Organized by Semeon Artamonov, Nicolai Reshetikhin, and Vera Serganova

Sep. 15 | Milen Yakimov | Poisson geometry of large quantum groups | Slides, Video |

Sep. 22 | Bernhard Keller | Tate-Hochschild cohomology, the singularity category and applications | Slides, Video |

Oct. 20 | Theo Johnson-Freyd | 3+1d topological orders with (only) an emergent fermion | Slides, Video |

Oct. 27 | Maxime Fairon | Generalised RS systems from cyclic quivers | Slides, Video |

Nov. 3 | Meredith Shea | A generalized Gelfand-Yaglom formula in the discrete and continuous settings | Slides, Video |

Nov. 10 | Michael Gekhtman | Generalized cluster structures related to the Drinfeld double of GL(n) | Slides, Video |

Nov. 17 | Arkady Berenstein | Geometric multiplicities | Slides, Video |

Nov. 24 | No Seminar | ||

Dec. 1 | |||

Dec. 8 |

Spring-Summer 2020, Fall 2019, Spring 2019, Fall 2018, Fall 2017

In a celebrated sequence of works from the 1990s, De Concini, Kac and Procesi constructed a Poisson geometric framework for the study of the irreducible representations of big quantum groups at roots of unity. We will describe an extension of this framework to a large family of Drinfeld doubles in the setting of Nicholas algebras, which includes as a special case the family of all big quantum supergroups. This is done by a new method, based on perfect pairings between restricted and non-restricted integral forms, which does not rely on any direct computations of Poisson brackets and reductions to low rank cases. We will provide an intuitive introduction to all of the above notions. This is a joint work with Nicolas Andruskiewitsch and Ivan Angiono (University of Cordoba).

There are exactly two bosonic 3+1d topological orders whose only nontrivial quasiparticle is an emergent fermion (and exactly one whose only nontrivial quasiparticle is an emergent boson). I will explain the meaning of this sentence: I will explain what a "3+1d topological order" is, and how I know that these are the complete list. Time permitting, I will tell you some details about these specific topological orders, and say what this classification has to do with "minimal modular extensions".

In 2015, Chalykh and Silantyev observed that generalisations of the Calogero-Moser system with different types of spins can be constructed on quiver varieties associated to cyclic quivers. I will give an overview of their work where I will emphasise how the system can be encoded directly at the level of the path algebra of the quiver. This approach uses a noncommutative version of Poisson geometry based on double Poisson brackets due to Van den Bergh. I will then outline how this construction can be adapted to obtain generalisations of the Ruijsenaars-Schneider system if one uses Van den Bergh's formalism of double quasi-Poisson brackets associated to the same quivers.

The Gelfand-Yaglom formula relates the regularized determinant of a differential operator to the solution of an initial value problem. Here we develop a generalized Gelfand-Yaglom formula for a Hamiltonian system with Lagrangian boundary conditions in the discrete and continuous settings. Later we analyze the convergence of the discretized Hamilton-Jacobi operator and propose a lattice regularization for the determinant.

I will present a unifying approach to a construction of several generalized cluster structures of geometric type. Examples include the Drinfeld double of GL(n), spaces of periodic difference operators and generalized cluster structures in GL(n) compatible with a certain subclass of Belavin-Drinfeld Poisson-Lie brackets. Based on a joint work with M. Shapiro and A. Vainshtein.

The goal of my talk (based on joint paper with Yanpeng Li) is to introduce geometric multiplicities, which are positive varieties with potential fibered over the Cartan subgroup T of a reductive group G. They form a (unitless) monoidal category Mult_G and we construct a monoidal functor from Mult_G to the representation category of the Langlands dual group G^\vee of G. Using this, we explicitly compute various multiplicities in G^\vee-modules in many ways. In particular, we recover the formulas for tensor product multiplicities obtained jointly with Andrei Zelevinsky in 2001 and generalize them in several directions. In the case when our geometric multiplicity X an algebra in Mult_G hence the corresponding G^\vee-module is an algebra as well, we expect that the spectrum of this algebra is an affine G^\vee-variety X^\vee, and thus the correspondence X\mapsto X^\vee has a flavor of both the Langlands duality and mirror symmetry.