Summer School 2021: Enumerative Geometry, Physics and Representation Theory
The main theme of this Summer School is enumerative geometry, with particular emphasis on connections with mathematical physics and representation theory. As its core, enumerative geometry is about counting geometric objects. The subject has a history of more than 2 000 years and has enjoyed many wonderful breakthroughs in the golden years of classical algebraic geometry, but we will be interested in more recent developments. This Summer School will focus on the following main subjects: - counting curves and sheaves (Gromov-Witten theory, Donaldson-Thomas and related theories), - gauge theory enumerative geometry (3d gauge theories and Coulomb branches, 4d gauge theories, and Vafa-Witten invariants, etc), - applications of enumerative geometry to categorification and low-dimensional topology, - Hall algebras and their refined versions (cohomological, K-theoretic, derived categories)
39
2021
214
1 Tag 20 Stunden
Ergebnisse 1-36 von 39
1:03:23
2Thomas, RichardThis course has 4 sections split over 5 lectures. The first section will be the longest, and hopefully useful for the other courses. - Sheaves, moduli and virtual cycles, - Vafa-Witten invariants: stable and semistable cases, - Techniques for calculation --- virtual degeneracy loci, cosection localisation and a vanishing theorem, - Refined Vafa-Witten invariants
2021Institut des Hautes Études Scientifiques (IHÉS)
1:02:53
3Thomas, RichardThis course has 4 sections split over 5 lectures. The first section will be the longest, and hopefully useful for the other courses. - Sheaves, moduli and virtual cycles, - Vafa-Witten invariants: stable and semistable cases, - Techniques for calculation --- virtual degeneracy loci, cosection localisation and a vanishing theorem, - Refined Vafa-Witten invariants
2021Institut des Hautes Études Scientifiques (IHÉS)
1:12:17
Thomas, Richard1. 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
2021Institut des Hautes Études Scientifiques (IHÉS)
1:08:31
6Bousseau, PierrickThe 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:12:04
6Göttsche, LotharWe 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:03:43
9Kamnitzer, JoelIn 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:12:21
3Kamnitzer, JoelIn 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:12:34
11Kamnitzer, JoelIn 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:05:37
10Gorsky, EugeneKhovanov 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
59:26
3Gorsky, EugeneKhovanov 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:13:36
5Gorsky, EugeneKhovanov 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:16:23
11Dimofte, TudorTopological 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.)
2021Institut des Hautes Études Scientifiques (IHÉS)
1:13:09
3Thomas, RichardThis 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
2021Institut des Hautes Études Scientifiques (IHÉS)
1:05:16
3Thomas, RichardThis 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
2021Institut des Hautes Études Scientifiques (IHÉS)
1:03:22
5Rimányi, RichárdThere 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).
2021Institut des Hautes Études Scientifiques (IHÉS)
1:09:19
8Pandharipande, RahulThe 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:14:57
5Pandharipande, RahulThe 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:04:14
7Pandharipande, RahulThe 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:03:35
3Pandharipande, RahulThe 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:07:55
1Pandharipande, RahulThe 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:07:17
6Shan, PengI 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:10:14
9Groechenig, MichaelLet 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).
2021Institut des Hautes Études Scientifiques (IHÉS)
1:15:22
7Reineke, MarkusWe 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:05:10
6Reineke, MarkusWe 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:09:06
6Reineke, MarkusWe 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:18:07
4Reineke, MarkusWe 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:06:17
4Kamnitzer, JoelIn 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:09:00
5Kamnitzer, JoelIn 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:06:31
4Gorsky, EugeneKhovanov 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:04:03
1Maulik, DaveshIn 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 χ-independence phenomena in this setting (joint with J. Shen).
2021Institut des Hautes Études Scientifiques (IHÉS)
1:04:47
1Maulik, DaveshIn 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 χ-independence phenomena in this setting (joint with J. Shen).
2021Institut des Hautes Études Scientifiques (IHÉS)
59:46
Maulik, DaveshIn 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 χ-independence phenomena in this setting (joint with J. Shen).
2021Institut des Hautes Études Scientifiques (IHÉS)
1:03:05
2Maulik, DaveshIn 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 χ-independence phenomena in this setting (joint with J. Shen).
2021Institut des Hautes Études Scientifiques (IHÉS)
1:04:43
1Maulik, DaveshIn 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 χ-independence phenomena in this setting (joint with J. Shen).
2021Institut des Hautes Études Scientifiques (IHÉS)
1:09:33
15Braverman, AlexanderI 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.
2021Institut des Hautes Études Scientifiques (IHÉS)
1:09:19
10Zvonkine, DimitriWe call generalized Airy functions particular solutions of the differential equations f(n)(x)=xaf(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.
2021Institut des Hautes Études Scientifiques (IHÉS)