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Enumerative Geometry, Physics and Representation Theory
from
Monday, July 5, 2021 (11:00 AM)
to
Friday, July 16, 2021 (6:30 PM)
Monday, July 5, 2021
11:00 AM
VafaWitten Invariants of Projective Surfaces (1/5)

Richard THOMAS
(
Imperial College London
)
VafaWitten Invariants of Projective Surfaces (1/5)
Richard THOMAS
(
Imperial College London
)
11:00 AM  12:00 PM
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. VafaWitten invariants: stable and semistable cases 3. Techniques for calculation  virtual degeneracy loci, cosection localisation and a vanishing theorem 4. Refined VafaWitten invariants
12:00 PM
Lunch Break
Lunch Break
12:00 PM  1:30 PM
Room: Marilyn and James Simons Conference Center
1:30 PM
Algebra and Geometry of Link Homology (1/5)

Eugene GORSKY
(
University of California at Davis
)
Algebra and Geometry of Link Homology (1/5)
Eugene GORSKY
(
University of California at Davis
)
1:30 PM  2:30 PM
Room: Marilyn and James Simons Conference Center
Khovanov and Rozansky defined a link homology theory which categorifies the HOMFLYPT polynomial. This homology is relatively easy to define, but notoriously hard to compute. I will discuss recent breakthroughs in understanding and computing KhovanovRozansky homology, focusing on connections to the algebraic geometry of Hilbert schemes of points, affine Springer fibers and braid varieties.
2:30 PM
Break
Break
2:30 PM  3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM
Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (1/5)

Joel KAMNITZER
(
University of Toronto
)
Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (1/5)
Joel KAMNITZER
(
University of Toronto
)
3:00 PM  4:00 PM
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 BradenLicataProudfootWebster, 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, BravermanFinkelbergNakajima 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.
4:00 PM
Break
Break
4:00 PM  5:30 PM
Room: Marilyn and James Simons Conference Center
5:30 PM
The Skein Algebra of the 4punctured Sphere from Curve Counting

Pierrick BOUSSEAU
(
CNRS and Université ParisSaclay
)
The Skein Algebra of the 4punctured Sphere from Curve Counting
Pierrick BOUSSEAU
(
CNRS and Université ParisSaclay
)
5:30 PM  6:30 PM
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 4punctured sphere as the output of a mirror symmetry construction based on higher genus GromovWitten invariants of a log CalabiYau cubic surface. This leads to the proof of a previously conjectured positivity property of the bracelets bases of the skein algebras of the 4punctured sphere and of the 2punctured torus.
Tuesday, July 6, 2021
11:00 AM
VafaWitten Invariants of Projective Surfaces (2/5)

Richard THOMAS
(
Imperial College London
)
VafaWitten Invariants of Projective Surfaces (2/5)
Richard THOMAS
(
Imperial College London
)
11:00 AM  12:00 PM
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. VafaWitten invariants: stable and semistable cases 3. Techniques for calculation  virtual degeneracy loci, cosection localisation and a vanishing theorem 4. Refined VafaWitten invariants
12:00 PM
Lunch Break
Lunch Break
12:00 PM  1:30 PM
Room: Marilyn and James Simons Conference Center
1:30 PM
Algebra and Geometry of Link Homology (2/5)

Eugene GORSKY
(
University of California at Davis
)
Algebra and Geometry of Link Homology (2/5)
Eugene GORSKY
(
University of California at Davis
)
1:30 PM  2:30 PM
Room: Marilyn and James Simons Conference Center
Khovanov and Rozansky defined a link homology theory which categorifies the HOMFLYPT polynomial. This homology is relatively easy to define, but notoriously hard to compute. I will discuss recent breakthroughs in understanding and computing KhovanovRozansky homology, focusing on connections to the algebraic geometry of Hilbert schemes of points, affine Springer fibers and braid varieties.
2:30 PM
Break
Break
2:30 PM  3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM
Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (2/5)

Joel KAMNITZER
(
University of Toronto
)
Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (2/5)
Joel KAMNITZER
(
University of Toronto
)
3:00 PM  4:00 PM
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 BradenLicataProudfootWebster, 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, BravermanFinkelbergNakajima 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.
4:00 PM
Exercise / Q&A Session
Exercise / Q&A Session
4:00 PM  5:30 PM
Room: Marilyn and James Simons Conference Center
5:30 PM
SU(r) VafaWitten Invariants and Continued Fractions

Lothar GOTTSCHE
(
ICTP
)
SU(r) VafaWitten Invariants and Continued Fractions
Lothar GOTTSCHE
(
ICTP
)
5:30 PM  6:30 PM
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) VafaWitten invariants of surfaces with a canonical curve, generalizing a similar formula proven by Laarakker for the monopole contribution. This expresses the VafaWitten invariants in terms of some universal power series and SeibergWitten 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_{r1} lattice and Ramanujan's continued fractions.
Wednesday, July 7, 2021
11:00 AM
VafaWitten Invariants of Projective Surfaces (3/5)

Richard THOMAS
(
Imperial College London
)
VafaWitten Invariants of Projective Surfaces (3/5)
Richard THOMAS
(
Imperial College London
)
11:00 AM  12:00 PM
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. VafaWitten invariants: stable and semistable cases 3. Techniques for calculation  virtual degeneracy loci, cosection localisation and a vanishing theorem 4. Refined VafaWitten invariants
12:00 PM
Lunch Break
Lunch Break
12:00 PM  1:30 PM
Room: Marilyn and James Simons Conference Center
1:30 PM
Algebra and Geometry of Link Homology (3/5)

Eugene GORSKY
(
University of California at Davis
)
Algebra and Geometry of Link Homology (3/5)
Eugene GORSKY
(
University of California at Davis
)
1:30 PM  2:30 PM
Room: Marilyn and James Simons Conference Center
Khovanov and Rozansky defined a link homology theory which categorifies the HOMFLYPT polynomial. This homology is relatively easy to define, but notoriously hard to compute. I will discuss recent breakthroughs in understanding and computing KhovanovRozansky homology, focusing on connections to the algebraic geometry of Hilbert schemes of points, affine Springer fibers and braid varieties.
2:30 PM
Break
Break
2:30 PM  3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM
Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (3/5)

Joel KAMNITZER
(
University of Toronto
)
Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (3/5)
Joel KAMNITZER
(
University of Toronto
)
3:00 PM  4:00 PM
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 BradenLicataProudfootWebster, 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, BravermanFinkelbergNakajima 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.
4:00 PM
Exercise / Q&A Session
Exercise / Q&A Session
4:00 PM  5:30 PM
Room: Marilyn and James Simons Conference Center
5:30 PM
Multiple Cover Formula for the Stable Pairs Theory of K3xE

Georg OBERDIECK
(
Mathematisches Institut der Universität Bonn
)
Multiple Cover Formula for the Stable Pairs Theory of K3xE
Georg OBERDIECK
(
Mathematisches Institut der Universität Bonn
)
5:30 PM  6:30 PM
Room: Marilyn and James Simons Conference Center
The count of stable pairs (PandharipandeThomas theory) on K3 x E is wellunderstood 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.
Thursday, July 8, 2021
11:00 AM
VafaWitten Invariants of Projective Surfaces (4/5)

Richard THOMAS
(
Imperial College London
)
VafaWitten Invariants of Projective Surfaces (4/5)
Richard THOMAS
(
Imperial College London
)
11:00 AM  12:00 PM
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. VafaWitten invariants: stable and semistable cases 3. Techniques for calculation  virtual degeneracy loci, cosection localisation and a vanishing theorem 4. Refined VafaWitten invariants
12:00 PM
Lunch Break
Lunch Break
12:00 PM  1:30 PM
Room: Marilyn and James Simons Conference Center
1:30 PM
Algebra and geometry of link homology (4/5)

Eugene GORSKY
(
University of California at Davis
)
Algebra and geometry of link homology (4/5)
Eugene GORSKY
(
University of California at Davis
)
1:30 PM  2:30 PM
Room: Marilyn and James Simons Conference Center
Khovanov and Rozansky defined a link homology theory which categorifies the HOMFLYPT polynomial. This homology is relatively easy to define, but notoriously hard to compute. I will discuss recent breakthroughs in understanding and computing KhovanovRozansky homology, focusing on connections to the algebraic geometry of Hilbert schemes of points, affine Springer fibers and braid varieties.
2:30 PM
Break
Break
2:30 PM  3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM
Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (4/5)

Joel KAMNITZER
(
University of Toronto
)
Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (4/5)
Joel KAMNITZER
(
University of Toronto
)
3:00 PM  4:00 PM
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 BradenLicataProudfootWebster, 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, BravermanFinkelbergNakajima 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.
4:00 PM
Exercise / Q&A Session
Exercise / Q&A Session
4:00 PM  5:30 PM
Room: Marilyn and James Simons Conference Center
5:30 PM
Gaiotto Conjectures for Quantum Supergroups

Alexander BRAVERMAN
(
University of Toronto and Perimeter Institute for Theoretical Physics
)
Gaiotto Conjectures for Quantum Supergroups
Alexander BRAVERMAN
(
University of Toronto and Perimeter Institute for Theoretical Physics
)
5:30 PM  6:30 PM
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 supergroups (such as U_q(gl(MN)) via the affine Grassmannian of certain (purely even) algebraic groups. These conjectures generalize both the wellknown 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(NN1) (for generic q), based on a joint work with Finkelberg and Travkin.
Friday, July 9, 2021
11:00 AM
VafaWitten Invariants of Projective Surfaces (5/5)

Richard THOMAS
(
Imperial College London
)
VafaWitten Invariants of Projective Surfaces (5/5)
Richard THOMAS
(
Imperial College London
)
11:00 AM  12:00 PM
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. VafaWitten invariants: stable and semistable cases 3. Techniques for calculation  virtual degeneracy loci, cosection localisation and a vanishing theorem 4. Refined VafaWitten invariants
12:00 PM
Lunch Break
Lunch Break
12:00 PM  1:30 PM
Room: Marilyn and James Simons Conference Center
1:30 PM
Algebra and Geometry of Link Homology (5/5)

Eugene GORSKY
(
University of California at Davis
)
Algebra and Geometry of Link Homology (5/5)
Eugene GORSKY
(
University of California at Davis
)
1:30 PM  2:30 PM
Room: Marilyn and James Simons Conference Center
Khovanov and Rozansky defined a link homology theory which categorifies the HOMFLYPT polynomial. This homology is relatively easy to define, but notoriously hard to compute. I will discuss recent breakthroughs in understanding and computing KhovanovRozansky homology, focusing on connections to the algebraic geometry of Hilbert schemes of points, affine Springer fibers and braid varieties.
2:30 PM
Break
Break
2:30 PM  3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM
Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (5/5)

Joel KAMNITZER
(
University of Toronto
)
Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry (5/5)
Joel KAMNITZER
(
University of Toronto
)
3:00 PM  4:00 PM
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 BradenLicataProudfootWebster, 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, BravermanFinkelbergNakajima 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.
4:00 PM
Exercise / Q&A Session
Exercise / Q&A Session
4:00 PM  5:30 PM
Room: Marilyn and James Simons Conference Center
5:30 PM
3d SUSY Gauge Theory and Quantum Groups at Roots of Unity

Tudor DIMOFTE
(
University of California at Davis and University of Edinburgh
)
3d SUSY Gauge Theory and Quantum Groups at Roots of Unity
Tudor DIMOFTE
(
University of California at Davis and University of Edinburgh
)
5:30 PM  6:30 PM
Room: Marilyn and James Simons Conference Center
Topological twists of 3d N=4 gauge theories naturally give rise to nonsemisimple 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 CostantinoGeerPatureau Mirand and collaborators. I will introduce a family of physical 3d quantum field theories that (conjecturally) reproduce these classic nonsemisimple TQFT's. The physical theories combine ChernSimonslike and 3d N=4like sectors. They are also related to FeiginTipunin vertex algebras, much the same way that ChernSimons theory is related to WZW vertex algebras. (Based on work with T. Creutzig, N. Garner, and N. Geer.)
Saturday, July 10, 2021
Sunday, July 11, 2021
Monday, July 12, 2021
11:00 AM
Enumerative Geometry of Curves, Maps, and Sheaves (1/5)

Rahul PANDHARIPANDE
(
ETH Zürich
)
Enumerative Geometry of Curves, Maps, and Sheaves (1/5)
Rahul PANDHARIPANDE
(
ETH Zürich
)
11:00 AM  12:00 PM
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 GromovWitten theory, and the enumeration of sheaves via DonaldsonThomas 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.
12:00 PM
Lunch Break
Lunch Break
12:00 PM  1:30 PM
Room: Marilyn and James Simons Conference Center
1:30 PM
Cohomological Hall Algebras and Motivic Invariants for Quivers (1/4)

Markus REINEKE
(
RuhrUniversität Bochum
)
Cohomological Hall Algebras and Motivic Invariants for Quivers (1/4)
Markus REINEKE
(
RuhrUniversität Bochum
)
1:30 PM  2:30 PM
Room: Marilyn and James Simons Conference Center
We motivate, define and study DonaldsonThomas invariants and Cohomological Hall algebras associated to quivers, relate them to the geometry of moduli spaces of quiver representations and (in special cases) to GromovWitten invariants, and discuss the algebraic structure of Cohomological Hall algebras.
2:30 PM
Break
Break
2:30 PM  3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM
Generalized Airy Functions and Givental's Rmatrices for Projective Spaces and Witten's Class

Dimitri ZVONKINE
(
Laboratoire Mathématiques de Versailles
)
Generalized Airy Functions and Givental's Rmatrices for Projective Spaces and Witten's Class
Dimitri ZVONKINE
(
Laboratoire Mathématiques de Versailles
)
3:00 PM  4:00 PM
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 Rmatrices both for GromovWitten invariants of projective spaces and for Witten's rspin classes. Joint work with Sybille Rosset.
4:00 PM
Break
Break
4:00 PM  5:30 PM
Room: Marilyn and James Simons Conference Center
5:30 PM
Stable Pairs and GopakumarVafa Invariants (1/5)

Davesh MAULIK
(
Massachusetts Institute of Technology
)
Stable Pairs and GopakumarVafa Invariants (1/5)
Davesh MAULIK
(
Massachusetts Institute of Technology
)
5:30 PM  6:30 PM
Room: Marilyn and James Simons Conference Center
In the first part of the course, I will give an overview of DonaldsonThomas theory for CalabiYau threefold geometries, and its cohomological refinement. In the second part, I will explain a conjectural ansatz (from joint work with Y. Toda) for defining GopakumarVafa invariants via moduli of onedimensional sheaves, emphasizing some examples where we can understand how they relate to curvecounting via stable pairs. If time permits, I will discuss some recent work on $\chi$\independence phenomena in this setting (joint with J. Shen).
Tuesday, July 13, 2021
11:00 AM
Enumerative Geometry of Curves, Maps, and Sheaves (2/5)

Rahul PANDHARIPANDE
(
ETH Zürich
)
Enumerative Geometry of Curves, Maps, and Sheaves (2/5)
Rahul PANDHARIPANDE
(
ETH Zürich
)
11:00 AM  12:00 PM
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 GromovWitten theory, and the enumeration of sheaves via DonaldsonThomas 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.
12:00 PM
Lunch Break
Lunch Break
12:00 PM  1:30 PM
Room: Marilyn and James Simons Conference Center
1:30 PM
Cohomological Hall Algebras and Motivic Invariants for Quivers (2/4)

Markus REINEKE
(
RuhrUniversität Bochum
)
Cohomological Hall Algebras and Motivic Invariants for Quivers (2/4)
Markus REINEKE
(
RuhrUniversität Bochum
)
1:30 PM  2:30 PM
Room: Marilyn and James Simons Conference Center
We motivate, define and study DonaldsonThomas invariants and Cohomological Hall algebras associated to quivers, relate them to the geometry of moduli spaces of quiver representations and (in special cases) to GromovWitten invariants, and discuss the algebraic structure of Cohomological Hall algebras.
2:30 PM
Break
Break
2:30 PM  3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM
Complex Ktheory of Dual Hitchin Systems

Michael GROECHENIG
(
University of Toronto
)
Complex Ktheory of Dual Hitchin Systems
Michael GROECHENIG
(
University of Toronto
)
3:00 PM  4:00 PM
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 DonagiPantev, the generic fibres of the moduli spaces of GHiggs bundles and G’Higgs bundles are dual abelian varieties and are therefore derivedequivalent. 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 Ktheoretic shadow thereof: natural equivalences between complex Ktheory spectra for certain moduli spaces of Higgs bundles (in type A).
4:00 PM
Exercise / Q&A Session
Exercise / Q&A Session
4:00 PM  5:30 PM
Room: Marilyn and James Simons Conference Center
5:30 PM
Stable Pairs and GopakumarVafa Invariants (2/5)

Davesh MAULIK
(
Massachusetts Institute of Technology
)
Stable Pairs and GopakumarVafa Invariants (2/5)
Davesh MAULIK
(
Massachusetts Institute of Technology
)
5:30 PM  6:30 PM
Room: Marilyn and James Simons Conference Center
In the first part of the course, I will give an overview of DonaldsonThomas theory for CalabiYau threefold geometries, and its cohomological refinement. In the second part, I will explain a conjectural ansatz (from joint work with Y. Toda) for defining GopakumarVafa invariants via moduli of onedimensional sheaves, emphasizing some examples where we can understand how they relate to curvecounting via stable pairs. If time permits, I will discuss some recent work on $\chi$\independence phenomena in this setting (joint with J. Shen).
Wednesday, July 14, 2021
11:00 AM
Enumerative Geometry of Curves, Maps, and Sheaves (3/5)

Rahul PANDHARIPANDE
(
ETH Zürich
)
Enumerative Geometry of Curves, Maps, and Sheaves (3/5)
Rahul PANDHARIPANDE
(
ETH Zürich
)
11:00 AM  12:00 PM
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 GromovWitten theory, and the enumeration of sheaves via DonaldsonThomas 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.
12:00 PM
Lunch Break
Lunch Break
12:00 PM  1:30 PM
Room: Marilyn and James Simons Conference Center
1:30 PM
Cohomological Hall Algebras and Motivic Invariants for Quivers (3/4)

Markus REINEKE
(
RuhrUniversität Bochum
)
Cohomological Hall Algebras and Motivic Invariants for Quivers (3/4)
Markus REINEKE
(
RuhrUniversität Bochum
)
1:30 PM  2:30 PM
Room: Marilyn and James Simons Conference Center
We motivate, define and study DonaldsonThomas invariants and Cohomological Hall algebras associated to quivers, relate them to the geometry of moduli spaces of quiver representations and (in special cases) to GromovWitten invariants, and discuss the algebraic structure of Cohomological Hall algebras.
2:30 PM
Break
Break
2:30 PM  3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM
Stable Envelopes, Bow Varieties, 3d Mirror Symmetry

Richard RIMANYI
(
University of North Carolina at Chapel Hill
)
Stable Envelopes, Bow Varieties, 3d Mirror Symmetry
Richard RIMANYI
(
University of North Carolina at Chapel Hill
)
3:00 PM  4:00 PM
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 MaulikOkounkov, Okounkov, AganagicOkounkov, is the "stable envelope (class)". The stable envelope fits into the story of characteristic classes of singularities as a 1parameter 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 Legendretransform extension of bow varieties (joint work with L. Rozansky), the geometric counterpart of passing from Yangian Rmatrices of Lie algebras gl(n) to Lie superalgebras gl(nm).
4:00 PM
Exercise / Q&A Session
Exercise / Q&A Session
4:00 PM  5:30 PM
Room: Marilyn and James Simons Conference Center
5:30 PM
Stable Pairs and GopakumarVafa Invariants (3/5)

Davesh MAULIK
(
Massachusetts Institute of Technology
)
Stable Pairs and GopakumarVafa Invariants (3/5)
Davesh MAULIK
(
Massachusetts Institute of Technology
)
5:30 PM  6:30 PM
Room: Marilyn and James Simons Conference Center
In the first part of the course, I will give an overview of DonaldsonThomas theory for CalabiYau threefold geometries, and its cohomological refinement. In the second part, I will explain a conjectural ansatz (from joint work with Y. Toda) for defining GopakumarVafa invariants via moduli of onedimensional sheaves, emphasizing some examples where we can understand how they relate to curvecounting via stable pairs. If time permits, I will discuss some recent work on $\chi$\independence phenomena in this setting (joint with J. Shen).
Thursday, July 15, 2021
11:00 AM
Cohomology of Affine Springer Fibres and Centre of Small Quantum Groups

Peng SHAN
(
Tsinghua University
)
Cohomology of Affine Springer Fibres and Centre of Small Quantum Groups
Peng SHAN
(
Tsinghua University
)
11:00 AM  12:00 PM
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. BoixedaAlvarez and E. Vasserot.
12:00 PM
Lunch Break
Lunch Break
12:00 PM  1:30 PM
Room: Marilyn and James Simons Conference Center
1:30 PM
Enumerative Geometry of Curves, Maps, and Sheaves (4/5)

Rahul PANDHARIPANDE
(
ETH Zürich
)
Enumerative Geometry of Curves, Maps, and Sheaves (4/5)
Rahul PANDHARIPANDE
(
ETH Zürich
)
1:30 PM  2:30 PM
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 GromovWitten theory, and the enumeration of sheaves via DonaldsonThomas 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.
2:30 PM
Break
Break
2:30 PM  3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM
Cohomological Hall Algebras and Motivic Invariants for Quivers (4/4)

Markus REINEKE
(
RuhrUniversität Bochum
)
Cohomological Hall Algebras and Motivic Invariants for Quivers (4/4)
Markus REINEKE
(
RuhrUniversität Bochum
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
We motivate, define and study DonaldsonThomas invariants and Cohomological Hall algebras associated to quivers, relate them to the geometry of moduli spaces of quiver representations and (in special cases) to GromovWitten invariants, and discuss the algebraic structure of Cohomological Hall algebras.
4:00 PM
Exercise / Q&A Session
Exercise / Q&A Session
4:00 PM  5:30 PM
Room: Marilyn and James Simons Conference Center
5:30 PM
Stable Pairs and GopakumarVafa Invariants (4/5)

Davesh MAULIK
(
Massachusetts Institute of Technology
)
Stable Pairs and GopakumarVafa Invariants (4/5)
Davesh MAULIK
(
Massachusetts Institute of Technology
)
5:30 PM  6:30 PM
Room: Marilyn and James Simons Conference Center
In the first part of the course, I will give an overview of DonaldsonThomas theory for CalabiYau threefold geometries, and its cohomological refinement. In the second part, I will explain a conjectural ansatz (from joint work with Y. Toda) for defining GopakumarVafa invariants via moduli of onedimensional sheaves, emphasizing some examples where we can understand how they relate to curvecounting via stable pairs. If time permits, I will discuss some recent work on $\chi$\independence phenomena in this setting (joint with J. Shen).
Friday, July 16, 2021
11:00 AM
Enumerative Geometry of Curves, Maps, and Sheaves (5/5)

Rahul PANDHARIPANDE
(
ETH Zürich
)
Enumerative Geometry of Curves, Maps, and Sheaves (5/5)
Rahul PANDHARIPANDE
(
ETH Zürich
)
11:00 AM  12:00 PM
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 GromovWitten theory, and the enumeration of sheaves via DonaldsonThomas 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.
12:00 PM
Lunch Break
Lunch Break
12:00 PM  1:30 PM
Room: Marilyn and James Simons Conference Center
1:30 PM
BPS Counting and Pseudoperiodic Topology

Maxim KONTSEVICH
(
IHES
)
BPS Counting and Pseudoperiodic Topology
Maxim KONTSEVICH
(
IHES
)
1:30 PM  2:30 PM
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 3dimensional CalabiYau category. As a corollary, the counting of such geodesics gives a wallcrossing structure in the Lie algebra of Hamiltonian vector fields on a symplectic algebraic torus. I will explain that essentially the same numbers give wallcrossing structure in a different graded Lie algebra, of matrixvalued 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 1forms on complex varieties of arbitrary dimension.
2:30 PM
Break
Break
2:30 PM  3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM
Stable Pairs and GopakumarVafa Invariants (5/5)

Davesh MAULIK
(
Massachusetts Institute of Technology
)
Stable Pairs and GopakumarVafa Invariants (5/5)
Davesh MAULIK
(
Massachusetts Institute of Technology
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
In the first part of the course, I will give an overview of DonaldsonThomas theory for CalabiYau threefold geometries, and its cohomological refinement. In the second part, I will explain a conjectural ansatz (from joint work with Y. Toda) for defining GopakumarVafa invariants via moduli of onedimensional sheaves, emphasizing some examples where we can understand how they relate to curvecounting via stable pairs. If time permits, I will discuss some recent work on $\chi$\independence phenomena in this setting (joint with J. Shen).