Séminaire "Groupes de Lie et espaces des modules"
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Section de Mathématiques, Université de GenèveRue du Lièvre 2-4, Genève |
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Section de Mathématiques, Université de GenèveRue du Lièvre 2-4, Genève |
Alejandro Cabrera (Rio de Janeiro)
Some results relating quantization and symplectic groupoids
In this talk, we plan to give an overview of some recent results involving concrete links between quantization of Poisson manifolds M and the underlying Lie-theoretic integration of M into (local) symplectic groupoids. In particular, we will discuss explicit relations to Kontsevich's star product for coordinate M and to its field-theoretic realization through the Poisson sigma model. Finally, we shall comment on missing links and further developments.
Zoom data will be communicated one day before the event
Tuesday, February 15, 2022, 15:30, Rue du Conseil Général 7-9, Room 1-07 and Zoom
Vladimir Bazhanov (Australian National University)
Scaling limit of the six-vertex model and two-dimensional black holes
In this talk I will report a detailed study of the scaling limit of a certain critical, integrable inhomogeneous six-vertex model subject to twisted boundary conditions. It is based on a numerical analysis of the Bethe ansatz equations as well as the powerful analytic technique of the ODE/IQFT correspondence. The results indicate that the critical behaviour of the lattice system is described by the gauged SL(2) WZW model with certain boundary and reality conditions imposed on the fields. Our proposal revises and extends the conjectured relation between the lattice system and the Euclidean black hole non-linear sigma model that was made Ikhlef, Jacobsen and Saleur in the 2011. This talk based on joint works with Gleb Kotousov, Sergei Lukyanov and Sergii Koval.
Tuesday, February 01, 2022, 15:00, Zoom
Arkady Berenstein (University of Oregon)
Geometric multiplicities
The goal of my talk (based on a joint paper with Yanpeng Li) is to introduce geometric multiplicities, which are positive varieties with potential fibered over the Cartan subgroup of a split 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 is an algebra in Mult_G (hence, the corresponding G^\vee-module is an algebra as well), we expect that the spectrum of the latter 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.
Tuesday, January 25, 2022, 15:30, Rue du Conseil Général 7-9, Room 1-07
Ilia Gaiur (Birmingham)
Hamiltonian description of isomonodromic deformations with irregular singularities
My talk will be mostly about my recent results obtained in collaboration with M. Mazzocco and V. Roubtsov (arXiv:2106.13760). We study isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the Takiff algebra (i.e. truncated current algebra). In my talk I will explain how to choose isomonodromic times in irregular situations, and how this choice may be explained from the Poisson point of view. Such choice covers a wide class of isomonodromic systems and is related to the classical Painlevé transcendents as well as to higher order and matrix Painlevé systems. I will also introduce a general formula for Hamiltonians of isomonodromic flows. In the end of the talk, I wish to discuss some questions about quantization of obtained systems and its connection with isomonodromic tau-function.
Zoom data will be sent one day before the event. Be aware of the unusual time!
Tuesday, December 21, 2021, 14:30, Rue du Conseil Général 7-9, Room 1-07 and Zoom
Donald Youmans (Bern)
Drinfeld-Sokolov reduction of 2d BF theory and the Schwarzian theory
In this talk we report on the holographic correspondence between 2d BF theory on the (punctured) disk with gauge group PSL(2,R) and the Schwarzian theory on the boundary. We realize the duality in terms of a Drinfeld-Sokolov reduction which is equivalent to enforcing a first class constraint. The constrained path integral localizes over (exceptional) Virasoro coadjoint orbits and can be computed by a formal Duistermaat-Heckman integration.
The talk is based on joint work with F. Valach.
Tuesday, November 30, 2021, 15:30, Rue du Conseil Général 7-9, Room 1-07 and Zoom
Jean Douçot (UniGE)
Towards the classification of wild character varieties
Moduli spaces of irregular meromorphic connections give rise, in their Betti description, to spaces of generalized monodromy data known as wild character varieties. Remarkably, it happens that wild character varieties associated to connections with different ranks, number of singularities, and irregular types may be isomorphic, which raises the question of their classification.
An approach towards this problem consists in associating to a connection a diagram encoding its singularity data, such that the same diagram can be read in several different ways as coming from different connections. I will review this story and discuss how to generalize known results to the case of connections with several irregular singularities.
Tuesday, November 09, 2021, 15:30, Rue du Conseil Général 7-9, Room 1-07
Vasily Golyshev (Moscow, Bures-sur-Yvette)
On character surfaces related to Markov's numbers
I will report on recent joint work with A. Mellit, V. Rubtsov, and D. van Straten. We study character identities that link GL_2-representations of the free group F_3 to representations into the group of units of an order in a quaternionic algebra, and find the kernel of an integral transform that relates these representations.
Tuesday, November 02, 2021, 15:30, Rue du Conseil Général 7-9, Room 1-07 and Zoom
Olga Trapeznikova (UniGE)
The Verlinde formula and parabolic Hecke correspondence
The Verlinde formula for the Hilbert function of the moduli space of vector bundles on a Riemann surface is one of the most beautiful results in enumerative geometry. I will present a new proof of this theorem (joint work with Andras Szenes) based on the wall-crossing technique and the Hecke correspondences on curves.
Tuesday, October 26, 2021, 15:30, Rue du Conseil Général 7-9
Aissa Wade (Penn State)
Modular classes of Poisson manifolds and Jacobi bundles
In the first part of the talk, we will review modular classes of Poisson manifolds and Lie algebroids. The study of modular classes of Poisson manifolds goes back to Koszul’s work in 1985. The modular class of a Poisson manifold is a class in the first Lichnerowicz-Poisson cohomology group that determines the obstruction to the existence of a volume form which is invariant under the flows of all Hamiltonian vector fields. In 1999, Evens, Lu and Weinstein defined the modular class of a Lie algebroid and they explained the relationship between the modular class of the cotangent Lie algebroid of a Poisson manifold and the modular class of its Poisson tensor.
In the second part of the talk, we will start by introducing the notion a Jacobi bundle, this is a line bundle L over a smooth manifold M equipped with a Lie bracket on sections of L which is a derivation with respect to each of its two entries. This is also known as Kirillov’s local algebra structure. When the line bundle is trivial, one recovers Lichnerowicz's notion of a Jacobi manifold. We will discuss modular classes of Jacobi bundles including the non-trivial case.
Hybrid: a Zoom link will be communicated one day before the event
Tuesday, June 01, 2021, 15:30, Rue du Conseil Général 7-9, Room 1-15 and Zoom
Shamil Shakirov (Geneva)
Genus two generalization of double affine Hecke algebra (DAHA)
TQFT provides algebraic representations for various interesting objects in Topology, for example, for mapping class groups of two-dimensional surfaces. This gives rise to two types of representations: by operators on finite-dimensional vector spaces, and by automorphisms of certain algebras on those vector spaces. We show that sometimes both these algebras and the mapping class group action admit an extra 1-parameter deformation. We give two examples for genus 1 and 2, and explain how this generalizes the spherical double affine Hecke algebra (DAHA).
Tuesday, May 18, 2021, 15:30, Zoom