Quantum structures in algebra and geometry

Schedule

Monday, Aug 26

9:20am - 9:30am		Opening Remarks
9:30am - 10:30am	Nakajima	Quiver gauge theories with symmetrizers Abstract We generalize the mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N}=4$ SUSY quiver gauge theories in arXiv:1503.03676, arXiv:1601.03686, arXiv:1604.03625 to the cases with symmetrizers. We obtain generalized affine Grassmannian slices of type BCFG as examples of the construction, and their deformation quantizations via truncated shifted Yangians. Finally, we study modules over these quantizations and relate them to the lower triangular part of the quantized enveloping algebra of type ADE. (Based on the joint work with Alex Weekes, arXiv:1907.06522.) Lecture Video Whiteboard Images (PDF)
10:30am - 11:00am		Coffee Break
11:00am - 12:00pm	Frenkel	An analytic version of the Langlands correspondence for complex curves Abstract The Langlands correspondence for complex curves is traditionally formulated in terms of sheaves rather than functions. Recently, Langlands asked whether it is possible to construct a function-theoretic version. In this talk, I will report on a joint work with P. Etingof and D. Kazhdan in which we use the algebra of global differential operators (both holomorphic and anti-holomorphic) on the moduli of G-bundles of a complex algebraic curve to develop such a theory, in which we link the spectrum of a self-adjoint extension of this algebra with the set of opers for the Langlands dual group of G satisfying a certain reality condition. Lecture Video
1:45pm - 2:45pm	Aganagic	Knot Categorification from Geometry and String Theory Abstract I will describe two geometric approaches to categorification of quantum invariants of knots, for any simple Lie algebra. In both cases, the fact that upon decategorification, one recovers the quantum knot invariants one started with, is manifest. The relation between the two approaches is two dimensional equivariant mirror symmetry. The two geometric approaches have the same string theory origin as the approach to based on a five dimensional gauge theory, due to Witten. The talk is based on joint work to appear, with Andrei Okounkov. Lecture Slides Lecture Video
3:00pm - 4:00pm	Ostrik	Tensor categories in positive characteristic Abstract A remarkable theorem of Deligne describes all symmetric tensor categories of subexponential growth over an algebraically closed field of characteristic zero in terms of affine supergroups. In this talk I will review what is known and what is not known in this direction over the fields of positive characteristic. Lecture Video
4:00pm - 4:30pm		Coffee Break
4:30pm - 5:30pm	Finkelberg	Geometric Satake equivalence for some supergroups Abstract We prove a particular case of a degenerate version of Gaiotto's conjecture, relating spherical perverse sheaves on the mirabolic affine Grassmannian to representations of gl(N\|N). This is a work of R.Travkin, V.Ginzburg and A.Braverman. Lecture Video

Tuesday, Aug 27

9:30am - 10:30am	S. Smirnov	2d Percolation Revisited Lecture Video
10:30am - 11:00am		Coffee Break
11:00am - 12:00pm	Schedler	D-modules on Poisson varieties Abstract To a complex whose differentials are differential operators one naturally associates a complex of D-modules. As an example, Poisson (co)homology is given by cohomology of D-modules. I will survey this theory and applications such as establishing finite-dimensionality, and (conjectural) ways to recover the de Rham cohomology of symplectic resolutions and special polynomials (Kostka, Tutte, q-weight multiplicities...). This includes joint work with Pavel Etingof and with Brent Pym. Lecture Video
1:45pm - 2:45pm	Braverman	From quantum field theories to equivalences of categories Abstract The purpose of this talk is to explain yet another application of QFT to geometric representation theory. Namely, I am going give a list of some (quite surprising from mathematical point of view) mostly conjectural equivalrences of categories all of which can be deduced from either 3d mirror symmetry for 3d N=4 gauge theories or from S-duality for 4d gauge theories. The equivalences in question include some pretty far going generalizatons of Laumon's local geometric class field theories as well as generalizations of "mirabolic Satake equivalence" which will be dicsussed in Finkelberg's talk. Lecture Video
3:00pm - 4:00pm	Ginzburg	Parabolic restriction and the Harish-Chandra D-module Abstract Let G be a connected reductive group and L a Levi subgroup. Parabolic induction and restriction is a pair of adjoint functors between Ad G-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) D-modules on G and L, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where the Levi is a maximal torus T. We give a simple interpretation of parabolic induction/restriction of D-modules in terms of Hamiltonian reduction and also discuss connections with Whittaker D-modules and results of Tsao-Hsien Chen on central sheaves. The link between geometric and algebraic approaches is provided by the Harish-Chandra D-module. Lecture Video
4:00pm - 4:30pm		Coffee Break
4:30pm - 5:30pm	Carlsson	Nabla positivity and Tor groups Abstract I will describe some new formulas with A. Mellit for the nabla operator from Haiman-Garsia-Macdonald theory in two different modified elementary symmetric function bases. I will explain their geometric origin, and pose some new conjectures for which these formulas would be a corollary, that would imply the conjectured positivity of this operator in the Schur basis, and are also related to certain Tor groups studied by Goresky, Kottwitz, and Macpherson related to the affine Springer fiber in type A. Lecture Slides

Wednesday, Aug 28

9:30am - 10:30am	Gaitsgory	An extension of Kazhdan-Lusztig equivalence via factorization algebras Abstract Last year, I proposed a conjecture that extends the Kazhdan-Lusztig equivalence beween G(O)-integrable representations of an affine Kac-Moody Lie algebra and modules over the quantum group. This conjecture says that the category of Iwahori-integrable Kac-Moody modules is equivalent to the category of modules over the "mixed" quantum group (it has Lusztig's positive part and Kac-DeConcini negative part). In this talk, I'll report on a possible approach to prove this conjecture, using factorization algebras. Lecture Video
10:30am - 11:00am		Coffee Break
11:00am - 12:00pm	Gorsky	From Hecke category to the Hilbert scheme of points Abstract Jones, Ocneanu and others defined a trace map from the type A Hecke algebra to symmetric functions. I will categorify this map by describing two constructions of trace-like functors from the Hecke category (in the form of Soergel bimodules) to the derived category of the Hilbert scheme of points on the plane. The talk is based on joint works with Hogancamp, Negut, Rasmussen and Wedrich. Lecture Video
1:45pm - 2:45pm	Kenyon	Gradient models and the Hessian Abstract This is joint work with Istvan Prause. We study models of surfaces in R^3 minimizing a surface tension function depending only on slope. Examples are limit shapes arising in the dimer model and six-vertex model. Under certain conditions, originally studied by Darboux, the resulting Euler-Lagrange equation can be solved explicitly, and solutions can be parameterized using analytic functions. This was successfully carried out by Kenyon and Okounkov for the dimer model. For the 5-vertex model, which generalizes the dimer model, the Hessian of the surface tension is the fourth power of a harmonic function in the natural conformal coordinate. This condition allows explicit analytic solutions of the EL equations as above, and leads us to a general solution of the ``isoradial" 5-vertex model. Lecture Slides Lecture Video
3:00pm - 4:00pm	Negut	R-matrices and shuffle algebras Abstract Feigin-Odesskii-type shuffle algebras are certain spaces of Laurent polynomials, endowed with a shuffle product that is "twisted" by a certain rational function. Among their uses is the fact that they provide models for the "left" and "right" halves of quantum affinizations, such as U_{q,q'}(gl_n^^). We present a different kind of shuffle algebra, defined as a certain space of matrix-valued Laurent polynomials, and endowed with a shuffle product that is "twisted" by a certain R-matrix. We show that these algebras provide models for the "top" and "bottom" halves of U_{q,q'}(gl_n^^). As an application, we can define a topological coproduct on the quantum toroidal algebra which extends the usual Drinfeld-Jimbo coproduct on the horizontal subalgebra U_q(gl_n^). Lecture Video [Part 1] Lecture Video [Part 2]
4:00pm - 4:30pm		Coffee Break
4:30pm - 5:30pm	A. Smirnov	Elliptic stable envelope for $\widehat{A}_n$-quiver varieties Abstract The elliptic stable envelopes, defined recently by M.Aganagic and A.Okounkov, is a geometric approach to representation theory of elliptic dynamical quantum groups associated with quiver varieties. In this talk I discuss explicit construction of elliptic stable envelopes in the case of $\widehat{A}_n$-quivers. If time permits, I will discuss applications to the theory of symmetric functions (in particular, elliptic dynamical Macdonald polynomials) and Bethe ansatz for quantum toroidal algebras. Lecture Slides

Thursday, Aug 29

9:30am - 10:30am	Nekrasov	Old and new correspondences in quantum and classical through correspondence with Andrei and Pasha Lecture Video
10:30am - 11:00am		Coffee Break
11:00am - 12:00pm	Melissa Liu	Gromov-Witten invariants of quintic Calabi-Yau threefolds Abstract In this talk, I will survey some conjectures and results on higher genus Gromov-Witten invariants of quintic Calabi-Yau threefolds.
12:15pm - 1:15pm	Oblomkov	Matrix factorizations, Soergel bimodules and HOMFLYPT homology. Abstract In the joint recent papers with Lev Rozansky we developed a theory of particular monoidal 2-categories that allows us to interpret HOMFLYPT homology of link as a vector space of global sections of the corresponding coherent sheaf on $Hilb_n(C^2)$. This geometric interpretation provides a simple proof of the categorified $q->1/q$ symmetry of the HOMFLYPT homology of knots as well as a simple explanation of the relevance of the Jucy-Murphy algebra for the localization formulas for the knot homology. A starting point of the construction of the above mentioned homology is a monoidal category of equivariant matrix factorizations. The first construction of HOMFLYPT homology due to Khovanov and Rozansky uses Soergel bimodules. In my talk I explain why the matrix factorization HOMFLYPT homology matches with the Soergel bimodule HOMFLYPT homology. Based on the joint project with Lev Rozansky. Lecture Video

Friday, Aug 30

9:30am - 10:30am	Rains	Elliptic double affine Hecke algebras Abstract One of the early applications of double affine Hecke algebras (DAHAs) was in the theory of Macdonald polynomials for general root systems. Over the past decade-plus, much of the theory for type $C$ (Koornwinder polynomials) has been extended to "elliptic" analogues (in which the multiplicative group is replaced by an elliptic curve), strongly suggesting that there should be a corresponding elliptic DAHA. I'll describe how to associate an affine Hecke algebra to any Coxeter group (possibly infinite) with a suitable action on an abelian variety, which in the case of an affine Weyl group gives an elliptic DAHA. (The connection to elliptic Macdonald polynomials remains open!) As an application, I'll show how the $C_n$ case gives rise to noncommutative deformations of the symmetric powers (and conjecturally the $n$-point Hilbert schemes) of any rational surface with a smooth anticanonical curve. Lecture Slides Lecture Video
10:30am - 11:00am		Coffee Break
11:00am - 12:00pm	Halpern-Leistner	Infinite dimensional GIT and gauged Gromov-Witten theory Abstract Harder-Narasimhan (HN) theory gives a structure theorem for principal G bundles on a smooth projective curve. A bundle is either “semistable”, or it admits a canonical “filtration” whose associated graded bundle is semistable in a graded sense. After reviewing recent advances in extending HN theory to arbitrary algebraic stacks, I will discuss work in progress with Pablo Solis and Eduardo Gonzalez to apply this general machinery to the stack of "gauged" maps from a curve C to a projective G-scheme X, where G is a reductive group. We formulate the problem as an example of an infinite dimensional analog of the usual set up of geometric invariant theory. Our main application is to use HN theory for gauged maps to compute generating functions for K-theoretic gauged Gromov-Witten invariants. In principle, these can be related via wall-crossing formulas to the K-theoretic Gromov-Witten invariants of the GIT quotient of X. Lecture Video
12:15pm - 1:15pm	Borodin	Algebraic Fourier bases and probability Lecture Slides Lecture Video

Schedule

Monday, Aug 26

9:20am - 9:30am

Opening Remarks
9:30am - 10:30am

Nakajima

Quiver gauge theories with symmetrizers

Abstract
We generalize the mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N}=4$ SUSY quiver gauge theories in arXiv:1503.03676, arXiv:1601.03686, arXiv:1604.03625 to the cases with symmetrizers. We obtain generalized affine Grassmannian slices of type BCFG as examples of the construction, and their deformation quantizations via truncated shifted Yangians. Finally, we study modules over these quantizations and relate them to the lower triangular part of the quantized enveloping algebra of type ADE. (Based on the joint work with Alex Weekes, arXiv:1907.06522.)

Lecture Video

Whiteboard Images (PDF)
10:30am - 11:00am

Coffee Break
11:00am - 12:00pm

Frenkel

An analytic version of the Langlands correspondence for complex curves

Abstract
The Langlands correspondence for complex curves is traditionally formulated in terms of sheaves rather than functions. Recently, Langlands asked whether it is possible to construct a function-theoretic version. In this talk, I will report on a joint work with P. Etingof and D. Kazhdan in which we use the algebra of global differential operators (both holomorphic and anti-holomorphic) on the moduli of G-bundles of a complex algebraic curve to develop such a theory, in which we link the spectrum of a self-adjoint extension of this algebra with the set of opers for the Langlands dual group of G satisfying a certain reality condition.

Lecture Video
1:45pm - 2:45pm

Aganagic

Knot Categorification from Geometry and String Theory

Abstract
I will describe two geometric approaches to categorification of quantum invariants of knots, for any simple Lie algebra. In both cases, the fact that upon decategorification, one recovers the quantum knot invariants one started with, is manifest. The relation between the two approaches is two dimensional equivariant mirror symmetry. The two geometric approaches have the same string theory origin as the approach to based on a five dimensional gauge theory, due to Witten. The talk is based on joint work to appear, with Andrei Okounkov.

Lecture Slides
Lecture Video
3:00pm - 4:00pm

Ostrik

Tensor categories in positive characteristic

Abstract
A remarkable theorem of Deligne describes all symmetric tensor categories of subexponential growth over an algebraically closed field of characteristic zero in terms of affine supergroups. In this talk I will review what is known and what is not known in this direction over the fields of positive characteristic.

Lecture Video
4:00pm - 4:30pm

Coffee Break
4:30pm - 5:30pm

Finkelberg

Geometric Satake equivalence for some supergroups

Abstract
We prove a particular case of a degenerate version of Gaiotto's conjecture, relating spherical perverse sheaves on the mirabolic affine Grassmannian to representations of gl(N|N). This is a work of R.Travkin, V.Ginzburg and A.Braverman.

Lecture Video

Tuesday, Aug 27

9:30am - 10:30am

S. Smirnov

2d Percolation Revisited

Lecture Video
10:30am - 11:00am

Coffee Break
11:00am - 12:00pm

Schedler

D-modules on Poisson varieties

Abstract
To a complex whose differentials are differential operators one naturally associates a complex of D-modules. As an example, Poisson (co)homology is given by cohomology of D-modules. I will survey this theory and applications such as establishing finite-dimensionality, and (conjectural) ways to recover the de Rham cohomology of symplectic resolutions and special polynomials (Kostka, Tutte, q-weight multiplicities...). This includes joint work with Pavel Etingof and with Brent Pym.

Lecture Video
1:45pm - 2:45pm

Braverman

From quantum field theories to equivalences of categories

Abstract
The purpose of this talk is to explain yet another application of QFT to geometric representation theory. Namely, I am going give a list of some (quite surprising from mathematical point of view) mostly conjectural equivalrences of categories all of which can be deduced from either 3d mirror symmetry for 3d N=4 gauge theories or from S-duality for 4d gauge theories. The equivalences in question include some pretty far going generalizatons of Laumon's local geometric class field theories as well as generalizations of "mirabolic Satake equivalence" which will be dicsussed in Finkelberg's talk.

Lecture Video
3:00pm - 4:00pm

Ginzburg

Parabolic restriction and the Harish-Chandra D-module

Abstract
Let G be a connected reductive group and L a Levi subgroup. Parabolic induction and restriction is a pair of adjoint functors between Ad G-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) D-modules on G and L, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where the Levi is a maximal torus T. We give a simple interpretation of parabolic induction/restriction of D-modules in terms of Hamiltonian reduction and also discuss connections with Whittaker D-modules and results of Tsao-Hsien Chen on central sheaves. The link between geometric and algebraic approaches is provided by the Harish-Chandra D-module.

Lecture Video
4:00pm - 4:30pm

Coffee Break
4:30pm - 5:30pm

Carlsson

Nabla positivity and Tor groups

Abstract
I will describe some new formulas with A. Mellit for the nabla operator from Haiman-Garsia-Macdonald theory in two different modified elementary symmetric function bases. I will explain their geometric origin, and pose some new conjectures for which these formulas would be a corollary, that would imply the conjectured positivity of this operator in the Schur basis, and are also related to certain Tor groups studied by Goresky, Kottwitz, and Macpherson related to the affine Springer fiber in type A.
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Lecture Slides

Wednesday, Aug 28

9:30am - 10:30am

Gaitsgory

An extension of Kazhdan-Lusztig equivalence via factorization algebras

Abstract
Last year, I proposed a conjecture that extends the Kazhdan-Lusztig equivalence beween G(O)-integrable representations of an affine Kac-Moody Lie algebra and modules over the quantum group. This conjecture says that the category of Iwahori-integrable Kac-Moody modules is equivalent to the category of modules over the "mixed" quantum group (it has Lusztig's positive part and Kac-DeConcini negative part). In this talk, I'll report on a possible approach to prove this conjecture, using factorization algebras.

Lecture Video
10:30am - 11:00am

Coffee Break
11:00am - 12:00pm

Gorsky

From Hecke category to the Hilbert scheme of points

Abstract
Jones, Ocneanu and others defined a trace map from the type A Hecke algebra to symmetric functions. I will categorify this map by describing two constructions of trace-like functors from the Hecke category (in the form of Soergel bimodules) to the derived category of the Hilbert scheme of points on the plane. The talk is based on joint works with Hogancamp, Negut, Rasmussen and Wedrich.

Lecture Video
1:45pm - 2:45pm

Kenyon

Gradient models and the Hessian

Abstract
This is joint work with Istvan Prause. We study models of surfaces in R^3 minimizing a surface tension function depending only on slope. Examples are limit shapes arising in the dimer model and six-vertex model. Under certain conditions, originally studied by Darboux, the resulting Euler-Lagrange equation can be solved explicitly, and solutions can be parameterized using analytic functions. This was successfully carried out by Kenyon and Okounkov for the dimer model.

For the 5-vertex model, which generalizes the dimer model, the Hessian of the surface tension is the fourth power of a harmonic function in the natural conformal coordinate. This condition allows explicit analytic solutions of the EL equations as above, and leads us to a general solution of the ``isoradial" 5-vertex model.

Lecture Slides

Lecture Video
3:00pm - 4:00pm

Negut

R-matrices and shuffle algebras

Abstract
Feigin-Odesskii-type shuffle algebras are certain spaces of Laurent polynomials, endowed with a shuffle product that is "twisted" by a certain rational function. Among their uses is the fact that they provide models for the "left" and "right" halves of quantum affinizations, such as U_{q,q'}(gl_n^^). We present a different kind of shuffle algebra, defined as a certain space of matrix-valued Laurent polynomials, and endowed with a shuffle product that is "twisted" by a certain R-matrix. We show that these algebras provide models for the "top" and "bottom" halves of U_{q,q'}(gl_n^^). As an application, we can define a topological coproduct on the quantum toroidal algebra which extends the usual Drinfeld-Jimbo coproduct on the horizontal subalgebra U_q(gl_n^).

Lecture Video [Part 1]

Lecture Video [Part 2]
4:00pm - 4:30pm

Coffee Break
4:30pm - 5:30pm

A. Smirnov

Elliptic stable envelope for $\widehat{A}_n$-quiver varieties.

Abstract
The elliptic stable envelopes, defined recently by M.Aganagic and A.Okounkov, is a geometric approach to representation theory of elliptic dynamical quantum groups associated with quiver varieties. In this talk I discuss explicit construction of elliptic stable envelopes in the case of $\widehat{A}_n$-quivers. If time permits, I will discuss applications to the theory of symmetric functions (in particular, elliptic dynamical Macdonald polynomials) and Bethe ansatz for quantum toroidal algebras.

Lecture Slides

Thursday, Aug 29

9:30am - 10:30am

Nekrasov

Old and new correspondences in quantum and classical through correspondence with Andrei and Pasha

Lecture Video
10:30am - 11:00am

Coffee Break
11:00am - 12:00pm

Melissa Liu

Gromov-Witten invariants of quintic Calabi-Yau threefolds

Abstract
In this talk, I will survey some conjectures and results on higher genus Gromov-Witten invariants of quintic Calabi-Yau threefolds.
12:15pm - 1:15pm

Oblomkov

Matrix factorizations, Soergel bimodules and HOMFLYPT homology

Abstract
In the joint recent papers with Lev Rozansky we developed a theory of particular monoidal 2-categories that allows us to interpret HOMFLYPT homology of link as a vector space of global sections of the corresponding coherent sheaf on $Hilb_n(C^2)$. This geometric interpretation provides a simple proof of the categorified $q->1/q$ symmetry of the HOMFLYPT homology of knots as well as a simple explanation of the relevance of the Jucy-Murphy algebra for the localization formulas for the knot homology. A starting point of the construction of the above mentioned homology is a monoidal category of equivariant matrix factorizations. The first construction of HOMFLYPT homology due to Khovanov and Rozansky uses Soergel bimodules. In my talk I explain why the matrix factorization HOMFLYPT homology matches with the Soergel bimodule HOMFLYPT homology. Based on the joint project with Lev Rozansky.

Lecture Video

Friday, Aug 30

9:30am - 10:30am

Rains

Elliptic double affine Hecke algebras

Abstract
One of the early applications of double affine Hecke algebras (DAHAs) was in the theory of Macdonald polynomials for general root systems. Over the past decade-plus, much of the theory for type $C$ (Koornwinder polynomials) has been extended to "elliptic" analogues (in which the multiplicative group is replaced by an elliptic curve), strongly suggesting that there should be a corresponding elliptic DAHA. I'll describe how to associate an affine Hecke algebra to any Coxeter group (possibly infinite) with a suitable action on an abelian variety, which in the case of an affine Weyl group gives an elliptic DAHA. (The connection to elliptic Macdonald polynomials remains open!) As an application, I'll show how the $C_n$ case gives rise to noncommutative deformations of the symmetric powers (and conjecturally the $n$-point Hilbert schemes) of any rational surface with a smooth anticanonical curve.

Lecture Slides

Lecture Video
10:30am - 11:00am

Coffee Break
11:00am - 12:00pm

Halpern-Leistner

Infinite dimensional GIT and gauged Gromov-Witten theory

Abstract
Harder-Narasimhan (HN) theory gives a structure theorem for principal G bundles on a smooth projective curve. A bundle is either “semistable”, or it admits a canonical “filtration” whose associated graded bundle is semistable in a graded sense. After reviewing recent advances in extending HN theory to arbitrary algebraic stacks, I will discuss work in progress with Pablo Solis and Eduardo Gonzalez to apply this general machinery to the stack of "gauged" maps from a curve C to a projective G-scheme X, where G is a reductive group. We formulate the problem as an example of an infinite dimensional analog of the usual set up of geometric invariant theory. Our main application is to use HN theory for gauged maps to compute generating functions for K-theoretic gauged Gromov-Witten invariants. In principle, these can be related via wall-crossing formulas to the K-theoretic Gromov-Witten invariants of the GIT quotient of X.

Lecture Video
12:15pm - 1:15pm

Borodin

Algebraic Fourier bases and probability

Lecture Slides

Lecture Video

Quantum structures in algebra and geometry

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Schedule

Monday, Aug 26

Tuesday, Aug 27

Wednesday, Aug 28

Thursday, Aug 29

Friday, Aug 30

Schedule

Monday, Aug 26

Tuesday, Aug 27

Wednesday, Aug 28

Thursday, Aug 29

Friday, Aug 30

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