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Documents  Ben-Zvi, David | enregistrements trouvés : 22

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Research talks;Algebraic and Complex Geometry;Mathematical Physics

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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Research talks

$T$-structures on derived categories of coherent sheaves are an important tool to encode both representation-theoretic and geometric information. Unfortunately there are only a limited amount of tools available for the constructions of such $t$-structures. We show how certain geometric/categorical quantum affine algebra actions naturally induce $t$-structures on the categories underlying the action. In particular we recover the categories of exotic sheaves of Bezrukavnikov and Mirkovic.
This is joint work with Sabin Cautis.
$T$-structures on derived categories of coherent sheaves are an important tool to encode both representation-theoretic and geometric information. Unfortunately there are only a limited amount of tools available for the constructions of such $t$-structures. We show how certain geometric/categorical quantum affine algebra actions naturally induce $t$-structures on the categories underlying the action. In particular we recover the categories of ...

14F05 ; 16E35

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Research talks

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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Research talks

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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Research talks

I will explain a relation between the center of the category of G1T modules and the cohomology of affine Springer fibers, and I'll discuss several related conjectures.
This is a joint work with P. Shan.

20C08 ; 14M15

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Research talks

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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Research talks

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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Research talks

We generalize the mathematical definition of Coulomb branches of 3-dimensional N = 4 SUSY quiver gauge theories to the cases of symmetrizable ones. We obtain generalized slices in affine Grassmannian of type BCFG as examples of the construction.
This is a joint work with Alex Weekes.

81T13 ; 81T60 ; 16G20

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Research talks

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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Research talks

The algebra $U(gl_n)$ contains a famous and beautiful commutative subalgebra, called the Gelfand-Tsetlin subalgebra. One problem which has attracted great attention over the recent decades is to classify the simple modules on which this subalgebra acts locally finitely (the Gelfand-Tsetlin modules). In investigating this question, Futorny and Ovsienko expanded attention to a generalization of these algebras, saddled with the unfortunate name of “principal Galois orders”. I’ll explain how all interesting known examples of these (and some unknown ones, such as the rational Cherednik algebras of $G(l,p,n)!)$ are the Coulomb branches of N = 4 3D gauge theories, and how this perspective allows us to classify the simple Gelfand-Tsetlin modules for $U(gl_n)$ and Cherednik algebras and explain the Koszul duality between Higgs and Coulomb categories O. The algebra $U(gl_n)$ contains a famous and beautiful commutative subalgebra, called the Gelfand-Tsetlin subalgebra. One problem which has attracted great attention over the recent decades is to classify the simple modules on which this subalgebra acts locally finitely (the Gelfand-Tsetlin modules). In investigating this question, Futorny and Ovsienko expanded attention to a generalization of these algebras, saddled with the unfortunate name of ...

17B10 ; 17B37

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Research talks;Algebraic and Complex Geometry;Mathematical Physics

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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Research talks

Heegaard-Floer theory is a 4-dimensional topological fi theory. It has been partially extended down to dimension 2: Lipshitz-Oszvath-Thurston constructed a differential algebra for a surface equipped with some extra structure. Douglas and Manolescu started building a partial extension down to dimension 1. I will discuss joint work with Andy Manion where we explain a gluing mechanism for surfaces. This is based on the construction of a monoidal 2-category of 2-representations of $gl(1|1)^+$. Heegaard-Floer theory is a 4-dimensional topological fi theory. It has been partially extended down to dimension 2: Lipshitz-Oszvath-Thurston constructed a differential algebra for a surface equipped with some extra structure. Douglas and Manolescu started building a partial extension down to dimension 1. I will discuss joint work with Andy Manion where we explain a gluing mechanism for surfaces. This is based on the construction of a monoidal ...

57R58 ; 57M27 ; 17B37

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Research talks

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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Research talks

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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Research talks

17B05 ; 22E46

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Research talks

We present here a bunch of questions (but almost no answers...) about partial resolutions/deformations of varieties of the form $(V × V^∗)/W$, where $W$ is a complex reflection groups, which are inspired by analogies with the representation theory of finite reductive groups.
Joint work with Raphaël Rouquier.

14L30 ; 20C33 ; 20G05 ; 20G40

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Research talks

Character varieties of closed surfaces have a natural Poisson structure whose quantization may be constructed in terms of the corresponding quantum group. When the quantum parameter is a root of unity, this quantization carries a central subalgebra isomorphic to the algebra of functions on the classical character variety. In this talk I will describe a procedure which allows one to obtain Azumaya algebras via quantum Hamiltonian reduction. As an application, I will show that quantizations of character varieties at roots of unity are Azumaya over the corresponding classical character varieties.
This is a report on joint work with Iordan Ganev and David Jordan.
Character varieties of closed surfaces have a natural Poisson structure whose quantization may be constructed in terms of the corresponding quantum group. When the quantum parameter is a root of unity, this quantization carries a central subalgebra isomorphic to the algebra of functions on the classical character variety. In this talk I will describe a procedure which allows one to obtain Azumaya algebras via quantum Hamiltonian reduction. As an ...

17B63 ; 14F05 ; 14L24 ; 16T20

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Research talks

Puzzles are combinatorial objects developed by Knutson and Tao for computing the expansion of the product of two Grassmannian Schubert classes. I will describe how selfdual puzzles give the restriction of a Grassmannian Schubert class to the symplectic Grassmannian in equivariant cohomology. The proof uses the machinery of quantum integrable systems. Time permitting, I will also discuss some ideas about how to interpret and generalize this result using Lagrangian correspondences and Maulik-Okounkov stable classes.
This is joint work in progress with Allen Knutson and Paul Zinn-Justin.
Puzzles are combinatorial objects developed by Knutson and Tao for computing the expansion of the product of two Grassmannian Schubert classes. I will describe how selfdual puzzles give the restriction of a Grassmannian Schubert class to the symplectic Grassmannian in equivariant cohomology. The proof uses the machinery of quantum integrable systems. Time permitting, I will also discuss some ideas about how to interpret and generalize this ...

14M15 ; 05E10

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Research talks

Skein algebras are certain diagrammatically defined algebras spanned by tangles drawn on the cylinder of a surface, with multiplication given by stacking diagrams. Quantum cluster algebras are certain systems of mutually birational quantum tori whose defining relations are encoded in a quiver drawn on the surface. The category of quantum character sheaves is a $q$-deformation of the category of ad-equivariant $D$-modules on the group $G$, expressed through an algebra $D_q (G)$ of “q-difference” operators on $G$.
In this I talk I will explain that these are in fact three sides of the same coin - namely they each arise as different flavors of factorization homology, and hence fit in the framework of four-dimensional topological field theory.
Skein algebras are certain diagrammatically defined algebras spanned by tangles drawn on the cylinder of a surface, with multiplication given by stacking diagrams. Quantum cluster algebras are certain systems of mutually birational quantum tori whose defining relations are encoded in a quiver drawn on the surface. The category of quantum character sheaves is a $q$-deformation of the category of ad-equivariant $D$-modules on the group $G$, ...

13F60 ; 16TXX ; 17B37 ; 58B32

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Research talks

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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