliminary knowledge of the representation theory of rational Cherednik algebras or of highest weight categories is required. |

September 15, 3-5 p.m. | Andrei Negut (MIT) |
W-algebras, moduli of sheaves on surfaces, and AGT Abstract. To a smooth surface, we associate the W-algebra of type gl_r with two deformation parameters equal to the Chern roots of the cotangent bundle of S. We expect that the resulting algebra acts on the K-theory groups of moduli spaces of semistable rank r sheaves on S, and one can compute commutation relations between the algebra and the Carlsson-Okounkov Ext operator. When the surface is S=A^2, this allows one to present the Ext operator as a vertex operator for deformed W-algebras, thus yielding a mathematical proof of the 5d AGT relations with matter for the gauge group U(r) |

September 22, 3-5 p.m. | Tomoyuki Arakawa (MIT) |
Vertex algebras and Higgs branche conjecture Four dimensional N=2 superconformal fields theories in physics produce some interesting mathematical invariants such as Schur indices and Higgs branches. In my talk I will explain some remarkable relations of these invariants with vertex algebras and their consequences. |

September 29, 3-5 p.m. | No Seminar | |

October 6 | Retreat | |

October 13, 3-5 p.m. | Yasuyuki Kawahigashi (University of Tokyo) |
From vertex operator algebras to conformal nets and back A vertex operator algebra and a local conformal net (of operator algebras) are two mathematical formulations of chiral conformal field theory. We present a construction of the latter from the former, with unitarity, and going back to the former. We also discuss representation theoretic aspects of this construction. |

October 20, 3-5 p.m. | Roman Bezrukavnikov (MIT) |
Stable envelops and Verma modules We will discuss some results about modules over quantized symplectic resolutions from a joint project with A. Okounkov aimed at describing automorphisms of their derived categories. |

October 27, 3-5 p.m. | Davide Gaiotto (Perimeter Institute) |
Gauge theory and vertex operator algebras I will describe some gauge theory constructions of Vertex Operator Algebras. Dualities between gauge theories imply non-trivial relations for the corresponding VOAs, such as Feigin-Frenkel duality or coset constructions of W-algebras. I will discuss the implications of these constructions for the Geometric Langlands program and Symplectic Duality |

November 3, 3-5 p.m. | Antun Milas (SUNY-Albany) |
Partial and false theta functions in representation theory An important problem in vertex algebra theory is to study modular
properties of characters of representations. By now this is well-understand for
regular (after Y.Zhu) and C_2-cofinite (after Miyamoto) vertex algebras. But
there are vertex algebras that do not belong to either group yet their
characters satisfy interesting modular properties. An important family of
examples come from integrable irreducible highest weight modules over affine
superalgebras (after Kac and Wakimoto). These characters are known to be
related to mock theta functions of Ramanujan. |

November 10 | Veterans Day | |

November 17, 3-5 p.m. | No Seminar | |

November 24 | Thanksgiving | |

December 1, 3-5 p.m. | Denis Gaitsgory (Harvard) |
A chiral algebra responsible for quantum Langlands correspondence In this talk we will report on a recent progress made by B.Feigin and the speaker of constructing a certain chiral algebra A that carries a Kac-Moody symmetry for a G and its Langlands dual G^L (at Langlands dual levels \kappa and \kappa^L). The localization of A on Bun_G\times Bun_{G^L} provides a kernel for the global quantum Langlands equivalence D_{\kappa}(Bun_G) -> D_{\kappa^L}(Bun_{G^L}). The category of chiral modules over A is a category acted on by the product of the loop groups G((t)) \times G^L((t)) and provides a kernel for the local quantum Langlands equivalence G((t))-Cat_{\kappa} -> G^L((t))-Cat_{\kappa^L}. |

December 8, 3-5 p.m. | Zhengwei Liu (harvard) |
A new one-parameter centralizer algebra and higher Dynkin diagrams We introduce a new one-parameter centralizer algebra as a Fourier cousin of the Birman-Wenzl-Murakami algebra and prove unitarity at roots of unity. We find a new type of Schur-Weyl duality between this centralizer algebra and two families of quantum subgroups. We construct its irreducible representations and compute their quantum dimensions in a closed form. Its Bratteli diagram gives two families of Dynkin diagrams for Ocneanu's higher representation theory. |