2023 Winter School in Mathematical Physics

8 Jan 2023, 19:00 → 13 Jan 2023, 14:00 Europe/Zurich

Maison des Congrès

Anton Alexeev (Universite de Geneve (CH)), Elise Raphael, Giovanni Felder

Mini courses by

• Tudor Dimofte (The University of Edinburgh)
• Alexander Braverman (University of Toronto)
• Nicolas Orantin (UNIGE)

 

Organized by

• Anton Alekseev (UNIGE)
• Alberto Cattaneo (UZH)
• Giovanni Felder (ETH Zürich)
• Maria Podkopaeva (IHES)
• Thomas Strobl (U. Lyon 1)
• Andras Szenes (UNIGE)

 

Registration deadline: December 4, 2022

 

*Note that this event won't be streamed.

 

Braverman_problems 1 .pdf

Dimofte_problems 1- new version.pdf

Dimofte_problems1_sol.pdf

Dimofte_problems2.pdf

Orantin_Exercises 1.pdf

Orantin_Exercises 2.pdf

Sunday 8 January

19:15 → 20:30 Dinner

Monday 9 January

09:00 → 09:50 Alexander Braverman, "Affine Grassmannian and symplectic duality": Lecture 1

The purpose of this mini-course is two-fold.
In the first half I plan to start by (very briefly!) recalling some basic notions of the theory of constructible sheaves on complex algebraic varieties (such as, for example, equivariant derived categories of constructible sheaves, perverse sheaves etc.) and then apply this formalism to the study of certain derived category of equivariant sheaves on the affine Grassmannian of a complex reductive group G (here the main goal will be to explain the formulation of the so called derived geometric Satake equivalence due to Drinfled and Bezrukavnikov-Finkelberg).
In the 2nd half I plan to explain applications of the above to the subject of symplectic duality. I plan to discuss topics such as (conical) symplectic resolutions and their quantizations, formulations of (various aspects of) symplectic duality,Braverman-Finkelberg-Nakajima construction of Coulomb branches of 3d N=4 gauge theories via the affine Grassmannian and its relation to symplectic duality.
Some further topics might be discussed depending on how much time is left.

09:50 → 10:10 Coffee Break

10:10 → 11:00 Alexander Braverman, "Affine Grassmannian and symplectic duality": Lecture 2

The purpose of this mini-course is two-fold.
In the first half I plan to start by (very briefly!) recalling some basic notions of the theory of constructible sheaves on complex algebraic varieties (such as, for example, equivariant derived categories of constructible sheaves, perverse sheaves etc.) and then apply this formalism to the study of certain derived category of equivariant sheaves on the affine Grassmannian of a complex reductive group G (here the main goal will be to explain the formulation of the so called derived geometric Satake equivalence due to Drinfled and Bezrukavnikov-Finkelberg).
In the 2nd half I plan to explain applications of the above to the subject of symplectic duality. I plan to discuss topics such as (conical) symplectic resolutions and their quantizations, formulations of (various aspects of) symplectic duality,Braverman-Finkelberg-Nakajima construction of Coulomb branches of 3d N=4 gauge theories via the affine Grassmannian and its relation to symplectic duality.
Some further topics might be discussed depending on how much time is left.

11:10 → 12:00 Tudor Dimofte, "Algebra, geometry, and twists in 3d N=4 gauge theory": Lecture 1

Three dimensional N=4 gauge theories have enjoyed a rich interplay with geometry and representation theory since the 90's, particularly motivated by the discovery of 3d mirror symmetry. An initial prediction of 3d mirror symmetry was the existence of many pairs of hyperkahler cones "M_H" and "M_C" (Higgs and Coulomb branches), with the rank of the hyperkahler isometry group of M_H equal to the dimension of the resolution space of M_C, and vice versa. This may be seen as an analogue of 2d mirror symmetry relating pairs Calabi-Yau manifolds whose Betti numbers are swapped in an appropriate way.

In the last decade, the interplay between the physics of 3d N=4 gauge theories and mathematics has greatly intensified, in part due to the advent of new methods to access the structure of local and extended operators in these 3d QFT's. A broad goal of my lectures will be to review some of the recent developments, especially those coming from mathematics, while connecting them directly to fundamental ideas in 3d physics. I will organize many of these developments within the framework of topological (and occasionally holomorphic) "twists" of 3d N=4 gauge theories, and the extended TQFT's (topological quantum field theories) that they predict to exist. This leads into a major current research goal: defining and proving a top-level "3d homological mirror symmetry" equivalence of 2-categories, that would encompass all other mirror equivalences.

More specifically, I will touch on: * the physical definition of 3d N=4 gauge theories, and their topological twists * algebraic and geometric structures predicted by these twists -- within the framework of 3d TQFT * what we know (and do not know) about 3d mirror symmetry * constructions of 3d Coulomb branches, e.g. in work of Braverman-Finkelberg-Nakajima * holomorphic twists and elliptic stable envelopes (in brief) * a current frontier: braided tensor categories of line operators and 2-categories of boundary conditions * the appearance and utility of vertex operator algebras in describing line operators * algebraic vs. analytic/symplectic geometry in making mathematical sense of physical structure * relations between 3d mirror symmetry, 2d mirror symmetry, and 4d S-duality

12:15 → 13:30 Lunch

16:30 → 17:00 Coffee Break

17:15 → 18:00 Alexander Braverman, "Affine Grassmannian and symplectic duality": Exercises 1

The purpose of this mini-course is two-fold.
In the first half I plan to start by (very briefly!) recalling some basic notions of the theory of constructible sheaves on complex algebraic varieties (such as, for example, equivariant derived categories of constructible sheaves, perverse sheaves etc.) and then apply this formalism to the study of certain derived category of equivariant sheaves on the affine Grassmannian of a complex reductive group G (here the main goal will be to explain the formulation of the so called derived geometric Satake equivalence due to Drinfled and Bezrukavnikov-Finkelberg).
In the 2nd half I plan to explain applications of the above to the subject of symplectic duality. I plan to discuss topics such as (conical) symplectic resolutions and their quantizations, formulations of (various aspects of) symplectic duality,Braverman-Finkelberg-Nakajima construction of Coulomb branches of 3d N=4 gauge theories via the affine Grassmannian and its relation to symplectic duality.
Some further topics might be discussed depending on how much time is left.

18:15 → 19:00 Tudor Dimofte, "Algebra, geometry, and twists in 3d N=4 gauge theory": Exercises 1

Three dimensional N=4 gauge theories have enjoyed a rich interplay with geometry and representation theory since the 90's, particularly motivated by the discovery of 3d mirror symmetry. An initial prediction of 3d mirror symmetry was the existence of many pairs of hyperkahler cones "M_H" and "M_C" (Higgs and Coulomb branches), with the rank of the hyperkahler isometry group of M_H equal to the dimension of the resolution space of M_C, and vice versa. This may be seen as an analogue of 2d mirror symmetry relating pairs Calabi-Yau manifolds whose Betti numbers are swapped in an appropriate way.

In the last decade, the interplay between the physics of 3d N=4 gauge theories and mathematics has greatly intensified, in part due to the advent of new methods to access the structure of local and extended operators in these 3d QFT's. A broad goal of my lectures will be to review some of the recent developments, especially those coming from mathematics, while connecting them directly to fundamental ideas in 3d physics. I will organize many of these developments within the framework of topological (and occasionally holomorphic) "twists" of 3d N=4 gauge theories, and the extended TQFT's (topological quantum field theories) that they predict to exist. This leads into a major current research goal: defining and proving a top-level "3d homological mirror symmetry" equivalence of 2-categories, that would encompass all other mirror equivalences.

More specifically, I will touch on: * the physical definition of 3d N=4 gauge theories, and their topological twists * algebraic and geometric structures predicted by these twists -- within the framework of 3d TQFT * what we know (and do not know) about 3d mirror symmetry * constructions of 3d Coulomb branches, e.g. in work of Braverman-Finkelberg-Nakajima * holomorphic twists and elliptic stable envelopes (in brief) * a current frontier: braided tensor categories of line operators and 2-categories of boundary conditions * the appearance and utility of vertex operator algebras in describing line operators * algebraic vs. analytic/symplectic geometry in making mathematical sense of physical structure * relations between 3d mirror symmetry, 2d mirror symmetry, and 4d S-duality

19:15 → 20:30 Dinner

Tuesday 10 January

09:00 → 09:50 Nicolas Orantin, "From enumerative geometry to differential equations and isomonodromic systems via topological recursion": Lecture 1

The study of random matrices has a fascinating connection with different fields of mathematics and physics ranging from number theory to topological string theories. Among such applications, the computation of moments of random matrices can be interpreted as defining generating series of discrete surfaces built by gluing together polygons by their edges. Trying to solve such a combinatorial problem, together with Chekhov and Eynard, we found a beautiful formula allowing to enumerate such surfaces by induction on their Euler characteristic. This inductive procedure, later called topological recursion, turns out to provide a universal solution to a large class of enumerative problems such as the study of statistical systems on a random lattice or the computation of integrals over the moduli space of Riemann surfaces with respect to different measures (including Gromov-Witten invariants or volumes of such moduli spaces).

In a first part of these lectures, I will introduce the formalism of topological recursion by explaining how one can transform the combinatorial problem of enumerating discrete surfaces into the
computation of residue integrals on an associated Riemann surface thanks to the definition of good generating series. The general idea is that, if one can find a generating series for the number of discs such that it defines a function on a Riemann surface, then applying topological recursion with this Riemann surface as initial data produces generating functions for the number of surfaces with any topology by induction on their Euler characteristic.

I will then give some more examples of application of the recursive formula discovered in this context to other enumeration problems including Hurwitz numbers, some volumes of the moduli space of Riemann surfaces and the expression of correlators of any semi-simple cohomological field theories.

In a second part of these lecture, I will present a completely different flavour of the topological recursion. Taking as input an algebraic curve defined by a polynomial equation E(x,y) = 0, topological recursion allows to build an eigenfunction of the operator $E(x,\hat{y})$ obtained by quantising $(x,y)->(x,\hat{y})$ where $\hat{y} = \hbar d/dx$. I will explain this result and show how it gives rise to a large family of isomonodromic tau functions, providing for example solutions to Painleve equations. The simplest example of this phenomenon is that the Kontsevich's generating function of intersection numbers on the moduli space of Riemann surfaces is a formal solution to the Airy equation.

09:50 → 10:10 Coffee Break

10:10 → 11:00 Nicolas Orantin, "From enumerative geometry to differential equations and isomonodromic systems via topological recursion": Lecture 2

The study of random matrices has a fascinating connection with different fields of mathematics and physics ranging from number theory to topological string theories. Among such applications, the computation of moments of random matrices can be interpreted as defining generating series of discrete surfaces built by gluing together polygons by their edges. Trying to solve such a combinatorial problem, together with Chekhov and Eynard, we found a beautiful formula allowing to enumerate such surfaces by induction on their Euler characteristic. This inductive procedure, later called topological recursion, turns out to provide a universal solution to a large class of enumerative problems such as the study of statistical systems on a random lattice or the computation of integrals over the moduli space of Riemann surfaces with respect to different measures (including Gromov-Witten invariants or volumes of such moduli spaces).

In a first part of these lectures, I will introduce the formalism of topological recursion by explaining how one can transform the combinatorial problem of enumerating discrete surfaces into the
computation of residue integrals on an associated Riemann surface thanks to the definition of good generating series. The general idea is that, if one can find a generating series for the number of discs such that it defines a function on a Riemann surface, then applying topological recursion with this Riemann surface as initial data produces generating functions for the number of surfaces with any topology by induction on their Euler characteristic.

I will then give some more examples of application of the recursive formula discovered in this context to other enumeration problems including Hurwitz numbers, some volumes of the moduli space of Riemann surfaces and the expression of correlators of any semi-simple cohomological field theories.

In a second part of these lecture, I will present a completely different flavour of the topological recursion. Taking as input an algebraic curve defined by a polynomial equation E(x,y) = 0, topological recursion allows to build an eigenfunction of the operator $E(x,\hat{y})$ obtained by quantising $(x,y)->(x,\hat{y})$ where $\hat{y} = \hbar d/dx$. I will explain this result and show how it gives rise to a large family of isomonodromic tau functions, providing for example solutions to Painleve equations. The simplest example of this phenomenon is that the Kontsevich's generating function of intersection numbers on the moduli space of Riemann surfaces is a formal solution to the Airy equation.

11:10 → 12:00 Tudor Dimofte, "Algebra, geometry, and twists in 3d N=4 gauge theory": Lecture 2

Three dimensional N=4 gauge theories have enjoyed a rich interplay with geometry and representation theory since the 90's, particularly motivated by the discovery of 3d mirror symmetry. An initial prediction of 3d mirror symmetry was the existence of many pairs of hyperkahler cones "M_H" and "M_C" (Higgs and Coulomb branches), with the rank of the hyperkahler isometry group of M_H equal to the dimension of the resolution space of M_C, and vice versa. This may be seen as an analogue of 2d mirror symmetry relating pairs Calabi-Yau manifolds whose Betti numbers are swapped in an appropriate way.

In the last decade, the interplay between the physics of 3d N=4 gauge theories and mathematics has greatly intensified, in part due to the advent of new methods to access the structure of local and extended operators in these 3d QFT's. A broad goal of my lectures will be to review some of the recent developments, especially those coming from mathematics, while connecting them directly to fundamental ideas in 3d physics. I will organize many of these developments within the framework of topological (and occasionally holomorphic) "twists" of 3d N=4 gauge theories, and the extended TQFT's (topological quantum field theories) that they predict to exist. This leads into a major current research goal: defining and proving a top-level "3d homological mirror symmetry" equivalence of 2-categories, that would encompass all other mirror equivalences.

More specifically, I will touch on: * the physical definition of 3d N=4 gauge theories, and their topological twists * algebraic and geometric structures predicted by these twists -- within the framework of 3d TQFT * what we know (and do not know) about 3d mirror symmetry * constructions of 3d Coulomb branches, e.g. in work of Braverman-Finkelberg-Nakajima * holomorphic twists and elliptic stable envelopes (in brief) * a current frontier: braided tensor categories of line operators and 2-categories of boundary conditions * the appearance and utility of vertex operator algebras in describing line operators * algebraic vs. analytic/symplectic geometry in making mathematical sense of physical structure * relations between 3d mirror symmetry, 2d mirror symmetry, and 4d S-duality

12:15 → 13:30 Lunch

16:30 → 17:00 Coffee Break

17:15 → 18:00 Nicolas Orantin, "From enumerative geometry to differential equations and isomonodromic systems via topological recursion": Exercices 1

The study of random matrices has a fascinating connection with different fields of mathematics and physics ranging from number theory to topological string theories. Among such applications, the computation of moments of random matrices can be interpreted as defining generating series of discrete surfaces built by gluing together polygons by their edges. Trying to solve such a combinatorial problem, together with Chekhov and Eynard, we found a beautiful formula allowing to enumerate such surfaces by induction on their Euler characteristic. This inductive procedure, later called topological recursion, turns out to provide a universal solution to a large class of enumerative problems such as the study of statistical systems on a random lattice or the computation of integrals over the moduli space of Riemann surfaces with respect to different measures (including Gromov-Witten invariants or volumes of such moduli spaces).

In a first part of these lectures, I will introduce the formalism of topological recursion by explaining how one can transform the combinatorial problem of enumerating discrete surfaces into the
computation of residue integrals on an associated Riemann surface thanks to the definition of good generating series. The general idea is that, if one can find a generating series for the number of discs such that it defines a function on a Riemann surface, then applying topological recursion with this Riemann surface as initial data produces generating functions for the number of surfaces with any topology by induction on their Euler characteristic.

I will then give some more examples of application of the recursive formula discovered in this context to other enumeration problems including Hurwitz numbers, some volumes of the moduli space of Riemann surfaces and the expression of correlators of any semi-simple cohomological field theories.

In a second part of these lecture, I will present a completely different flavour of the topological recursion. Taking as input an algebraic curve defined by a polynomial equation E(x,y) = 0, topological recursion allows to build an eigenfunction of the operator $E(x,\hat{y})$ obtained by quantising $(x,y)->(x,\hat{y})$ where $\hat{y} = \hbar d/dx$. I will explain this result and show how it gives rise to a large family of isomonodromic tau functions, providing for example solutions to Painleve equations. The simplest example of this phenomenon is that the Kontsevich's generating function of intersection numbers on the moduli space of Riemann surfaces is a formal solution to the Airy equation.

18:10 → 19:00 Alexander Braverman, "Affine Grassmannian and symplectic duality": Lecture 3

The purpose of this mini-course is two-fold.
In the first half I plan to start by (very briefly!) recalling some basic notions of the theory of constructible sheaves on complex algebraic varieties (such as, for example, equivariant derived categories of constructible sheaves, perverse sheaves etc.) and then apply this formalism to the study of certain derived category of equivariant sheaves on the affine Grassmannian of a complex reductive group G (here the main goal will be to explain the formulation of the so called derived geometric Satake equivalence due to Drinfled and Bezrukavnikov-Finkelberg).
In the 2nd half I plan to explain applications of the above to the subject of symplectic duality. I plan to discuss topics such as (conical) symplectic resolutions and their quantizations, formulations of (various aspects of) symplectic duality,Braverman-Finkelberg-Nakajima construction of Coulomb branches of 3d N=4 gauge theories via the affine Grassmannian and its relation to symplectic duality.
Some further topics might be discussed depending on how much time is left.

19:15 → 20:30 Dinner

Wednesday 11 January

09:00 → 09:25 Short talk: Lie bialgebra structures on loop algebras and torsion-free sheaves of Lie algebras on nodal irreducible cubic curves

Lie bialgebra structures are fundamental to the theory of quantum groups proposed by Drinfeld in the late eighties. One of the most important examples is the standard bialgebra structure on a symmetrizable Kac-Moody algebra. In the affine case, this structure induces a Lie bialgebra structure on the underlying loop algebra. In this talk, I will relate all twistings of this Lie bialgebra structure to torsion-free sheaves on nodal irreducible cubic curves and to trigonometric solutions of the classical Yang-Baxter equation. This connections result in a classification of these objects.

Speaker: Raschid Abedin

09:30 → 09:55 Short talk: Bow varieties, stable envelopes and 3d-mirror symmetry

Mirror symmetry for 3d N=4 supersymmetric gauge theories has recently received plenty of attention in both representation theory and mathematical physics. It predicts that Higgs and Coulomb branches of a pair of dual theories are interchanged, and hence that both pairs of homologous branches (Higgs-Higgs and Coulomb-Coulomb) share exceptional topological and geometric properties. One of the predictions of mirror symmetry is that elliptic stable envelopes, which are certain topological classes intimately related with elliptic quantum groups, are the same after appropriate identifications. In this talk I will focus on Coulomb and Higgs branches of type A, which are collectively described by a class of varieties known as Cherkis bow varieties, and I will discuss the main ideas behind the proof of mirror symmetry of sable envelopes (joint work in progress with Richard Rimanyi).

Speaker: Tommaso Maria Botta (ETH Zurich)

09:55 → 10:20 Coffee Break

10:20 → 10:45 Short talk: Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism

In this talk I will describe a local Poisson structure (up to homotopy) associated to corners in four-dimensional gravity in the coframe (Palatini--Cartan) formalism. This is achieved through the use of the BFV formalism. This is a joint work with A. S. Cattaneo

Speaker: Giovanni Canepa (CPT Marseille)

10:50 → 11:15 Short talk: A homotopy Poisson structure from Poisson reduction

Applying the BFV-BRST techniques from field theory to the hamiltonian reduction of degree one graded symplectic manifolds, we obtain a homotopy version of the classical Konstant-Sternberg BRST algebra in a generalized hamiltonian context. This is based on the correspondence between hamiltonian symplectic degree one manifolds and Poisson manifolds, due to Roytenberg, and the relation between degree one graded reduction and standard Poisson reduction explored by Cattaneo and Zambon.

Speaker: Pedro Henrique Carvalho Silva (UZH)

11:20 → 11:45 Short talk: Hamiltonian dynamics and multiplicative groupoids

We use local symplectic Lie groupoids to construct Poisson integrators for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a neighborhood of the unit manifold, that, in turn give Poisson integrators. We also insist on the role of the Magnus formula, in the context of Poisson geometry, for the backward analysis of such integrators.
The talk is based on the preprint "Symplectic groupoids for Poisson integrators" (Cosserat, 2022, arXiv:2205.04838).

Speaker: Oscar Cosserat (La Rochelle Université)

12:15 → 13:30 Lunch

17:00 → 17:30 Coffee Break

17:30 → 17:55 Short talk: Deformations of holographic symmetry algebras

The celestial holography program centres around a conjectural holographic duality between a QFT in an asymptotically flat 4d bulk and a 2d CFT on the celestial sphere, the "CCFT". There is no dynamical evidence for this conjecture to date but symmetries of the bulk theories restrict the CCFT. For (selfdual) gravity in the bulk, Strominger et al found a symmetry algebra closely related to $w_{1+\infty}$. From a 2d CFT point of view, this algebra has well-known deformations which were conjectured to correspond to quantum effects in the bulk. In arXiv:2208.13750v2, we showed that such deformations actually arise when turning on a Moyal deformation in the 4d bulk.

Speaker: Simon Heuveline (University of Cambridge)

18:00 → 18:25 Short talk: Hikita conjecture for Gieseker varieties

Symplectic duality is an observation that symplectic resolutions tend to come in pairs with matching geometric properties. Equivariant Hikita-Nakajima conjecture is one such statement, which connects the geometric and algebraic properties of symplectically dual pairs. In this talk I try to explain, what is usually meant by symplectic duality, provide some examples and state the conjecture I am working on. The talk is based on a joint work with Vasily Krylov (arXiv:2202.09934).

Speaker: Pavel Shlykov (University of Toronto)

18:30 → 18:55 Short talk: Polydifferential Lie operad and applications

I will present the operad $O(Lie_d)$ obtained by applying a functor constructed by S. Merkulov and T. Willwacher to the operad $Lie_d$ of (degree shifted) Lie algebras. Then I will show some applications and properties of said operad.

Speaker: Vincent Wolff (University of Luxembourg)

19:15 → 20:30 Dinner

Thursday 12 January

09:00 → 09:45 Alexander Braverman, "Affine Grassmannian and symplectic duality": Exercises 2

The purpose of this mini-course is two-fold.
In the first half I plan to start by (very briefly!) recalling some basic notions of the theory of constructible sheaves on complex algebraic varieties (such as, for example, equivariant derived categories of constructible sheaves, perverse sheaves etc.) and then apply this formalism to the study of certain derived category of equivariant sheaves on the affine Grassmannian of a complex reductive group G (here the main goal will be to explain the formulation of the so called derived geometric Satake equivalence due to Drinfled and Bezrukavnikov-Finkelberg).
In the 2nd half I plan to explain applications of the above to the subject of symplectic duality. I plan to discuss topics such as (conical) symplectic resolutions and their quantizations, formulations of (various aspects of) symplectic duality,Braverman-Finkelberg-Nakajima construction of Coulomb branches of 3d N=4 gauge theories via the affine Grassmannian and its relation to symplectic duality.
Some further topics might be discussed depending on how much time is left.

09:50 → 10:10 Coffee Break

10:10 → 11:00 Nicolas Orantin, "From enumerative geometry to differential equations and isomonodromic systems via topological recursion": Lecture 3

The study of random matrices has a fascinating connection with different fields of mathematics and physics ranging from number theory to topological string theories. Among such applications, the computation of moments of random matrices can be interpreted as defining generating series of discrete surfaces built by gluing together polygons by their edges. Trying to solve such a combinatorial problem, together with Chekhov and Eynard, we found a beautiful formula allowing to enumerate such surfaces by induction on their Euler characteristic. This inductive procedure, later called topological recursion, turns out to provide a universal solution to a large class of enumerative problems such as the study of statistical systems on a random lattice or the computation of integrals over the moduli space of Riemann surfaces with respect to different measures (including Gromov-Witten invariants or volumes of such moduli spaces).

In a first part of these lectures, I will introduce the formalism of topological recursion by explaining how one can transform the combinatorial problem of enumerating discrete surfaces into the
computation of residue integrals on an associated Riemann surface thanks to the definition of good generating series. The general idea is that, if one can find a generating series for the number of discs such that it defines a function on a Riemann surface, then applying topological recursion with this Riemann surface as initial data produces generating functions for the number of surfaces with any topology by induction on their Euler characteristic.

I will then give some more examples of application of the recursive formula discovered in this context to other enumeration problems including Hurwitz numbers, some volumes of the moduli space of Riemann surfaces and the expression of correlators of any semi-simple cohomological field theories.

In a second part of these lecture, I will present a completely different flavour of the topological recursion. Taking as input an algebraic curve defined by a polynomial equation E(x,y) = 0, topological recursion allows to build an eigenfunction of the operator $E(x,\hat{y})$ obtained by quantising $(x,y)->(x,\hat{y})$ where $\hat{y} = \hbar d/dx$. I will explain this result and show how it gives rise to a large family of isomonodromic tau functions, providing for example solutions to Painleve equations. The simplest example of this phenomenon is that the Kontsevich's generating function of intersection numbers on the moduli space of Riemann surfaces is a formal solution to the Airy equation.

11:10 → 12:00 Tudor Dimofte, "Algebra, geometry, and twists in 3d N=4 gauge theory": Lecture 3

Three dimensional N=4 gauge theories have enjoyed a rich interplay with geometry and representation theory since the 90's, particularly motivated by the discovery of 3d mirror symmetry. An initial prediction of 3d mirror symmetry was the existence of many pairs of hyperkahler cones "M_H" and "M_C" (Higgs and Coulomb branches), with the rank of the hyperkahler isometry group of M_H equal to the dimension of the resolution space of M_C, and vice versa. This may be seen as an analogue of 2d mirror symmetry relating pairs Calabi-Yau manifolds whose Betti numbers are swapped in an appropriate way.

In the last decade, the interplay between the physics of 3d N=4 gauge theories and mathematics has greatly intensified, in part due to the advent of new methods to access the structure of local and extended operators in these 3d QFT's. A broad goal of my lectures will be to review some of the recent developments, especially those coming from mathematics, while connecting them directly to fundamental ideas in 3d physics. I will organize many of these developments within the framework of topological (and occasionally holomorphic) "twists" of 3d N=4 gauge theories, and the extended TQFT's (topological quantum field theories) that they predict to exist. This leads into a major current research goal: defining and proving a top-level "3d homological mirror symmetry" equivalence of 2-categories, that would encompass all other mirror equivalences.

More specifically, I will touch on: * the physical definition of 3d N=4 gauge theories, and their topological twists * algebraic and geometric structures predicted by these twists -- within the framework of 3d TQFT * what we know (and do not know) about 3d mirror symmetry * constructions of 3d Coulomb branches, e.g. in work of Braverman-Finkelberg-Nakajima * holomorphic twists and elliptic stable envelopes (in brief) * a current frontier: braided tensor categories of line operators and 2-categories of boundary conditions * the appearance and utility of vertex operator algebras in describing line operators * algebraic vs. analytic/symplectic geometry in making mathematical sense of physical structure * relations between 3d mirror symmetry, 2d mirror symmetry, and 4d S-duality

12:15 → 13:30 Lunch

16:30 → 17:00 Coffee Break

17:15 → 18:00 Nicolas Orantin, "From enumerative geometry to differential equations and isomonodromic systems via topological recursion": Exercises 2

The study of random matrices has a fascinating connection with different fields of mathematics and physics ranging from number theory to topological string theories. Among such applications, the computation of moments of random matrices can be interpreted as defining generating series of discrete surfaces built by gluing together polygons by their edges. Trying to solve such a combinatorial problem, together with Chekhov and Eynard, we found a beautiful formula allowing to enumerate such surfaces by induction on their Euler characteristic. This inductive procedure, later called topological recursion, turns out to provide a universal solution to a large class of enumerative problems such as the study of statistical systems on a random lattice or the computation of integrals over the moduli space of Riemann surfaces with respect to different measures (including Gromov-Witten invariants or volumes of such moduli spaces).

In a first part of these lectures, I will introduce the formalism of topological recursion by explaining how one can transform the combinatorial problem of enumerating discrete surfaces into the
computation of residue integrals on an associated Riemann surface thanks to the definition of good generating series. The general idea is that, if one can find a generating series for the number of discs such that it defines a function on a Riemann surface, then applying topological recursion with this Riemann surface as initial data produces generating functions for the number of surfaces with any topology by induction on their Euler characteristic.

I will then give some more examples of application of the recursive formula discovered in this context to other enumeration problems including Hurwitz numbers, some volumes of the moduli space of Riemann surfaces and the expression of correlators of any semi-simple cohomological field theories.

In a second part of these lecture, I will present a completely different flavour of the topological recursion. Taking as input an algebraic curve defined by a polynomial equation E(x,y) = 0, topological recursion allows to build an eigenfunction of the operator $E(x,\hat{y})$ obtained by quantising $(x,y)->(x,\hat{y})$ where $\hat{y} = \hbar d/dx$. I will explain this result and show how it gives rise to a large family of isomonodromic tau functions, providing for example solutions to Painleve equations. The simplest example of this phenomenon is that the Kontsevich's generating function of intersection numbers on the moduli space of Riemann surfaces is a formal solution to the Airy equation.

18:10 → 19:00 Alexander Braverman, "Affine Grassmannian and symplectic duality": Lecture 4

The purpose of this mini-course is two-fold.
In the first half I plan to start by (very briefly!) recalling some basic notions of the theory of constructible sheaves on complex algebraic varieties (such as, for example, equivariant derived categories of constructible sheaves, perverse sheaves etc.) and then apply this formalism to the study of certain derived category of equivariant sheaves on the affine Grassmannian of a complex reductive group G (here the main goal will be to explain the formulation of the so called derived geometric Satake equivalence due to Drinfled and Bezrukavnikov-Finkelberg).
In the 2nd half I plan to explain applications of the above to the subject of symplectic duality. I plan to discuss topics such as (conical) symplectic resolutions and their quantizations, formulations of (various aspects of) symplectic duality,Braverman-Finkelberg-Nakajima construction of Coulomb branches of 3d N=4 gauge theories via the affine Grassmannian and its relation to symplectic duality.
Some further topics might be discussed depending on how much time is left.

19:15 → 19:35 Dinner

Friday 13 January

09:00 → 09:45 Tudor Dimofte, "Algebra, geometry, and twists in 3d N=4 gauge theory": Exercises 2

Three dimensional N=4 gauge theories have enjoyed a rich interplay with geometry and representation theory since the 90's, particularly motivated by the discovery of 3d mirror symmetry. An initial prediction of 3d mirror symmetry was the existence of many pairs of hyperkahler cones "M_H" and "M_C" (Higgs and Coulomb branches), with the rank of the hyperkahler isometry group of M_H equal to the dimension of the resolution space of M_C, and vice versa. This may be seen as an analogue of 2d mirror symmetry relating pairs Calabi-Yau manifolds whose Betti numbers are swapped in an appropriate way.

In the last decade, the interplay between the physics of 3d N=4 gauge theories and mathematics has greatly intensified, in part due to the advent of new methods to access the structure of local and extended operators in these 3d QFT's. A broad goal of my lectures will be to review some of the recent developments, especially those coming from mathematics, while connecting them directly to fundamental ideas in 3d physics. I will organize many of these developments within the framework of topological (and occasionally holomorphic) "twists" of 3d N=4 gauge theories, and the extended TQFT's (topological quantum field theories) that they predict to exist. This leads into a major current research goal: defining and proving a top-level "3d homological mirror symmetry" equivalence of 2-categories, that would encompass all other mirror equivalences.

More specifically, I will touch on: * the physical definition of 3d N=4 gauge theories, and their topological twists * algebraic and geometric structures predicted by these twists -- within the framework of 3d TQFT * what we know (and do not know) about 3d mirror symmetry * constructions of 3d Coulomb branches, e.g. in work of Braverman-Finkelberg-Nakajima * holomorphic twists and elliptic stable envelopes (in brief) * a current frontier: braided tensor categories of line operators and 2-categories of boundary conditions * the appearance and utility of vertex operator algebras in describing line operators * algebraic vs. analytic/symplectic geometry in making mathematical sense of physical structure * relations between 3d mirror symmetry, 2d mirror symmetry, and 4d S-duality

09:50 → 10:10 Coffee Break

10:10 → 11:00 Tudor Dimofte, "Algebra, geometry, and twists in 3d N=4 gauge theory": Lecture 4

Three dimensional N=4 gauge theories have enjoyed a rich interplay with geometry and representation theory since the 90's, particularly motivated by the discovery of 3d mirror symmetry. An initial prediction of 3d mirror symmetry was the existence of many pairs of hyperkahler cones "M_H" and "M_C" (Higgs and Coulomb branches), with the rank of the hyperkahler isometry group of M_H equal to the dimension of the resolution space of M_C, and vice versa. This may be seen as an analogue of 2d mirror symmetry relating pairs Calabi-Yau manifolds whose Betti numbers are swapped in an appropriate way.

In the last decade, the interplay between the physics of 3d N=4 gauge theories and mathematics has greatly intensified, in part due to the advent of new methods to access the structure of local and extended operators in these 3d QFT's. A broad goal of my lectures will be to review some of the recent developments, especially those coming from mathematics, while connecting them directly to fundamental ideas in 3d physics. I will organize many of these developments within the framework of topological (and occasionally holomorphic) "twists" of 3d N=4 gauge theories, and the extended TQFT's (topological quantum field theories) that they predict to exist. This leads into a major current research goal: defining and proving a top-level "3d homological mirror symmetry" equivalence of 2-categories, that would encompass all other mirror equivalences.

More specifically, I will touch on: * the physical definition of 3d N=4 gauge theories, and their topological twists * algebraic and geometric structures predicted by these twists -- within the framework of 3d TQFT * what we know (and do not know) about 3d mirror symmetry * constructions of 3d Coulomb branches, e.g. in work of Braverman-Finkelberg-Nakajima * holomorphic twists and elliptic stable envelopes (in brief) * a current frontier: braided tensor categories of line operators and 2-categories of boundary conditions * the appearance and utility of vertex operator algebras in describing line operators * algebraic vs. analytic/symplectic geometry in making mathematical sense of physical structure * relations between 3d mirror symmetry, 2d mirror symmetry, and 4d S-duality

11:10 → 12:00 Nicolas Orantin, "From enumerative geometry to differential equations and isomonodromic systems via topological recursion": Lecture 4

The study of random matrices has a fascinating connection with different fields of mathematics and physics ranging from number theory to topological string theories. Among such applications, the computation of moments of random matrices can be interpreted as defining generating series of discrete surfaces built by gluing together polygons by their edges. Trying to solve such a combinatorial problem, together with Chekhov and Eynard, we found a beautiful formula allowing to enumerate such surfaces by induction on their Euler characteristic. This inductive procedure, later called topological recursion, turns out to provide a universal solution to a large class of enumerative problems such as the study of statistical systems on a random lattice or the computation of integrals over the moduli space of Riemann surfaces with respect to different measures (including Gromov-Witten invariants or volumes of such moduli spaces).

In a first part of these lectures, I will introduce the formalism of topological recursion by explaining how one can transform the combinatorial problem of enumerating discrete surfaces into the
computation of residue integrals on an associated Riemann surface thanks to the definition of good generating series. The general idea is that, if one can find a generating series for the number of discs such that it defines a function on a Riemann surface, then applying topological recursion with this Riemann surface as initial data produces generating functions for the number of surfaces with any topology by induction on their Euler characteristic.

I will then give some more examples of application of the recursive formula discovered in this context to other enumeration problems including Hurwitz numbers, some volumes of the moduli space of Riemann surfaces and the expression of correlators of any semi-simple cohomological field theories.

In a second part of these lecture, I will present a completely different flavour of the topological recursion. Taking as input an algebraic curve defined by a polynomial equation E(x,y) = 0, topological recursion allows to build an eigenfunction of the operator $E(x,\hat{y})$ obtained by quantising $(x,y)->(x,\hat{y})$ where $\hat{y} = \hbar d/dx$. I will explain this result and show how it gives rise to a large family of isomonodromic tau functions, providing for example solutions to Painleve equations. The simplest example of this phenomenon is that the Kontsevich's generating function of intersection numbers on the moduli space of Riemann surfaces is a formal solution to the Airy equation.

12:15 → 13:30 Lunch