*WP = Watch Party
Time  Speaker  Title  Resources  

13:50 to 14:00  Rajesh Gopakumar (ICTSTIFR, India)  Welcome Remarks  
14:00 to 14:10  Aswin Balasubramanian (Rutgers University, USA)  Introduction by organiser  
14:10 to 15:30  Ana PeonNieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) 
Geometry of the Global Nilpotent Cone (Lecture 1) Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this minicourse, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss DonagiPantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci. Outline (1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2] (2) Applications in a nutshell (3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP] References: [BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989), 169–179. [BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. [DP1] R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189 (2012), no. 3, 653–735. [DP2] R.Y. Donagi and T.B. Pantev,Geometric Langlands and nonabelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009. [DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG] [Fa] G. Faltings, Theta functions on moduli spaces of Gbundles, J. Algebraic Geom. 18 (2009), 309369. DOI: https://doi.org/10.1090/S1056391108004992. [G] V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 . [HT] T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229. [HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583. [Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114. [Hi2] N. Hitchin,The selfduality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. [La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. [NR1] M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51. [PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry. [PPe] C. Pauly and A. PeónNieto, Very stable bundles and properness of the Hitchin map, A. Geom. Dedicata (2018). DOI:10.1007/s1071101803336. Preprint arXiv:1710.10152 [math.AG]. [Pe] A. PeónNieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447. [S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129. [S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 . [Z] H. Zelaci On very stablity of principal Gbundles. Geom. Dedicata 204 (2020), 165–173. 

15:30 to 17:00  Madalena Lemos (Durham University, UK) 
A minicourse on vertex operator algebras of N= 2 Superconformal Field Theories (Lecture 1) Any fourdimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a twodimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for fourdimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the minicourse, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c]. References [a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280 [b] Ribault, Minimal lectures on twodimensional conformal field theory, arxiv:1609.09523 [c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892 

20:30 to 20:40  Aswin Balasubramanian (Rutgers University, USA)  Introduction by organiser  
20:40 to 22:00  Mathew Bullimore (Durham University, UK) 
A Mathematical Introduction to 3d N = 4 Gauge Theories (Lecture 1) This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians. References [a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf. [b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf. 

22:00 to 23:30  Ana PeonNieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) 
Geometry of the Global Nilpotent Cone 1 (WP) Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this minicourse, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss DonagiPantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci. Outline (1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2] (2) Applications in a nutshell (3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP] References: [BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989), 169–179. [BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. [DP1] R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189 (2012), no. 3, 653–735. [DP2] R.Y. Donagi and T.B. Pantev,Geometric Langlands and nonabelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009. [DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG] [Fa] G. Faltings, Theta functions on moduli spaces of Gbundles, J. Algebraic Geom. 18 (2009), 309369. DOI: https://doi.org/10.1090/S1056391108004992. [G] V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 . [HT] T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229. [HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583. [Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114. [Hi2] N. Hitchin,The selfduality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. [La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. [NR1] M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51. [PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry. [PPe] C. Pauly and A. PeónNieto, Very stable bundles and properness of the Hitchin map, A. Geom. Dedicata (2018). DOI:10.1007/s1071101803336. Preprint arXiv:1710.10152 [math.AG]. [Pe] A. PeónNieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447. [S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129. [S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 . [Z] H. Zelaci On very stablity of principal Gbundles. Geom. Dedicata 204 (2020), 165–173. 
Time  Speaker  Title  Resources  

00:30 to 02:00  Madalena Lemos (Durham University, UK) 
A minicourse on vertex operator algebras of N= 2 Superconformal Field Theories 1 (WP) Any fourdimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a twodimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for fourdimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the minicourse, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c]. References: [a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280 [b] Ribault, Minimal lectures on twodimensional conformal field theory, arxiv:1609.09523 [c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892 

11:00 to 12:30  Mathew Bullimore (Durham University, UK) 
A Mathematical Introduction to 3d N = 4 Gauge Theories 1 (WP) This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians. References [a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf. [b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf. 

15:30 to 17:00  Madalena Lemos (Durham University, UK) 
A minicourse on vertex operator algebras of N= 2 Superconformal Field Theories (Lecture 2) Any fourdimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a twodimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for fourdimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the minicourse, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c]. References: [a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280 [b] Ribault, Minimal lectures on twodimensional conformal field theory, arxiv:1609.09523 [c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892 

20:30 to 22:00  Mathew Bullimore (Durham University, UK) 
A Mathematical Introduction to 3d N = 4 Gauge Theories (Lecture 2) This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians. References [a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf. [b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf. 

22:00 to 23:30  Miroslav Rapcak (University of California, Berkeley, USA) 
The Miura operator at the M2M5 Intersection M5branes in the omega background give rise to a large class of vertex operator algebras. Analogously, M2branes in the omega background lead to a large class of Higgs branch/Coulomb branch algebras. I am going to give a new interpretation to the Miura operator, an important object in the theory of integrable hierarchies and vertex operator algebra, as an operator living at the M5M2 intersection and intertwining the action of the two algebras. References: [1] Gaiotto, Rapcak, Miura operators, degenerate fields and the M2M5 intersection, 2012.04118 [2] Gaiotto, Oh, Aspects of \Omegadeformed Mtheory, 1907.06495 [3] Prochazka, Rapcak, Walgebra modules, free fields, and GukovWitten defects, 1808.08837 
Time  Speaker  Title  Resources  

00:30 to 02:00  Madalena Lemos (Durham University, UK) 
A minicourse on vertex operator algebras of N= 2 Superconformal Field Theories 2 (WP) Any fourdimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a twodimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for fourdimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the minicourse, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c]. References: [a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280 [b] Ribault, Minimal lectures on twodimensional conformal field theory, arxiv:1609.09523 [c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892 

11:00 to 12:30  Mathew Bullimore (Durham University, UK) 
A Mathematical Introduction to 3d N = 4 Gauge Theories 2 (WP) This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians. References [a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf. [b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf. 

14:00 to 15:30  Ana PeonNieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) 
Geometry of the Global Nilpotent Cone (Lecture 2) Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this minicourse, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss DonagiPantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci. Outline (1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2] (2) Applications in a nutshell (3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP] References: [BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989), 169–179. [BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. [DP1] R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189 (2012), no. 3, 653–735. [DP2] R.Y. Donagi and T.B. Pantev,Geometric Langlands and nonabelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009. [DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG] [Fa] G. Faltings, Theta functions on moduli spaces of Gbundles, J. Algebraic Geom. 18 (2009), 309369. DOI: https://doi.org/10.1090/S1056391108004992. [G] V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 . [HT] T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229. [HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583. [Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114. [Hi2] N. Hitchin,The selfduality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. [La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. [NR1] M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51. [PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry. [PPe] C. Pauly and A. PeónNieto, Very stable bundles and properness of the Hitchin map, A. Geom. Dedicata (2018). DOI:10.1007/s1071101803336. Preprint arXiv:1710.10152 [math.AG]. [Pe] A. PeónNieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447. [S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129. [S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 . [Z] H. Zelaci On very stablity of principal Gbundles. Geom. Dedicata 204 (2020), 165–173. 

15:30 to 17:00  Miroslav Rapcak (University of California, Berkeley, USA) 
The Miura operator at the M2M5 Intersection (WP) M5branes in the omega background give rise to a large class of vertex operator algebras. Analogously, M2branes in the omega background lead to a large class of Higgs branch/Coulomb branch algebras. I am going to give a new interpretation to the Miura operator, an important object in the theory of integrable hierarchies and vertex operator algebra, as an operator living at the M5M2 intersection and intertwining the action of the two algebras. References: [1] Gaiotto, Rapcak, Miura operators, degenerate fields and the M2M5 intersection, 2012.04118 [2] Gaiotto, Oh, Aspects of \Omegadeformed Mtheory, 1907.06495 [3] Prochazka, Rapcak, Walgebra modules, free fields, and GukovWitten defects, 1808.08837 

20:30 to 22:00  Mathew Bullimore (Durham University, UK) 
A Mathematical Introduction to 3d N = 4 Gauge Theories (Lecture 3) This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians. References [a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf. [b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf. 

22:00 to 23:30  Jacques Hurtubise (McGill University, Canada) 
Geometry of the Hitchin integrable systems, and some variations (Lecture 1) The NarasimhanSeshadri (HitchinKobayashiYau...) correspondence. The basic idea (ASD YangMills), examples of various noncompact cases and reductions: monopoles on various spaces, YangMillsHiggs. The YangMIllsHiggs moduli. Higgs bundles, symplectic structures and integrable systems. The spectral curve and abelianization. Parabolic generalisations, examples. Darboux coordinates and geometry of Poisson surfaces. Variants: The Sklyanin brackets. Integrable systems and surfaces. Local geometry of integrable systems of Jacobians: rank two systems. Other groups: Cameral covers and Higgs. Sklyanin type systems; the analogue of the rank two condition. 
Time  Speaker  Title  Resources  

00:30 to 02:00  Ana PeonNieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) 
Geometry of the Global Nilpotent Cone 2 (WP) Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this minicourse, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss DonagiPantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci. Outline (1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2] (2) Applications in a nutshell (3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP] References: [BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989), 169–179. [BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. [DP1] R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189 (2012), no. 3, 653–735. [DP2] R.Y. Donagi and T.B. Pantev,Geometric Langlands and nonabelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009. [DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG] [Fa] G. Faltings, Theta functions on moduli spaces of Gbundles, J. Algebraic Geom. 18 (2009), 309369. DOI: https://doi.org/10.1090/S1056391108004992. [G] V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 . [HT] T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229. [HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583. [Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114. [Hi2] N. Hitchin,The selfduality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. [La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. [NR1] M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51. [PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry. [PPe] C. Pauly and A. PeónNieto, Very stable bundles and properness of the Hitchin map, A. Geom. Dedicata (2018). DOI:10.1007/s1071101803336. Preprint arXiv:1710.10152 [math.AG]. [Pe] A. PeónNieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447. [S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129. [S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 . [Z] H. Zelaci On very stablity of principal Gbundles. Geom. Dedicata 204 (2020), 165–173. 

11:00 to 12:30  Mathew Bullimore (Durham University, UK) 
A Mathematical Introduction to 3d N = 4 Gauge Theories 3 (WP) This mini course will introduce mathematical structures arising in 3d N = 4 gauge theories, with a focus on aspects that are important in geometric representation theory. The course will cover the supersymmetry algebra, central charges, mass and FI parameter deformations, walls and chambers, Higgs and Coulomb branch moduli spaces, and 3d mirror symmetry. This should provide a stepping stone to more advanced topics in the literature. I will try to explain how physicists think about these concepts in a way that is friendly to mathematicians. References [a] A quick summary of some aspects of 3d N = 4 gauge theories can be found in section 2 of https://arxiv.org/pdf/1503.04817.pdf. [b] For some motivation for mathematicians, see section 2 of https://arxiv.org/pdf/1706.05154.pdf. 

14:00 to 15:30  Dan Xie (Tsinghua University, China) 
Affine Springer fiber and representation theory of W algebra I will discuss the interesting map between fixed variety of affine Springer fiber and representation theory of W algebra. The corresponding affine Springer fiber and W algebra appear in the study of 4d ArgyresDouglas theory, and the above map might be thought of as a generalized 3d mirror symmetry. References: 1. Z.W Yun, Lectures on Springer theories and orbital integral, arXiv:1602.01451 2. V.Kac, Wakimoto, On rationality of W algebras, arXiv: 0711.2296 3. D. Xie, Generalized ArgyresDouglas theories, arXiv: 1204.2270 4. J. W. Song, D.Xie, W.B Yan, Vertex operator algebras of ArgyresDouglas theories from M5 branes, arXiv: 1706.01607 

15:30 to 17:00  Madalena Lemos (Durham University, UK) 
A minicourse on vertex operator algebras of N= 2 Superconformal Field Theories (Lecture 3) Any fourdimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a twodimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for fourdimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the minicourse, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c]. References: [a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280 [b] Ribault, Minimal lectures on twodimensional conformal field theory, arxiv:1609.09523 [c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892 

20:30 to 22:00  Edward Witten (Institute for Advanced Study, Princeton, USA) 
Quantization, Gauge Theory, and the Analytic Approach to Geometric Langlands (Lecture 1) The first lecture will be a general introduction to the problem of quantization. Contrary to the impression that one may get from textbooks, there is not a satisfactory general answer to the problem of quantizing a symplectic manifold. I will explain two partial answers: geometric quantization and ``quantization by branes.'' In the second lecture, I will give an introduction to the gauge theory approach to the geometric Langlands correspondence. The third lecture is part of a special program in memory of M. S. Narasimhan and C. S. Seshadri. Much of their work has provided important foundations for the sort of mathematical physics that is represented at this meeting  including my lectures. Finally, the fourth lecture will explain work with D. Gaiotto in which we use ``quantization by branes'' to elucidate a new twist on geometric Langlands that has been discovered by P. Etinghof, E. Frenkel, and D. Kazhdan. References: On quantization by branes: https://arxiv.org/abs/0809.0305 For a brief introduction to the subject of lecture 2, see the first three sections of https://arxiv.org/abs/0809.0305 (If you have time, you can look at the more detailed paper https://arxiv.org/pdf/hepth/0604151.pdf.) It is hard to give references for my lecture at the event in honor of Narasimhan and Seshadri, but I do recommend this famous article by Atiyah and Bott where some of their work is important background: http://www.math.toronto.edu/mgualt/Morse%20Theory/AtiyahBott.pdf Finally, on my fourth lecture, the reference would be a paper that Gaiotto and I are writing that hopefully will appear before my lectures. 

22:00 to 23:30  Dan Xie (Tsinghua University, China) 
Affine Springer fiber and representation theory of W algebra (WP) I will discuss the interesting map between fixed variety of affine Springer fiber and representation theory of W algebra. The corresponding affine Springer fiber and W algebra appear in the study of 4d ArgyresDouglas theory, and the above map might be thought of as a generalized 3d mirror symmetry. References: 1. Z.W Yun, Lectures on Springer theories and orbital integral, arXiv:1602.01451 2. V.Kac, Wakimoto, On rationality of W algebras, arXiv: 0711.2296 3. D. Xie, Generalized ArgyresDouglas theories, arXiv: 1204.2270 4. J. W. Song, D.Xie, W.B Yan, Vertex operator algebras of ArgyresDouglas theories from M5 branes, arXiv: 1706.01607 
Time  Speaker  Title  Resources  

00:30 to 02:00  Madalena Lemos (Durham University, UK) 
A minicourse on vertex operator algebras of N= 2 Superconformal Field Theories 3 (WP) Any fourdimensional N>=2 superconformal field theory possess a protected subsector isomorphic to a twodimensional chiral algebra/vertex operator algebra [arxiv:1312.5344]. After a brief review of N=2 SCFTs we will describe the construction of this subsector, the consequences for fourdimensional physics, as well as a brief outlook of recent progress. Some basic familiarity with conformal field theories is desirable, for example with what is covered in [a] section 1,2,3 and the corresponding background material (the remaining sections of [a] provide details on 4d N=2 superconformal theories which is relevant for the minicourse, but detailed knowledge of that will not be assumed). Some familiarity with enhanced Virasoro algebra in 2d CFTs, along the lines of what is summarized in chapter 1 of [b] is also beneficial. The bulk of the lectures will proceed along the same lines as [c]. References: [a] Eberhardt, Superconformal symmetry and representations, arxiv:2006.13280 [b] Ribault, Minimal lectures on twodimensional conformal field theory, arxiv:1609.09523 [c] Lemos, Lectures on chiral algebras of N⩾2 superconformal field theories, arXiv:2006.13892 

11:00 to 12:30  Jacques Hurtubise (McGill University, Canada) 
Geometry of the Hitchin integrable systems, and some variations 1 (WP) The NarasimhanSeshadri (HitchinKobayashiYau...) correspondence. The basic idea (ASD YangMills), examples of various noncompact cases and reductions: monopoles on various spaces, YangMillsHiggs. The YangMIllsHiggs moduli. Higgs bundles, symplectic structures and integrable systems. The spectral curve and abelianization. Parabolic generalisations, examples. Darboux coordinates and geometry of Poisson surfaces. Variants: The Sklyanin brackets. Integrable systems and surfaces. Local geometry of integrable systems of Jacobians: rank two systems. Other groups: Cameral covers and Higgs. Sklyanin type systems; the analogue of the rank two condition. 

14:00 to 15:30  Ana PeonNieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) 
Geometry of the Global Nilpotent Cone (Lecture 3) Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this minicourse, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss DonagiPantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci. Outline (1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2] (2) Applications in a nutshell (3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP] References: [BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989), 169–179. [BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. [DP1] R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189 (2012), no. 3, 653–735. [DP2] R.Y. Donagi and T.B. Pantev,Geometric Langlands and nonabelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009. [DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG] [Fa] G. Faltings, Theta functions on moduli spaces of Gbundles, J. Algebraic Geom. 18 (2009), 309369. DOI: https://doi.org/10.1090/S1056391108004992. [G] V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 . [HT] T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229. [HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583. [Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114. [Hi2] N. Hitchin,The selfduality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. [La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. [NR1] M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51. [PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry. [PPe] C. Pauly and A. PeónNieto, Very stable bundles and properness of the Hitchin map, A. Geom. Dedicata (2018). DOI:10.1007/s1071101803336. Preprint arXiv:1710.10152 [math.AG]. [Pe] A. PeónNieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447. [S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129. [S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 . [Z] H. Zelaci On very stablity of principal Gbundles. Geom. Dedicata 204 (2020), 165–173. 

15:30 to 17:00  Edward Witten (Institute for Advanced Study, Princeton, USA) 
Quantization, Gauge Theory, and the Analytic Approach to Geometric Langlands 1 (WP) The first lecture will be a general introduction to the problem of quantization. Contrary to the impression that one may get from textbooks, there is not a satisfactory general answer to the problem of quantizing a symplectic manifold. I will explain two partial answers: geometric quantization and ``quantization by branes.'' In the second lecture, I will give an introduction to the gauge theory approach to the geometric Langlands correspondence. The third lecture is part of a special program in memory of M. S. Narasimhan and C. S. Seshadri. Much of their work has provided important foundations for the sort of mathematical physics that is represented at this meeting  including my lectures. Finally, the fourth lecture will explain work with D. Gaiotto in which we use ``quantization by branes'' to elucidate a new twist on geometric Langlands that has been discovered by P. Etinghof, E. Frenkel, and D. Kazhdan. References: On quantization by branes: https://arxiv.org/abs/0809.0305 For a brief introduction to the subject of lecture 2, see the first three sections of https://arxiv.org/abs/0809.0305 (If you have time, you can look at the more detailed paper https://arxiv.org/pdf/hepth/0604151.pdf.) It is hard to give references for my lecture at the event in honor of Narasimhan and Seshadri, but I do recommend this famous article by Atiyah and Bott where some of their work is important background: http://www.math.toronto.edu/mgualt/Morse%20Theory/AtiyahBott.pdf Finally, on my fourth lecture, the reference would be a paper that Gaiotto and I are writing that hopefully will appear before my lectures. 

20:30 to 22:00  Edward Witten (Institute for Advanced Study, Princeton, USA) 
Quantization by Branes and Geometric Langlands (Lecture 2) The first lecture will be a general introduction to the problem of quantization. Contrary to the impression that one may get from textbooks, there is not a satisfactory general answer to the problem of quantizing a symplectic manifold. I will explain two partial answers: geometric quantization and ``quantization by branes.'' In the second lecture, I will give an introduction to the gauge theory approach to the geometric Langlands correspondence. The third lecture is part of a special program in memory of M. S. Narasimhan and C. S. Seshadri. Much of their work has provided important foundations for the sort of mathematical physics that is represented at this meeting  including my lectures. Finally, the fourth lecture will explain work with D. Gaiotto in which we use ``quantization by branes'' to elucidate a new twist on geometric Langlands that has been discovered by P. Etinghof, E. Frenkel, and D. Kazhdan. References: On quantization by branes: https://arxiv.org/abs/0809.0305 For a brief introduction to the subject of lecture 2, see the first three sections of https://arxiv.org/abs/0809.0305 (If you have time, you can look at the more detailed paper https://arxiv.org/pdf/hepth/0604151.pdf.) It is hard to give references for my lecture at the event in honor of Narasimhan and Seshadri, but I do recommend this famous article by Atiyah and Bott where some of their work is important background: http://www.math.toronto.edu/mgualt/Morse%20Theory/AtiyahBott.pdf Finally, on my fourth lecture, the reference would be a paper that Gaiotto and I are writing that hopefully will appear before my lectures. 

22:00 to 23:30  Jacques Hurtubise (McGill University, Canada) 
Geometry of the Hitchin integrable systems, and some variations (Lecture 2) The NarasimhanSeshadri (HitchinKobayashiYau...) correspondence. The basic idea (ASD YangMills), examples of various noncompact cases and reductions: monopoles on various spaces, YangMillsHiggs. The YangMIllsHiggs moduli. Higgs bundles, symplectic structures and integrable systems. The spectral curve and abelianization. Parabolic generalisations, examples. Darboux coordinates and geometry of Poisson surfaces. Variants: The Sklyanin brackets. Integrable systems and surfaces. Local geometry of integrable systems of Jacobians: rank two systems. Other groups: Cameral covers and Higgs. Sklyanin type systems; the analogue of the rank two condition. 
Time  Speaker  Title  Resources  

00:30 to 02:00  Ana PeonNieto (University of Nice Sophia Antipolis, France and University of Birmingham, UK) 
Geometry of the Global Nilpotent Cone 3 (WP) Higgs bundles are central players in the current arena in mathematics and physics. Their applications to mirror symmetry and the geometric Langlands programme have captured a lot of attention in the past few years. In this minicourse, we will study some aspects of the geometry of the moduli space of these objects M_X , focusing on the nilpotent cone . Besides capturing the global topology of M_X , the nilpotent cone is at the heart of many applications that we will explore, always basing our discussion on manageable examples. More precisely, after a brief introduction, we will discuss DonagiPantev’s programme towards geometric Langlands through abelianisation, as well as discuss two important conjectures therein. One, due to the authors of the programme, concerns the equality of the wobbly and shaky loci. The other one, due to Drinfeld, concerns pure codimensionality of wobbly loci. Outline (1) Introduction to Higgs bundles and the non abelian Hodge correspondence. Refs: [Hi2, S1, S2] (2) Applications in a nutshell (3) The nilpotent cone. Refs:[BD, Fa, G, La, PaP] References: [BNR] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989), 169–179. [BD] A.A. Beilinson and V.G. Drinfeld, Quantization of Hitchin’s fibration and Langlands’ program. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. [DP1] R.Y. Donagi and T.B. Pantev,Langlands duality for Hitchin systems. Invent. Math. 189 (2012), no. 3, 653–735. [DP2] R.Y. Donagi and T.B. Pantev,Geometric Langlands and nonabelian Hodge theory. Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009. [DP3] R.Y. Donagi and T.B. Pantev,Parabolic Hecke eigensheaves. Preprint, arXiv:1910.02357 [math.AG] [Fa] G. Faltings, Theta functions on moduli spaces of Gbundles, J. Algebraic Geom. 18 (2009), 309369. DOI: https://doi.org/10.1090/S1056391108004992. [G] V. Ginzburg,The global nilpotent variety is Lagrangian. Duke Math. J. 109 (2001), no. 3, 511–519 . [HT] T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153 (2003), no. 1, 197–229. [HH] T. Hausel and N. Hitchin, Very stable Higgs bundles, equivariant multiplicity and mirror symmetry, Preprint, arXiv: 2101.08583. [Hi1] N. Hitchin,Stable bundles and integrable systems. Duke Math. J. 54 (1987), no. 1, 91–114. [Hi2] N. Hitchin,The selfduality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. [La] G. Laumon, Un analogue global du cône nilpotent, Duke Math. Jour. 57 (1988), 647–671. [NR1] M.S. Narasimhan and S. Ramanan, Moduli of Vector Bundles on a Compact Riemann Surface, Ann. Math., Second Series, Vol. 89, No. 1 (1969), 14–51. [PaP] S. Pal and C. Pauly, Wobbly divisors in the moduli space of rank two bundle, to appear in Adv. in Geometry. [PPe] C. Pauly and A. PeónNieto, Very stable bundles and properness of the Hitchin map, A. Geom. Dedicata (2018). DOI:10.1007/s1071101803336. Preprint arXiv:1710.10152 [math.AG]. [Pe] A. PeónNieto, Wobbly and shaky bundles and resolutions of rational maps, arXiv:2007.13447. [S1] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I and II. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129. [S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 . [Z] H. Zelaci On very stablity of principal Gbundles. Geom. Dedicata 204 (2020), 165–173. 
Time  Speaker  Title  Resources  

15:30 to 17:00  Vikraman Balaji (Chennai Mathematical Institute, India) 
Parahoric torsors and degeneration of moduli spaces I will briefly introduce ideas from my joint work with C.S. Seshadri on "Parahoric torsors" and indicate how this leads to an understanding and solution to the problem of degenerations of the moduli spaces of principal $G$bundles. References: V.BALAJI, C.S.SESHADRI, Moduli of parahoric $\mathcal G$torsors on a compact Riemann surface, {\em Journal of Algebraic Geometry}, Volume 24, pages 149, (2015). V. BALAJI, Torsors on semistable curves and degenerations, (math archiv) (to appear in the {\em Proceedings of the Indian Academy of Sciences}, 2021). 

19:00 to 20:30  Jacques Hurtubise (McGill University, Canada) 
Geometry of the Hitchin integrable systems, and some variations (Lecture 3) The NarasimhanSeshadri (HitchinKobayashiYau...) correspondence. The basic idea (ASD YangMills), examples of various noncompact cases and reductions: monopoles on various spaces, YangMillsHiggs. The YangMIllsHiggs moduli. Higgs bundles, symplectic structures and integrable systems. The spectral curve and abelianization. Parabolic generalisations, examples. Darboux coordinates and geometry of Poisson surfaces. Variants: The Sklyanin brackets. Integrable systems and surfaces. Local geometry of integrable systems of Jacobians: rank two systems. Other groups: Cameral covers and Higgs. Sklyanin type systems; the analogue of the rank two condition. 

20:30 to 22:00  Edward Witten (Institute for Advanced Study, Princeton, USA) 
On The Work Of Narasimhan and Seshadri (Lecture 3) The first lecture will be a general introduction to the problem of quantization. Contrary to the impression that one may get from textbooks, there is not a satisfactory general answer to the problem of quantizing a symplectic manifold. I will explain two partial answers: geometric quantization and ``quantization by branes.'' In the second lecture, I will give an introduction to the gauge theory approach to the geometric Langlands correspondence. The third lecture is part of a special program in memory of M. S. Narasimhan and C. S. Seshadri. Much of their work has provided important foundations for the sort of mathematical physics that is represented at this meeting  including my lectures. Finally, the fourth lecture will explain work with D. Gaiotto in which we use ``quantization by branes'' to elucidate a new twist on geometric Langlands that has been discovered by P. Etinghof, E. Frenkel, and D. Kazhdan. References: On quantization by branes: https://arxiv.org/abs/0809.0305 For a brief introduction to the subject of lecture 2, see the first three sections of https://arxiv.org/abs/0809.0305 (If you have time, you can look at the more detailed paper https://arxiv.org/pdf/hepth/0604151.pdf.) It is hard to give references for my lecture at the event in honor of Narasimhan and Seshadri, but I do recommend this famous article by Atiyah and Bott where some of their work is important background: http://www.math.toronto.edu/mgualt/Morse%20Theory/AtiyahBott.pdf Finally, on my fourth lecture, the reference would be a paper that Gaiotto and I are writing that hopefully will appear before my lectures. 

22:00 to 23:30  Shrawan Kumar (University of North Carolina at Chapel Hill, USA) 
Conformal blocks for Galois covers of algebraic curves This is a joint work with Jiuzu Hong. We study the space of twisted conformal blocks attached to Acurves S with marked Aorbits and an action of A on a simple Lie algebra g, where A is a finite group. We prove that if A stabilizes a Borel subalgebra of g, then Propogation Theorem and Factorization Theorem hold. We endow a projectively flat connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed Acurves; in particular,it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. We further identify the space of twisted conformal blocks with the space of global sections of certain line bundles on the stack of Aequivariant principal Gbundles over the curve S, G being the simplyconnected group with Lie algebra g. This generalizes the Verlinde theory of conformal blocks to the twisted setting. 
Time  Speaker  Title  Resources  

15:30 to 17:00  Joerg Teschner (University of Hamburg and DESY, Germany) 
Analytic geometric Langlandscorrespondence: Relations to conformal field theory and integrable models (Lecture 1) In the first of my three lectures I plan to present a review of the approach of Beilinson and Drinfeld to the geometric Langlands correspondence (GL), based on ideas from conformal field theory and the representation theory of the affine Lie algebras at the critical level. The goal of my second lecture will be to review some aspects of a recent variant of the GL called the analytic Langlands correspondence. The third lecture will discuss a natural deformation of the analytic Langlands correspondence related to the H3WZNW model, and its relation to a generalisation of the AGTcorrespondence. References: Lecture 1: E. Frenkel, Affine algebras, Langlands duality and Bethe ansatz, in Proc. of Int. Congress of Math. Phys. (Paris, 1994), ed. D. Iagolnitzer, pp. 606–642, International Press, 1995 (arXiv:qalg/9506003). E. Frenkel, Lectures on the Langlands program and conformal field theory, in Frontiers in number theory, physics, and geometry. II (P. Cartier, ed.), pp. 387–533. Springer, Berlin, 2007. (arXiv:hepth/0512172). Lecture 2: J. Teschner, Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence, in Geometry and Physics, in honour of Nigel Hitchin, Vol. I, eds. Dancer, e.a., pp. 347–375, Oxford University Press, 2018 (arXiv:1707.07873). P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves, in Integrability, Quantization, and Geometry, dedicated to Boris Dubrovin, Vol. II, eds. S. Novikov, e.a., pp. 137–202, Proc. Symp. Pure Math. 103.2, AMS, 2021 (arXiv:1908.09677). Lecture 3: Based on ideas going back to, J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011), no. 2, 471–564. [arXiv:1005.2846] and some unpublished work. 

20:30 to 22:00  Edward Witten (Institute for Advanced Study, Princeton, USA) 
Quantization by Branes and Geometric Langlands (Lecture 4) The first lecture will be a general introduction to the problem of quantization. Contrary to the impression that one may get from textbooks, there is not a satisfactory general answer to the problem of quantizing a symplectic manifold. I will explain two partial answers: geometric quantization and ``quantization by branes.'' In the second lecture, I will give an introduction to the gauge theory approach to the geometric Langlands correspondence. The third lecture is part of a special program in memory of M. S. Narasimhan and C. S. Seshadri. Much of their work has provided important foundations for the sort of mathematical physics that is represented at this meeting  including my lectures. Finally, the fourth lecture will explain work with D. Gaiotto in which we use ``quantization by branes'' to elucidate a new twist on geometric Langlands that has been discovered by P. Etinghof, E. Frenkel, and D. Kazhdan. References: On quantization by branes: https://arxiv.org/abs/0809.0305 For a brief introduction to the subject of lecture 2, see the first three sections of https://arxiv.org/abs/0809.0305 (If you have time, you can look at the more detailed paper https://arxiv.org/pdf/hepth/0604151.pdf.) It is hard to give references for my lecture at the event in honor of Narasimhan and Seshadri, but I do recommend this famous article by Atiyah and Bott where some of their work is important background: http://www.math.toronto.edu/mgualt/Morse%20Theory/AtiyahBott.pdf Finally, on my fourth lecture, the reference would be a paper that Gaiotto and I are writing that hopefully will appear before my lectures. 

22:00 to 23:30  Sam Raskin (University of Texas, USA) 
Geometric Langlands and 3d mirror symmetry (Lecture 1) In this series of talks, we will give an introduction to the geometric Langlands correspondence, emphasizing local aspects of the subject. We will describe expected compatibilities for the geometric Langlands conjectures in the language of boundary conditions. Many of these compatibility conjectures have emerged only recently, coming from work of BenZvi, Braverman, Costello, Dimofte, Finkelberg, Gaiotto, Hilburn, Sakellaridis, Venkatesh, Witten, and Yoo (and probably others) on uniting 3d mirror symmetry with geometric Langlands. Finally, we will provide an overview of recent joint work with Justin Hilburn establishing one of these recent conjectures in the abelian case. References: Beilinson Drinfeld  Quantization of Hitchin's integrable system and Hecke eigensheaves link Beilinson Drinfeld  Chiral algebras link Gaitsgory  Outline of the proof of the geometric Langlands conjecture for GL_2 link Gaiotto Witten  Sduality of boundary conditions in N = 4 super Yang–Mills theory link Hilburn Raskin  Tate's thesis in the de Rham setting link 
Time  Speaker  Title  Resources  

20:30 to 22:00  Heeyeon Kim (Rutgers University, USA) 
Supersymmetric Ground States of 3rd N=4 Theories on a Riemann surface I will discuss effective quantum mechanics obtained from a large class of 3d N=4 gauge theories compactified on a Riemann surface. In the first part of the talk, I will focus on the Witten index of the quantum mechanics, which can be identified as the virtual Euler characteristic of the moduli space of vortices on the Riemann surface. In particular, I will discuss 3d mirror symmetry that provides nontrivial relations among these invariants, and their wallcrossing formulae derived from a path integral point of view. Finally, I will comments on explicit construction of the space of supersymmetric ground states of these theories. 

22:00 to 23:30  Sam Raskin (University of Texas, USA) 
Geometric Langlands and 3d mirror symmetry (Lecture 2) In this series of talks, we will give an introduction to the geometric Langlands correspondence, emphasizing local aspects of the subject. We will describe expected compatibilities for the geometric Langlands conjectures in the language of boundary conditions. Many of these compatibility conjectures have emerged only recently, coming from work of BenZvi, Braverman, Costello, Dimofte, Finkelberg, Gaiotto, Hilburn, Sakellaridis, Venkatesh, Witten, and Yoo (and probably others) on uniting 3d mirror symmetry with geometric Langlands. Finally, we will provide an overview of recent joint work with Justin Hilburn establishing one of these recent conjectures in the abelian case. References: Beilinson Drinfeld  Quantization of Hitchin's integrable system and Hecke eigensheaves link Beilinson Drinfeld  Chiral algebras link Gaitsgory  Outline of the proof of the geometric Langlands conjecture for GL_2 link Gaiotto Witten  Sduality of boundary conditions in N = 4 super Yang–Mills theory link Hilburn Raskin  Tate's thesis in the de Rham setting link 
Time  Speaker  Title  Resources  

14:00 to 15:30  Dario Beraldo (University College London, United Kingdom) 
On the Ramanujan conjecture in Geometric Langlands After discussing the notion of temperedness arising in the geometric Langlands program, I’ll sketch a proof of a version of the Ramanujan conjecture in this setting. Essential ingredients for the formulation and the proof are the derived Satake equivalence and (geometric analogues of) the DeligneLusztig duality functors. References: [1] D. Arinkin, D. Gaitsgory, Singular support of coherent sheaves and the geometric Langlands conjecture. Selecta Math. (N.S.) 21 (2015), no. 1, 1199. [2] D. Arinkin, D. Gaitsgory, The category of singularities as a crystal and global Springer fibers. J. Amer. Math. Soc. 31 (2018), no. 1, 135214. [3] D. Beraldo. On the geometric Ramanujan conjecture, https://arxiv.org/abs/2103.17211 

15:30 to 17:00  Joerg Teschner (University of Hamburg and DESY, Germany) 
Analytic geometric Langlandscorrespondence: Relations to conformal field theory and integrable models (Lecture 2) In the first of my three lectures I plan to present a review of the approach of Beilinson and Drinfeld to the geometric Langlands correspondence (GL), based on ideas from conformal field theory and the representation theory of the affine Lie algebras at the critical level. The goal of my second lecture will be to review some aspects of a recent variant of the GL called the analytic Langlands correspondence. The third lecture will discuss a natural deformation of the analytic Langlands correspondence related to the H3WZNW model, and its relation to a generalisation of the AGTcorrespondence. References: Lecture 1: E. Frenkel, Affine algebras, Langlands duality and Bethe ansatz, in Proc. of Int. Congress of Math. Phys. (Paris, 1994), ed. D. Iagolnitzer, pp. 606–642, International Press, 1995 (arXiv:qalg/9506003). E. Frenkel, Lectures on the Langlands program and conformal field theory, in Frontiers in number theory, physics, and geometry. II (P. Cartier, ed.), pp. 387–533. Springer, Berlin, 2007. (arXiv:hepth/0512172). Lecture 2: J. Teschner, Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence, in Geometry and Physics, in honour of Nigel Hitchin, Vol. I, eds. Dancer, e.a., pp. 347–375, Oxford University Press, 2018 (arXiv:1707.07873). P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves, in Integrability, Quantization, and Geometry, dedicated to Boris Dubrovin, Vol. II, eds. S. Novikov, e.a., pp. 137–202, Proc. Symp. Pure Math. 103.2, AMS, 2021 (arXiv:1908.09677). Lecture 3: Based on ideas going back to, J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011), no. 2, 471–564. [arXiv:1005.2846] and some unpublished work. 

20:30 to 22:00  Amihay Hanany (Imperial College, UK) 
Magnetic Quivers and Phase Diagrams  New ways of thinking about moduli spaces of supersymmetric gauge theories (Lecture 1) This lecture series will focus on two new developments which significantly change the way we think and analyze moduli spaces of supersymmetric gauge theories. The magnetic quiver is a tool to solve long standing problems on the dynamics of strongly coupled theories like 4d AD (Argyres Douglas) points, 5d fixed points and 6d tensionless strings. The Hasse diagram neatly encodes phases of the theory with fixed sets of massless states and allows to understand the complexity of given moduli spaces. We will introduce the new concepts, look at simple examples and discuss the exciting ways they change our thinking of moduli spaces. References: Branes, Quivers and the Affine Grassmannian, arXiv:2102.06190 

22:00 to 23:30  Emily Cliff (University of Sydney, Australia) 
Moduli spaces of principal 2group bundles and a categorification of the FreedQuinn line bundle A 2group is a higher categorical analogue of a group, while a smooth 2group is a higher categorical analogue of a Lie group. An important example is the string 2group, defined by SchommerPries. We study the notion of principal bundles for smooth 2groups, and investigate the moduli "space" of such objects. In particular in the case of flat principal bundles for a finite 2group over a Riemann surface, we prove that the moduli space gives a categorification of the FreedQuinn line bundle. This line bundle has as its global sections the state space of ChernSimons theory for the underlying finite group. We can also use our results to better understand the notion of geometric string structures (as previously studied by Waldorf and StolzTeichner). This is joint work with Dan BerwickEvans, Laura Murray, Apurva Nakade, and Emma Phillips. References on 2groups: https://arxiv.org/abs/1407.6849 (Ganter, Usher: Representation and character theory of finite categorical groups) https://arxiv.org/abs/math/0307200 (Baez, Lauda: Higherdimensional algebra V: 2groups) 
Time  Speaker  Title  Resources  

15:30 to 17:00  Joerg Teschner (University of Hamburg and DESY, Germany) 
Analytic geometric Langlandscorrespondence: Relations to conformal field theory and integrable models (Lecture 3) In the first of my three lectures I plan to present a review of the approach of Beilinson and Drinfeld to the geometric Langlands correspondence (GL), based on ideas from conformal field theory and the representation theory of the affine Lie algebras at the critical level. The goal of my second lecture will be to review some aspects of a recent variant of the GL called the analytic Langlands correspondence. The third lecture will discuss a natural deformation of the analytic Langlands correspondence related to the H3WZNW model, and its relation to a generalisation of the AGTcorrespondence. References: Lecture 1: E. Frenkel, Affine algebras, Langlands duality and Bethe ansatz, in Proc. of Int. Congress of Math. Phys. (Paris, 1994), ed. D. Iagolnitzer, pp. 606–642, International Press, 1995 (arXiv:qalg/9506003). E. Frenkel, Lectures on the Langlands program and conformal field theory, in Frontiers in number theory, physics, and geometry. II (P. Cartier, ed.), pp. 387–533. Springer, Berlin, 2007. (arXiv:hepth/0512172). Lecture 2: J. Teschner, Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence, in Geometry and Physics, in honour of Nigel Hitchin, Vol. I, eds. Dancer, e.a., pp. 347–375, Oxford University Press, 2018 (arXiv:1707.07873). P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves, in Integrability, Quantization, and Geometry, dedicated to Boris Dubrovin, Vol. II, eds. S. Novikov, e.a., pp. 137–202, Proc. Symp. Pure Math. 103.2, AMS, 2021 (arXiv:1908.09677). Lecture 3: Based on ideas going back to, J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011), no. 2, 471–564. [arXiv:1005.2846] and some unpublished work. 

20:30 to 22:00  Amihay Hanany (Imperial College, UK) 
Magnetic Quivers and Phase Diagrams  New ways of thinking about moduli spaces of supersymmetric gauge theories (Lecture 2) This lecture series will focus on two new developments which significantly change the way we think and analyze moduli spaces of supersymmetric gauge theories. The magnetic quiver is a tool to solve long standing problems on the dynamics of strongly coupled theories like 4d AD (Argyres Douglas) points, 5d fixed points and 6d tensionless strings. The Hasse diagram neatly encodes phases of the theory with fixed sets of massless states and allows to understand the complexity of given moduli spaces. We will introduce the new concepts, look at simple examples and discuss the exciting ways they change our thinking of moduli spaces. References: Branes, Quivers and the Affine Grassmannian, arXiv:2102.06190 

22:00 to 23:30  Sam Raskin (University of Texas, USA) 
Geometric Langlands and 3d mirror symmetry (Lecture 3) In this series of talks, we will give an introduction to the geometric Langlands correspondence, emphasizing local aspects of the subject. We will describe expected compatibilities for the geometric Langlands conjectures in the language of boundary conditions. Many of these compatibility conjectures have emerged only recently, coming from work of BenZvi, Braverman, Costello, Dimofte, Finkelberg, Gaiotto, Hilburn, Sakellaridis, Venkatesh, Witten, and Yoo (and probably others) on uniting 3d mirror symmetry with geometric Langlands. Finally, we will provide an overview of recent joint work with Justin Hilburn establishing one of these recent conjectures in the abelian case. References: Beilinson Drinfeld  Quantization of Hitchin's integrable system and Hecke eigensheaves link Beilinson Drinfeld  Chiral algebras link Gaitsgory  Outline of the proof of the geometric Langlands conjecture for GL_2 link Gaiotto Witten  Sduality of boundary conditions in N = 4 super Yang–Mills theory link Hilburn Raskin  Tate's thesis in the de Rham setting link 
Time  Speaker  Title  Resources  

11:00 to 12:30  Hiraku Nakajima (Kavli IPMU / NAOJ, Tokyo, Japan) 
Geometric Satake correspondence for affine Lie groups (Lecture 1) Let be a complex reductive group, which is a complexification of a compact Lie group . Let be the affine Grassmannian associated with . It is a partial flag variety associated with the affine Lie group for , and related to moduli spaces of singular monopoles on . Geometric Satake correspondence gives a topological construction of representations of the Langlands dual group via . If we replace by , we cannot naively consider . Nevertheless we can still consider moduli spaces of instantons on the TaubNUT space divided by a finite cyclic group (multiTaubNUT space), and construct integrable representations of the Langlands dual group of . (By technical reasons, a rigorous proof is given only in type A.) References: I will give a pedagogic presentation of materials, sacrificing mathematically rigorous introduction of basic tools, e.g., perverse sheaves, the definition of affine Grassmannian, and a definition of Coulomb branches via equivariant BorelMoore homology. For mathematically oriented participants, I recommend two articles of Mark Andrea de Cataldo and Zinwen Zhu in Geometry of Moduli Spaces and Representation Theory, IAS Park City 24, AMS 2017 for the first two topics, and my expository article https://arxiv.org/abs/1706.05154 for Coulomb branches. An exposition of geometric Satake for affine Lie algebras is available at https://arxiv.org/abs/1812.11710. 

20:30 to 22:00  Amihay Hanany (Imperial College, UK) 
Magnetic Quivers and Phase Diagrams  New ways of thinking about moduli spaces of supersymmetric gauge theories (Lecture 3) This lecture series will focus on two new developments which significantly change the way we think and analyze moduli spaces of supersymmetric gauge theories. The magnetic quiver is a tool to solve long standing problems on the dynamics of strongly coupled theories like 4d AD (Argyres Douglas) points, 5d fixed points and 6d tensionless strings. The Hasse diagram neatly encodes phases of the theory with fixed sets of massless states and allows to understand the complexity of given moduli spaces. We will introduce the new concepts, look at simple examples and discuss the exciting ways they change our thinking of moduli spaces. References: Branes, Quivers and the Affine Grassmannian, arXiv:2102.06190 

22:00 to 23:30  Shlomo Razamat (Technion Israel Institute of Technology, Israel) 
SQCD and pairs of pants We will discuss a geometric reinterpretation of N=1 SQCD with special unitary gauge groups. We will argue that the 4d SU(M) SQCD in the middle of the conformal window can be engineered by compactifying certain 6d SCFTs on three punctured spheres. We will also discuss in this context the interplay between simple geometric and group theoretic considerations and field theoretic strong coupling phenomena. In particular we will show how many known and novel dualities and symmetry emergence phenomena of supersymmetric gauge theories, with simple and semisimple special unitary gauge groups, are related to the Weyl group of D_{6+2K}. We will also comment on the relation of this construction to integrable models. The talk will be based mainly on: https://arxiv.org/pdf/2006.03480.pdf https://arxiv.org/pdf/1906.05088.pdf 
Time  Speaker  Title  Resources  

11:00 to 12:30  Hiraku Nakajima (Kavli IPMU / NAOJ, Tokyo, Japan) 
Geometric Satake correspondence for affine Lie groups (Lecture 2) Let be a complex reductive group, which is a complexification of a compact Lie group . Let be the affine Grassmannian associated with . It is a partial flag variety associated with the affine Lie group for , and related to moduli spaces of singular monopoles on . Geometric Satake correspondence gives a topological construction of representations of the Langlands dual group via . If we replace by , we cannot naively consider . Nevertheless we can still consider moduli spaces of instantons on the TaubNUT space divided by a finite cyclic group (multiTaubNUT space), and construct integrable representations of the Langlands dual group of . (By technical reasons, a rigorous proof is given only in type A.) References: I will give a pedagogic presentation of materials, sacrificing mathematically rigorous introduction of basic tools, e.g., perverse sheaves, the definition of affine Grassmannian, and a definition of Coulomb branches via equivariant BorelMoore homology. For mathematically oriented participants, I recommend two articles of Mark Andrea de Cataldo and Zinwen Zhu in Geometry of Moduli Spaces and Representation Theory, IAS Park City 24, AMS 2017 for the first two topics, and my expository article https://arxiv.org/abs/1706.05154 for Coulomb branches. An exposition of geometric Satake for affine Lie algebras is available at https://arxiv.org/abs/1812.11710. 

20:30 to 22:00  Mina Aganagic (University of California, Berkeley, USA) 
Knot homologies from Mirror Symmetry (Lecture 2) Khovanov showed, more than 20 years ago, that the Jones polynomial emerges as an Euler characteristic of a homology theory. The knot categorification problem is to find a uniform description of this theory, for all gauge groups, which originates from physics, or from geometry. In these lectures, I will describe two solutions to this problem, which originate in string theory, and which are related by a version of homological mirror symmetry. In the first lecture, I will give an overview of the problem, the first of its solutions. The first approach (based on a category of Btype branes on the moduli spaces of singular monopoles on R^3, or Coulomb branches of 3d N=4 quiver gauge theories) was recently proven by Ben Webster to be equivalent to his earlier, purely algebraic approach by KRLW algebras. In the second. lecture, I will describe the second approach (based on a category of Abranes in a LandauGinsburg model) and the (equivariant) homological mirror symmetry that relates it to the first. The second theory is solvable explicitly, in terms of a simpler cousin of KRLW algebras. This leads to a new, geometric and algebraic, understanding of link homologies. In the third lecture, I will describe the string theory origin of these approaches. References: M. Aganagic, ``Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves,’' arXiv:2004.14518 M. Aganagic, ``Knot Categorification from Mirror Symmetry, Part II: Lagrangians,’' arXiv:2105.06039. 

22:00 to 23:30  Alex Weekes (University of Saskatchewan, Canada) 
Coulomb branches for quiver gauge theories with symmetrizers (Lecture 2) The Coulomb branches of 3d N=4 quiver gauge theories are very interesting from the perspective of representation theory because of their close relationship with affine Grassmannians and the Geometric Satake correspondence (see Prof. Nakajima's lectures.) One limitation of this connection is that quiver gauge theories only make sense for symmetric KacMoody types (such as finite ADE), but not symmetrizable types (such as finite BCFG). In this talk I will discuss joint work with Prof. Nakajima, which extends the (mathematical) definition of Coulomb branches to incorporate symmetrizable types. I will also discuss some simple examples and properties. References:

Time  Speaker  Title  Resources  

11:00 to 12:30  Hiraku Nakajima (Kavli IPMU / NAOJ, Tokyo, Japan) 
Geometric Satake correspondence for affine Lie groups (Lecture 3) Let be a complex reductive group, which is a complexification of a compact Lie group . Let be the affine Grassmannian associated with . It is a partial flag variety associated with the affine Lie group for , and related to moduli spaces of singular monopoles on . Geometric Satake correspondence gives a topological construction of representations of the Langlands dual group via . If we replace by , we cannot naively consider . Nevertheless we can still consider moduli spaces of instantons on the TaubNUT space divided by a finite cyclic group (multiTaubNUT space), and construct integrable representations of the Langlands dual group of . (By technical reasons, a rigorous proof is given only in type A.) References: I will give a pedagogic presentation of materials, sacrificing mathematically rigorous introduction of basic tools, e.g., perverse sheaves, the definition of affine Grassmannian, and a definition of Coulomb branches via equivariant BorelMoore homology. For mathematically oriented participants, I recommend two articles of Mark Andrea de Cataldo and Zinwen Zhu in Geometry of Moduli Spaces and Representation Theory, IAS Park City 24, AMS 2017 for the first two topics, and my expository article https://arxiv.org/abs/1706.05154 for Coulomb branches. An exposition of geometric Satake for affine Lie algebras is available at https://arxiv.org/abs/1812.11710. 