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Kapustin and Witten introduced a powerful perspective on the geometric Langlands correspondence as an aspect of electric-magnetic duality in four dimensional gauge theory. While the familiar (de Rham) correspondence is best seen as a statement in conformal field theory, much of the structure can be seen in the simpler (Betti) setting of topological field theory using Lurie's proof of the Cobordism Hypothesis. In these lectures I will explain this perspective and illustrate its applications to representation theory following joint work with Nadler as well as Brochier, Gunningham, Jordan and Preygel.[-]

Kapustin and Witten introduced a powerful perspective on the geometric Langlands correspondence as an aspect of electric-magnetic duality in four dimensional gauge theory. While the familiar (de Rham) correspondence is best seen as a statement in conformal field theory, much of the structure can be seen in the simpler (Betti) setting of topological field theory using Lurie's proof of the Cobordism Hypothesis. In these lectures I will explain ...[+]

14D24 ; 22E57 ; 22E46 ; 20G05

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Given a representation of a reductive group, Braverman-Finkelberg-Nakajima have defined a remarkable Poisson variety called the Coulomb branch. Their construction of this space was motivated by considerations from supersymmetric gauge theories and symplectic duality. The coordinate ring of this Coulomb branch is defined as a kind of cohomological Hall algebra; thus it makes sense to develop a type of “Springer theory” to define modules over this algebra. In this talk, we will explain this BFN Springer theory and give many examples. In the toric case, we will see a beautiful combinatorics of polytopes. In the quiver case, we will see connections to the representations of quivers over power series rings. In the general case, we will explore the relations between this Springer theory and quasimap spaces.[-]

Given a representation of a reductive group, Braverman-Finkelberg-Nakajima have defined a remarkable Poisson variety called the Coulomb branch. Their construction of this space was motivated by considerations from supersymmetric gauge theories and symplectic duality. The coordinate ring of this Coulomb branch is defined as a kind of cohomological Hall algebra; thus it makes sense to develop a type of “Springer theory” to define modules over this ...[+]

81T40 ; 81T60

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Let $G$ be a semisimple algebraic group of adjoint type. The universal centralizer is the family of centralizers in $G$ of regular elements in Lie$(G)$, parametrized by their conjugacy classes. It has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent bundle $T^∗ G$. We consider a partial compactification of the universal centralizer, where each centralizer fiber is replaced by its closure inside the wonderful compactification of $G$. The symplectic structure extends to a log-symplectic Poisson structure on this partial compactification, whose fibers are isomorphic to regular Hessenberg varieties.[-]

Let $G$ be a semisimple algebraic group of adjoint type. The universal centralizer is the family of centralizers in $G$ of regular elements in Lie$(G)$, parametrized by their conjugacy classes. It has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent bundle $T^∗ G$. We consider a partial compactification of the universal centralizer, where each centralizer fiber is replaced by its closure inside the wonderful ...[+]

20G05

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I will discuss applications of geometric representation theory to topological and quantum invariants of character stacks. In particular, I will explain how generalized Springer correspondence for class $D$-modules and Koszul duality for Hecke categories encode surprising structure underlying the homology of character stacks of surfaces (joint work with David Ben-Zvi and David Nadler). I will then report on some work in progress with David Jordan and Pavel Safronov concerning a q-analogue of these ideas. The applications include an approach towards Witten's conjecture on the fi dimensionality of skein modules, and methods for computing these dimensions in certain cases.[-]

I will discuss applications of geometric representation theory to topological and quantum invariants of character stacks. In particular, I will explain how generalized Springer correspondence for class $D$-modules and Koszul duality for Hecke categories encode surprising structure underlying the homology of character stacks of surfaces (joint work with David Ben-Zvi and David Nadler). I will then report on some work in progress with David Jordan ...[+]

14F10 ; 14D23

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In this talk I will give a short overview about fusion rings arising from quantum groups at odd and even roots of unities. These are Grothendieck rings of certain semisimple tensor categories. Then I will study these rings in more detail. The main focus of the talk will be an expectation by Cherednik that there is a certain DAHA action on these rings which can be used to describe the multiplication and semisimplicity of these rings. As a result we present a theorem which makes Cherednik's expectation rigorous.[-]

In this talk I will give a short overview about fusion rings arising from quantum groups at odd and even roots of unities. These are Grothendieck rings of certain semisimple tensor categories. Then I will study these rings in more detail. The main focus of the talk will be an expectation by Cherednik that there is a certain DAHA action on these rings which can be used to describe the multiplication and semisimplicity of these rings. As a result ...[+]

17B37 ; 20G42

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I will characterzize, among conical symplectic varieties, the nilpotent orbit closures of a complex semisimple Lie algebra and their finite coverings.

14E15 ; 14L30 ; 17B20

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S. Cautis and H. Williams identified the equivariant K-theory of the affine Grassmannian of $GL(n)$ with a quantum unipotent cell of $LSL(2)$. Under this identification the classes of irreducible equivariant perverse coherent sheaves go to the dual canonical basis.
This is a joint work with Ryo Fujita.

14M15 ; 13F60

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It is believed that certain physical duality underlies various versions of Langlands duality in its geometric incarnation. By setting up a mathematical model for relevant physical theories, we suggest a program that enriches mathematical subjects such as geometric Langlands theory and symplectic duality. This talk is based on several works, main parts of which are joint with Chris Elliott and with Justin Hilburn.

17B37 ; 22E57 ; 11R39 ; 53DXX

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Since Geordie Williamson showed that the exceptional primes for an algebraic group grow at least exponentially with the rank, the problem of calculating simple characters seems to be less approachable than ever before. In the talk I will give a short overview on recent results on simple characters, and I want to introduce a category that is rather elementary to define and still encodes the whole character problem.
This category is the result of joint work with Martina Lanini.[-]

Since Geordie Williamson showed that the exceptional primes for an algebraic group grow at least exponentially with the rank, the problem of calculating simple characters seems to be less approachable than ever before. In the talk I will give a short overview on recent results on simple characters, and I want to introduce a category that is rather elementary to define and still encodes the whole character problem.
This category is the result of ...[+]

17B15

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We explain how to use a Virasoro algebra to construct a solution to the Yang-Baxter equation acting in the tensor square of the cohomology of the Hilbert scheme of points on a generalsurface $S$. In the special case where the surface $S$ is $C^2$, the construction appears in work of Maulik and Okounkov on the quantum cohomology of symplectic resolutions and recovers their $R$-matrix constructed using stable envelopes.

17B62 ; 17B68 ; 17B05 ; 17B37

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Documents Ben-Zvi, David 21 résultats

Geometric Langlands correspondence and topological field theory - Part 2 - Ben-Zvi, David (Auteur de la Conférence) | CIRM H Nouveau

BFN Springer theory - Kamnitzer, Joel (Auteur de la Conférence) | CIRM H Nouveau

The partial compactification of the universal centralizer - Balibanu, Ana (Auteur de la Conférence) | CIRM H Nouveau

Character stacks and $(q-)$geometric representation theory - Gunningham, Sam (Auteur de la Conférence) | CIRM H Nouveau

Fusion rings from quantum groups and DAHA actions - Stroppel, Catharina (Auteur de la Conférence) | CIRM H Nouveau

Symplectic singularities and nilpotent orbits - Namikawa, Yoshinori (Auteur de la Conférence) | CIRM H Nouveau

Irreducible equivariant perverse coherent sheaves on affine Grassmannians of type $A$ and dual canonical bases - Finkelberg, Michael (Auteur de la Conférence) | CIRM H Nouveau

Langlands duality and quantum field theory - Yoo, Philsang (Auteur de la Conférence) | CIRM H Nouveau

A sheaf theoretic approach towards calculating simple characters of algebraic groups - Fiebig, Peter (Auteur de la Conférence) | CIRM H Nouveau

A geometric $R$-matrix for the Hilbert scheme of points on a general surface - Arbesfeld, Noah (Auteur de la Conférence) | CIRM H Nouveau