Vafa-Witten Invariants of Projective Surfaces (1/5)

• Richard THOMAS (Imperial College London)

Room: Marilyn and James Simons Conference Center This course has 4 sections split over 5 lectures. The first section will be the longest, and hopefully useful for the other courses. 1. Sheaves, moduli and virtual cycles 2. Vafa-Witten invariants: stable and semistable cases 3. Techniques for calculation --- virtual degeneracy loci, cosection localisation and a vanishing theorem 4. Refined Vafa-Witten invariants

Algebra and Geometry of Link Homology (1/5)

• Eugene GORSKY (University of California at Davis)

Room: Marilyn and James Simons Conference Center Khovanov and Rozansky defined a link homology theory which categorifies the HOMFLY-PT polynomial. This homology is relatively easy to define, but notoriously hard to compute. I will discuss recent breakthroughs in understanding and computing Khovanov-Rozansky homology, focusing on connections to the algebraic geometry of Hilbert schemes of points, affine Springer fibers and braid varieties.

Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (1/5)

• Joel KAMNITZER (University of Toronto)

Room: Marilyn and James Simons Conference Center In the 21st century, there has been a great interest in the study of symplectic resolutions, such as cotangent bundles of flag varieties, hypertoric varieties, quiver varieties, and affine Grassmannian slices. Mathematicians, especially Braden-Licata-Proudfoot-Webster, and physicists observed that these spaces come in dual pairs: this phenomenon is known as 3d mirror symmetry or symplectic duality. In physics, these dual pairs come from Higgs and Coulomb branches of 3d supersymmetric field theories. In a remarkable 2016 paper, Braverman-Finkelberg-Nakajima gave a mathematical definition of the Coulomb branch associated to a 3d gauge theory. We will discuss all these developments, as well as recent progress building on the work of BFN. We will particularly study the Coulomb branches associated to quiver gauge theories: these are known as generalized affine Grassmannian slices.

The Skein Algebra of the 4-punctured Sphere from Curve Counting

• Pierrick BOUSSEAU (CNRS and Université Paris-Saclay)

Room: Marilyn and James Simons Conference Center The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the SL_2 character of a topological surface. I will explain how to realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov-Witten invariants of a log Calabi-Yau cubic surface. This leads to the proof of a previously conjectured positivity property of the bracelets bases of the skein algebras of the 4-punctured sphere and of the 2-punctured torus.

Vafa-Witten Invariants of Projective Surfaces (2/5)

• Richard THOMAS (Imperial College London)

Room: Marilyn and James Simons Conference Center This course has 4 sections split over 5 lectures. The first section will be the longest, and hopefully useful for the other courses. 1. Sheaves, moduli and virtual cycles 2. Vafa-Witten invariants: stable and semistable cases 3. Techniques for calculation --- virtual degeneracy loci, cosection localisation and a vanishing theorem 4. Refined Vafa-Witten invariants

Algebra and Geometry of Link Homology (2/5)

• Eugene GORSKY (University of California at Davis)

Room: Marilyn and James Simons Conference Center Khovanov and Rozansky defined a link homology theory which categorifies the HOMFLY-PT polynomial. This homology is relatively easy to define, but notoriously hard to compute. I will discuss recent breakthroughs in understanding and computing Khovanov-Rozansky homology, focusing on connections to the algebraic geometry of Hilbert schemes of points, affine Springer fibers and braid varieties.

Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (2/5)

• Joel KAMNITZER (University of Toronto)

Room: Marilyn and James Simons Conference Center In the 21st century, there has been a great interest in the study of symplectic resolutions, such as cotangent bundles of flag varieties, hypertoric varieties, quiver varieties, and affine Grassmannian slices. Mathematicians, especially Braden-Licata-Proudfoot-Webster, and physicists observed that these spaces come in dual pairs: this phenomenon is known as 3d mirror symmetry or symplectic duality. In physics, these dual pairs come from Higgs and Coulomb branches of 3d supersymmetric field theories. In a remarkable 2016 paper, Braverman-Finkelberg-Nakajima gave a mathematical definition of the Coulomb branch associated to a 3d gauge theory. We will discuss all these developments, as well as recent progress building on the work of BFN. We will particularly study the Coulomb branches associated to quiver gauge theories: these are known as generalized affine Grassmannian slices.

Exercise / Q&A Session Room: Marilyn and James Simons Conference Center

SU(r) Vafa-Witten Invariants and Continued Fractions

• Lothar GOTTSCHE (ICTP)

Room: Marilyn and James Simons Conference Center This is joint work with Martijn Kool and Thies Laarakker. We conjecture a formula for the structure of SU(r) Vafa-Witten invariants of surfaces with a canonical curve, generalizing a similar formula proven by Laarakker for the monopole contribution. This expresses the Vafa-Witten invariants in terms of some universal power series and Seiberg-Witten invariants. Using computations on nested Hilbert schemes we conjecturally determine these universal power series for r at most 5 in terms of theta functions for the A_{r-1} lattice and Ramanujan's continued fractions.

Vafa-Witten Invariants of Projective Surfaces (3/5)

• Richard THOMAS (Imperial College London)

Room: Marilyn and James Simons Conference Center This course has 4 sections split over 5 lectures. The first section will be the longest, and hopefully useful for the other courses. 1. Sheaves, moduli and virtual cycles 2. Vafa-Witten invariants: stable and semistable cases 3. Techniques for calculation --- virtual degeneracy loci, cosection localisation and a vanishing theorem 4. Refined Vafa-Witten invariants

Algebra and Geometry of Link Homology (3/5)

• Eugene GORSKY (University of California at Davis)

Room: Marilyn and James Simons Conference Center Khovanov and Rozansky defined a link homology theory which categorifies the HOMFLY-PT polynomial. This homology is relatively easy to define, but notoriously hard to compute. I will discuss recent breakthroughs in understanding and computing Khovanov-Rozansky homology, focusing on connections to the algebraic geometry of Hilbert schemes of points, affine Springer fibers and braid varieties.

Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (3/5)

• Joel KAMNITZER (University of Toronto)

Room: Marilyn and James Simons Conference Center In the 21st century, there has been a great interest in the study of symplectic resolutions, such as cotangent bundles of flag varieties, hypertoric varieties, quiver varieties, and affine Grassmannian slices. Mathematicians, especially Braden-Licata-Proudfoot-Webster, and physicists observed that these spaces come in dual pairs: this phenomenon is known as 3d mirror symmetry or symplectic duality. In physics, these dual pairs come from Higgs and Coulomb branches of 3d supersymmetric field theories. In a remarkable 2016 paper, Braverman-Finkelberg-Nakajima gave a mathematical definition of the Coulomb branch associated to a 3d gauge theory. We will discuss all these developments, as well as recent progress building on the work of BFN. We will particularly study the Coulomb branches associated to quiver gauge theories: these are known as generalized affine Grassmannian slices.

Exercise / Q&A Session Room: Marilyn and James Simons Conference Center

Multiple Cover Formula for the Stable Pairs Theory of K3xE

• Georg OBERDIECK (Mathematisches Institut der Universität Bonn)

Room: Marilyn and James Simons Conference Center The count of stable pairs (Pandharipande-Thomas theory) on K3 x E is well-understood whenever the curve class is primitive over the K3 factor. I will explain how ideas of Pandharipande and Thomas can be used to remove the primitivity assumption.

Vafa-Witten Invariants of Projective Surfaces (4/5)

• Richard THOMAS (Imperial College London)

Room: Marilyn and James Simons Conference Center This course has 4 sections split over 5 lectures. The first section will be the longest, and hopefully useful for the other courses. 1. Sheaves, moduli and virtual cycles 2. Vafa-Witten invariants: stable and semistable cases 3. Techniques for calculation --- virtual degeneracy loci, cosection localisation and a vanishing theorem 4. Refined Vafa-Witten invariants

Algebra and geometry of link homology (4/5)

• Eugene GORSKY (University of California at Davis)

Room: Marilyn and James Simons Conference Center Khovanov and Rozansky defined a link homology theory which categorifies the HOMFLY-PT polynomial. This homology is relatively easy to define, but notoriously hard to compute. I will discuss recent breakthroughs in understanding and computing Khovanov-Rozansky homology, focusing on connections to the algebraic geometry of Hilbert schemes of points, affine Springer fibers and braid varieties.

Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (4/5)

• Joel KAMNITZER (University of Toronto)

Room: Marilyn and James Simons Conference Center In the 21st century, there has been a great interest in the study of symplectic resolutions, such as cotangent bundles of flag varieties, hypertoric varieties, quiver varieties, and affine Grassmannian slices. Mathematicians, especially Braden-Licata-Proudfoot-Webster, and physicists observed that these spaces come in dual pairs: this phenomenon is known as 3d mirror symmetry or symplectic duality. In physics, these dual pairs come from Higgs and Coulomb branches of 3d supersymmetric field theories. In a remarkable 2016 paper, Braverman-Finkelberg-Nakajima gave a mathematical definition of the Coulomb branch associated to a 3d gauge theory. We will discuss all these developments, as well as recent progress building on the work of BFN. We will particularly study the Coulomb branches associated to quiver gauge theories: these are known as generalized affine Grassmannian slices.

Exercise / Q&A Session Room: Marilyn and James Simons Conference Center

Gaiotto Conjectures for Quantum Super-groups

• Alexander BRAVERMAN (University of Toronto and Perimeter Institute for Theoretical Physics)

Room: Marilyn and James Simons Conference Center I am going to explain a series of conjectures due to D.Gaiotto which provide a geometric realization of categories of representations of certain quantum super-groups (such as U_q(gl(M|N)) via the affine Grassmannian of certain (purely even) algebraic groups. These conjectures generalize both the well-known geometric Satake equivalence and the so called Fundamental Local Equivalence of Gaitsgory and Lurie (which will be recalled in the talk). In the 2nd part of the talk I will explain a recent proof of this conjecture for U_q(N|N-1) (for generic q), based on a joint work with Finkelberg and Travkin.

Vafa-Witten Invariants of Projective Surfaces (5/5)

• Richard THOMAS (Imperial College London)

Room: Marilyn and James Simons Conference Center This course has 4 sections split over 5 lectures. The first section will be the longest, and hopefully useful for the other courses. 1. Sheaves, moduli and virtual cycles 2. Vafa-Witten invariants: stable and semistable cases 3. Techniques for calculation --- virtual degeneracy loci, cosection localisation and a vanishing theorem 4. Refined Vafa-Witten invariants

Algebra and Geometry of Link Homology (5/5)

• Eugene GORSKY (University of California at Davis)

Room: Marilyn and James Simons Conference Center Khovanov and Rozansky defined a link homology theory which categorifies the HOMFLY-PT polynomial. This homology is relatively easy to define, but notoriously hard to compute. I will discuss recent breakthroughs in understanding and computing Khovanov-Rozansky homology, focusing on connections to the algebraic geometry of Hilbert schemes of points, affine Springer fibers and braid varieties.

Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (5/5)

• Joel KAMNITZER (University of Toronto)

Room: Marilyn and James Simons Conference Center In the 21st century, there has been a great interest in the study of symplectic resolutions, such as cotangent bundles of flag varieties, hypertoric varieties, quiver varieties, and affine Grassmannian slices. Mathematicians, especially Braden-Licata-Proudfoot-Webster, and physicists observed that these spaces come in dual pairs: this phenomenon is known as 3d mirror symmetry or symplectic duality. In physics, these dual pairs come from Higgs and Coulomb branches of 3d supersymmetric field theories. In a remarkable 2016 paper, Braverman-Finkelberg-Nakajima gave a mathematical definition of the Coulomb branch associated to a 3d gauge theory. We will discuss all these developments, as well as recent progress building on the work of BFN. We will particularly study the Coulomb branches associated to quiver gauge theories: these are known as generalized affine Grassmannian slices.

Exercise / Q&A Session Room: Marilyn and James Simons Conference Center

3d SUSY Gauge Theory and Quantum Groups at Roots of Unity

• Tudor DIMOFTE (University of California at Davis and University of Edinburgh)

Room: Marilyn and James Simons Conference Center Topological twists of 3d N=4 gauge theories naturally give rise to non-semisimple 3d TQFT's. In mathematics, prototypical examples of the latter were constructed in the 90's (by Lyubashenko and others) from representation categories of small quantum groups at roots of unity; they were recently generalized in work of Costantino-Geer-Patureau Mirand and collaborators. I will introduce a family of physical 3d quantum field theories that (conjecturally) reproduce these classic non-semisimple TQFT's. The physical theories combine Chern-Simons-like and 3d N=4-like sectors. They are also related to Feigin-Tipunin vertex algebras, much the same way that Chern-Simons theory is related to WZW vertex algebras. (Based on work with T. Creutzig, N. Garner, and N. Geer.)

Enumerative Geometry of Curves, Maps, and Sheaves (1/5)

• Rahul PANDHARIPANDE (ETH Zürich)

Room: Marilyn and James Simons Conference Center The main topics will be the intersection theory of tautological classes on moduli space of curves, the enumeration of stable maps via Gromov-Witten theory, and the enumeration of sheaves via Donaldson-Thomas theory. I will cover a mix of classical and modern results. My goal will be, by the end, to explain recent progress on the Virasoro constraints on the sheaf side.

Cohomological Hall Algebras and Motivic Invariants for Quivers (1/4)

• Markus REINEKE (Ruhr-Universität Bochum)

Room: Marilyn and James Simons Conference Center We motivate, define and study Donaldson-Thomas invariants and Cohomological Hall algebras associated to quivers, relate them to the geometry of moduli spaces of quiver representations and (in special cases) to Gromov-Witten invariants, and discuss the algebraic structure of Cohomological Hall algebras.

Generalized Airy Functions and Givental's R-matrices for Projective Spaces and Witten's Class

• Dimitri ZVONKINE (Laboratoire Mathématiques de Versailles)

Room: Marilyn and James Simons Conference Center We call generalized Airy functions particular solutions of the differential equations $f^{(n)}(x) = x^a f(x)$. We show that asymptotic expansions of generalized Airy functions contain coefficients of Givental's R-matrices both for Gromov-Witten invariants of projective spaces and for Witten's r-spin classes. Joint work with Sybille Rosset.

Stable Pairs and Gopakumar-Vafa Invariants (1/5)

• Davesh MAULIK (Massachusetts Institute of Technology)

Room: Marilyn and James Simons Conference Center In the first part of the course, I will give an overview of Donaldson-Thomas theory for Calabi-Yau threefold geometries, and its cohomological refinement. In the second part, I will explain a conjectural ansatz (from joint work with Y. Toda) for defining Gopakumar-Vafa invariants via moduli of one-dimensional sheaves, emphasizing some examples where we can understand how they relate to curve-counting via stable pairs. If time permits, I will discuss some recent work on $\chi$\-independence phenomena in this setting (joint with J. Shen).

Enumerative Geometry of Curves, Maps, and Sheaves (2/5)

• Rahul PANDHARIPANDE (ETH Zürich)

Room: Marilyn and James Simons Conference Center The main topics will be the intersection theory of tautological classes on moduli space of curves, the enumeration of stable maps via Gromov-Witten theory, and the enumeration of sheaves via Donaldson-Thomas theory. I will cover a mix of classical and modern results. My goal will be, by the end, to explain recent progress on the Virasoro constraints on the sheaf side.

Cohomological Hall Algebras and Motivic Invariants for Quivers (2/4)

• Markus REINEKE (Ruhr-Universität Bochum)

Room: Marilyn and James Simons Conference Center We motivate, define and study Donaldson-Thomas invariants and Cohomological Hall algebras associated to quivers, relate them to the geometry of moduli spaces of quiver representations and (in special cases) to Gromov-Witten invariants, and discuss the algebraic structure of Cohomological Hall algebras.

Complex K-theory of Dual Hitchin Systems

• Michael GROECHENIG (University of Toronto)

Room: Marilyn and James Simons Conference Center Let G and G’ be Langlands dual reductive groups (e.g. SL(n) and PGL(n)). According to a theorem by Donagi-Pantev, the generic fibres of the moduli spaces of G-Higgs bundles and G’-Higgs bundles are dual abelian varieties and are therefore derived-equivalent. It is an interesting open problem to prove existence of a derived equivalence over the full Hitchin base. I will report on joint work in progress with Shiyu Shen, in which we construct a K-theoretic shadow thereof: natural equivalences between complex K-theory spectra for certain moduli spaces of Higgs bundles (in type A).

Exercise / Q&A Session Room: Marilyn and James Simons Conference Center

Stable Pairs and Gopakumar-Vafa Invariants (2/5)

• Davesh MAULIK (Massachusetts Institute of Technology)

Room: Marilyn and James Simons Conference Center In the first part of the course, I will give an overview of Donaldson-Thomas theory for Calabi-Yau threefold geometries, and its cohomological refinement. In the second part, I will explain a conjectural ansatz (from joint work with Y. Toda) for defining Gopakumar-Vafa invariants via moduli of one-dimensional sheaves, emphasizing some examples where we can understand how they relate to curve-counting via stable pairs. If time permits, I will discuss some recent work on $\chi$\-independence phenomena in this setting (joint with J. Shen).

Enumerative Geometry of Curves, Maps, and Sheaves (3/5)

• Rahul PANDHARIPANDE (ETH Zürich)

Room: Marilyn and James Simons Conference Center The main topics will be the intersection theory of tautological classes on moduli space of curves, the enumeration of stable maps via Gromov-Witten theory, and the enumeration of sheaves via Donaldson-Thomas theory. I will cover a mix of classical and modern results. My goal will be, by the end, to explain recent progress on the Virasoro constraints on the sheaf side.

Cohomological Hall Algebras and Motivic Invariants for Quivers (3/4)

• Markus REINEKE (Ruhr-Universität Bochum)

Room: Marilyn and James Simons Conference Center We motivate, define and study Donaldson-Thomas invariants and Cohomological Hall algebras associated to quivers, relate them to the geometry of moduli spaces of quiver representations and (in special cases) to Gromov-Witten invariants, and discuss the algebraic structure of Cohomological Hall algebras.

Stable Envelopes, Bow Varieties, 3d Mirror Symmetry

• Richard RIMANYI (University of North Carolina at Chapel Hill)

Room: Marilyn and James Simons Conference Center There are many bridges connecting geometry with representation theory. A key notion in one of these connections, defined by Maulik-Okounkov, Okounkov, Aganagic-Okounkov, is the "stable envelope (class)". The stable envelope fits into the story of characteristic classes of singularities as a 1-parameter deformation (ℏ) of the fundamental class of singularities. Special cases of the latter include Schubert classes on homogeneous spaces and Thom polynomials is singularity theory. While stable envelopes are traditionally defined for quiver varieties, we will present a larger pool of spaces called Cherkis bow varieties, and explore their geometry and combinatorics. There is a natural pairing among bow varieties called 3d mirror symmetry. One consequence is a ‘coincidence' between elliptic stable envelopes on 3d mirror dual bow varieties (a work in progress). We will also discuss the Legendre-transform extension of bow varieties (joint work with L. Rozansky), the geometric counterpart of passing from Yangian R-matrices of Lie algebras gl(n) to Lie superalgebras gl(n|m).

Exercise / Q&A Session Room: Marilyn and James Simons Conference Center

Stable Pairs and Gopakumar-Vafa Invariants (3/5)

• Davesh MAULIK (Massachusetts Institute of Technology)

Room: Marilyn and James Simons Conference Center In the first part of the course, I will give an overview of Donaldson-Thomas theory for Calabi-Yau threefold geometries, and its cohomological refinement. In the second part, I will explain a conjectural ansatz (from joint work with Y. Toda) for defining Gopakumar-Vafa invariants via moduli of one-dimensional sheaves, emphasizing some examples where we can understand how they relate to curve-counting via stable pairs. If time permits, I will discuss some recent work on $\chi$\-independence phenomena in this setting (joint with J. Shen).

Cohomology of Affine Springer Fibres and Centre of Small Quantum Groups

• Peng SHAN (Tsinghua University)

Room: Marilyn and James Simons Conference Center I will report some recent progress on relationship between cohomology of affine Springer fibres and centre of small quantum groups. This is based on joint work with R. Bezrukavnikov, P. Boixeda-Alvarez and E. Vasserot.

Enumerative Geometry of Curves, Maps, and Sheaves (4/5)

• Rahul PANDHARIPANDE (ETH Zürich)

Room: Marilyn and James Simons Conference Center The main topics will be the intersection theory of tautological classes on moduli space of curves, the enumeration of stable maps via Gromov-Witten theory, and the enumeration of sheaves via Donaldson-Thomas theory. I will cover a mix of classical and modern results. My goal will be, by the end, to explain recent progress on the Virasoro constraints on the sheaf side.

Cohomological Hall Algebras and Motivic Invariants for Quivers (4/4)

• Markus REINEKE (Ruhr-Universität Bochum)

Room: Marilyn and James Simons Conference Center We motivate, define and study Donaldson-Thomas invariants and Cohomological Hall algebras associated to quivers, relate them to the geometry of moduli spaces of quiver representations and (in special cases) to Gromov-Witten invariants, and discuss the algebraic structure of Cohomological Hall algebras.

Exercise / Q&A Session Room: Marilyn and James Simons Conference Center

Stable Pairs and Gopakumar-Vafa Invariants (4/5)

• Davesh MAULIK (Massachusetts Institute of Technology)

Room: Marilyn and James Simons Conference Center In the first part of the course, I will give an overview of Donaldson-Thomas theory for Calabi-Yau threefold geometries, and its cohomological refinement. In the second part, I will explain a conjectural ansatz (from joint work with Y. Toda) for defining Gopakumar-Vafa invariants via moduli of one-dimensional sheaves, emphasizing some examples where we can understand how they relate to curve-counting via stable pairs. If time permits, I will discuss some recent work on $\chi$\-independence phenomena in this setting (joint with J. Shen).

Enumerative Geometry of Curves, Maps, and Sheaves (5/5)

• Rahul PANDHARIPANDE (ETH Zürich)

Room: Marilyn and James Simons Conference Center The main topics will be the intersection theory of tautological classes on moduli space of curves, the enumeration of stable maps via Gromov-Witten theory, and the enumeration of sheaves via Donaldson-Thomas theory. I will cover a mix of classical and modern results. My goal will be, by the end, to explain recent progress on the Virasoro constraints on the sheaf side.

BPS Counting and Pseudoperiodic Topology

• Maxim KONTSEVICH (IHES)

Room: Marilyn and James Simons Conference Center A holomorphic quadratic differential on a complex curve defines a flat metric with conical singularities. In the case of simple zeroes, T. Bridgeland and I. Smith identified geodesic intervals connecting zeroes, as well as maximal geodesic cylinders, with stable objects in certain 3-dimensional Calabi-Yau category. As a corollary, the counting of such geodesics gives a wall-crossing structure in the Lie algebra of Hamiltonian vector fields on a symplectic algebraic torus. I will explain that essentially the same numbers give wall-crossing structure in a different graded Lie algebra, of matrix-valued functions on an algebraic torus (joint work with Y. Soibelman). This WCS makes sense for curves endowed with abelian differentials with zeroes of arbitrary order and can be generalized to closed holomorphic 1-forms on complex varieties of arbitrary dimension.

Stable Pairs and Gopakumar-Vafa Invariants (5/5)

• Davesh MAULIK (Massachusetts Institute of Technology)

Room: Marilyn and James Simons Conference Center In the first part of the course, I will give an overview of Donaldson-Thomas theory for Calabi-Yau threefold geometries, and its cohomological refinement. In the second part, I will explain a conjectural ansatz (from joint work with Y. Toda) for defining Gopakumar-Vafa invariants via moduli of one-dimensional sheaves, emphasizing some examples where we can understand how they relate to curve-counting via stable pairs. If time permits, I will discuss some recent work on $\chi$\-independence phenomena in this setting (joint with J. Shen).