29/06/2009, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Dietmar Salamon, ETH Zurich
Floer Homology
Floer homology groups in symplectic topology I
12/09/2008, 16:00 — 17:00 — Anfiteatro Pa1, Pavilhão de Matemática
James Sparks, Mathematical Institute, Oxford
Sasaki-Einstein Geometry and the AdS/CFT Correspondence - III
The aim of these lectures will be to describe some recent developments in Sasaki-Einstein geometry, and also to explain, in a way that is hopefully accessible to geometers, how these results are related to the AdS/CFT correspondence in string theory. I will begin with a general introduction to Sasakian geometry, which is the odd-dimensional cousin of Kahler geometry. I will then introduce Sasaki-Einstein geometry, and describe a number of different constructions of Sasaki-Einstein manifolds. In particular, I will develop the theory of toric Sasakian manifolds, culminating with the recent theorem of Futaki-Ono-Wang on the existence of toric Sasaki-Einstein metrics. Next I will describe a number of different obstructions to the existence of Sasaki-Einstein metrics, together with some simple examples. Finally, I will outline how Sasaki-Einstein manifolds arise as solutions to supergravity, and describe their role in the AdS/CFT correspondence. The latter conjectures that for each Sasaki-Einstein \(5\)-manifold there exists a corresponding conformal field theory on \(\mathbb{R}^4\) This map is only really understood in certain examples, and for concreteness I will focus mainly on the toric case. The conformal field theory is then (conjecturally) described by a gauge theory on \(\mathbb{R}^4\) that is determined from the algebraic geometry of the cone over the Sasaki-Einstein manifold. Mathematically, this data is encoded by a bipartite graph on a two-torus. I will conclude by explaining how AdS/CFT relates some of the properties of Sasaki-Einstein manifolds described earlier to this combinatorial structure.
References
The article arXiv:math/0701518 [math.DG] reviews much of the material that I will cover, and also contains references to the original literature.
12/09/2008, 14:00 — 15:00 — Anfiteatro Pa1, Pavilhão de Matemática
Anton Kapustin, California Institute of Technology
Gauge Theory and the Geometric Langlands Program - III
Recently, it has been shown that Geometric Langlands duality is closely related to properties of quantized supersymmetric Yang-Mills theory in four dimensions, as well as to Mirror Symmetry and Topological Field Theory. I will provide an introduction to these developments.
Literature
- A. Kapustin and E. Witten, Electric-Magnetic Duality and the Geometric Langlands Program, hep-th/0604151
- S.Gukov and E. Witten, Gauge Theory, Ramification, and the Geometric Langlands Program, hep-th/0612173
- A. Kapustin, Holomorphic reduction of \(N=2\) gauge theories, Wilson-'t Hooft operators, and S-duality, hep-th/0612119.
For some background, the following papers are very useful:
- E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353.
- E. Witten, Mirror manifolds and Topological Field Theory, hep-th/9112056.
11/09/2008, 15:00 — 16:00 — Anfiteatro Pa1, Pavilhão de Matemática
Anton Kapustin, California Institute of Technology
Gauge Theory and the Geometric Langlands Program - II
Recently, it has been shown that Geometric Langlands duality is closely related to properties of quantized supersymmetric Yang-Mills theory in four dimensions, as well as to Mirror Symmetry and Topological Field Theory. I will provide an introduction to these developments.
Literature
- A. Kapustin and E. Witten, Electric-Magnetic Duality and the Geometric Langlands Program, hep-th/0604151
- S.Gukov and E. Witten, Gauge Theory, Ramification, and the Geometric Langlands Program, hep-th/0612173
- A. Kapustin, Holomorphic reduction of \(N=2\) gauge theories, Wilson-'t Hooft operators, and S-duality, hep-th/0612119.
For some background, the following papers are very useful:
- E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353.
- E. Witten, Mirror manifolds and Topological Field Theory, hep-th/9112056.
11/09/2008, 11:00 — 12:00 — Anfiteatro Pa1, Pavilhão de Matemática
James Sparks, Mathematical Institute, Oxford
Sasaki-Einstein Geometry and the AdS/CFT Correspondence - II
The aim of these lectures will be to describe some recent developments in Sasaki-Einstein geometry, and also to explain, in a way that is hopefully accessible to geometers, how these results are related to the AdS/CFT correspondence in string theory. I will begin with a general introduction to Sasakian geometry, which is the odd-dimensional cousin of Kahler geometry. I will then introduce Sasaki-Einstein geometry, and describe a number of different constructions of Sasaki-Einstein manifolds. In particular, I will develop the theory of toric Sasakian manifolds, culminating with the recent theorem of Futaki-Ono-Wang on the existence of toric Sasaki-Einstein metrics. Next I will describe a number of different obstructions to the existence of Sasaki-Einstein metrics, together with some simple examples. Finally, I will outline how Sasaki-Einstein manifolds arise as solutions to supergravity, and describe their role in the AdS/CFT correspondence. The latter conjectures that for each Sasaki-Einstein \(5\)-manifold there exists a corresponding conformal field theory on \(\mathbb{R}^4\) This map is only really understood in certain examples, and for concreteness I will focus mainly on the toric case. The conformal field theory is then (conjecturally) described by a gauge theory on \(\mathbb{R}^4\) that is determined from the algebraic geometry of the cone over the Sasaki-Einstein manifold. Mathematically, this data is encoded by a bipartite graph on a two-torus. I will conclude by explaining how AdS/CFT relates some of the properties of Sasaki-Einstein manifolds described earlier to this combinatorial structure.
References
The article arXiv:math/0701518 [math.DG] reviews much of the material that I will cover, and also contains references to the original literature.
09/09/2008, 16:00 — 17:00 — Anfiteatro Pa1, Pavilhão de Matemática
James Sparks, Mathematical Institute, Oxford
Sasaki-Einstein Geometry and the AdS/CFT Correspondence - I
The aim of these lectures will be to describe some recent developments in Sasaki-Einstein geometry, and also to explain, in a way that is hopefully accessible to geometers, how these results are related to the AdS/CFT correspondence in string theory. I will begin with a general introduction to Sasakian geometry, which is the odd-dimensional cousin of Kahler geometry. I will then introduce Sasaki-Einstein geometry, and describe a number of different constructions of Sasaki-Einstein manifolds. In particular, I will develop the theory of toric Sasakian manifolds, culminating with the recent theorem of Futaki-Ono-Wang on the existence of toric Sasaki-Einstein metrics. Next I will describe a number of different obstructions to the existence of Sasaki-Einstein metrics, together with some simple examples. Finally, I will outline how Sasaki-Einstein manifolds arise as solutions to supergravity, and describe their role in the AdS/CFT correspondence. The latter conjectures that for each Sasaki-Einstein \(5\)-manifold there exists a corresponding conformal field theory on \(\mathbb{R}^4\) This map is only really understood in certain examples, and for concreteness I will focus mainly on the toric case. The conformal field theory is then (conjecturally) described by a gauge theory on \(\mathbb{R}^4\) that is determined from the algebraic geometry of the cone over the Sasaki-Einstein manifold. Mathematically, this data is encoded by a bipartite graph on a two-torus. I will conclude by explaining how AdS/CFT relates some of the properties of Sasaki-Einstein manifolds described earlier to this combinatorial structure.
References
The article arXiv:math/0701518 [math.DG] reviews much of the material that I will cover, and also contains references to the original literature.
09/09/2008, 11:00 — 12:00 — Anfiteatro Pa1, Pavilhão de Matemática
Anton Kapustin, California Institute of Technology
Gauge Theory and the Geometric Langlands Program - I
Recently, it has been shown that Geometric Langlands duality is closely related to properties of quantized supersymmetric Yang-Mills theory in four dimensions, as well as to Mirror Symmetry and Topological Field Theory. I will provide an introduction to these developments.
Literature
- A. Kapustin and E. Witten, Electric-Magnetic Duality and the Geometric Langlands Program, hep-th/0604151
- S.Gukov and E. Witten, Gauge Theory, Ramification, and the Geometric Langlands Program, hep-th/0612173
- A. Kapustin, Holomorphic reduction of \(N=2\) gauge theories, Wilson-'t Hooft operators, and S-duality, hep-th/0612119.
For some background, the following papers are very useful:
- E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353.
- E. Witten, Mirror manifolds and Topological Field Theory, hep-th/9112056.
14/06/2007, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Hansjörg Geiges, Mathematical Institute of the University of Cologne, Germany
Contact Manifolds — Classification results and applications to Geometric Topology (V)
Constructions of higher-dimensional contact manifolds
14/06/2007, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Hansjörg Geiges, Mathematical Institute of the University of Cologne, Germany
Contact Manifolds — Classification results and applications to Geometric Topology (IV)
Symplectic fillings and property \(\mathcal{P}\) for knots
12/06/2007, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Hansjörg Geiges, Mathematical Institute of the University of Cologne, Germany
Contact Manifolds — Classification results and applications to Geometric Topology (III)
Contact Dehn surgery
12/06/2007, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Hansjörg Geiges, Mathematical Institute of the University of Cologne, Germany
Contact Manifolds — Classification results and applications to Geometric Topology (II)
Tight contact structures and Cerf's Theorem
11/06/2007, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Hansjörg Geiges, Mathematical Institute of the University of Cologne, Germany
Contact Manifolds — Classification results and applications to Geometric Topology (I)
Legendrian knots and the Whitney-Graustein Theorem
The first lecture will be a colloquium-style talk. I shall give a very gentle introduction to some basic concepts of contact geometry, notably concerning knots in contact 3-manifolds. This will be used to give a contact geometric proof of the Whitney-Graustein theorem in planar geometry: immersions of the circle in the 2-plane are classified, up to regular homotopy, by their rotation number.
28/06/2006, 15:00 — 16:00 — Sala P8, Pavilhão de Matemática, IST
Jonathan Weitsman, University of California at Santa Cruz, USA
Equivariant Morse Theory, Old and New (IV)
28/06/2006, 13:30 — 14:30 — Sala P8, Pavilhão de Matemática, IST
Jonathan Weitsman, University of California at Santa Cruz, USA
Equivariant Morse Theory, Old and New (III)
27/06/2006, 14:30 — 15:30 — Sala P8, Pavilhão de Matemática, IST
Jonathan Weitsman, University of California at Santa Cruz, USA
Equivariant Morse Theory, Old and New (II)
26/06/2006, 14:30 — 15:30 — Sala P8, Pavilhão de Matemática, IST
Jonathan Weitsman, University of California at Santa Cruz, USA
Equivariant Morse Theory, Old and New (I)
17/06/2005, 12:00 — 13:00 — Sala P12, Pavilhão de Matemática
William Goldman, University of Maryland
Projective geometry on manifolds (V)
\(RP2\)-structures on surfaces
- Coxeter groups.
- Convex projective structures; Hitchin's conjecture.
- Choi's convex decomposition theorem.
- Regularity of the boundary, bulging deformations.
- Survey of recent progress: higher dimensions and other geometries.
17/06/2005, 11:00 — 12:00 — Sala P12, Pavilhão de Matemática
Jonathan Weitsman, University of California at Santa Cruz
Measures on Banach manifolds and supersymmetric quantum field theory (V)
17/06/2005, 09:30 — 10:30 — Sala P12, Pavilhão de Matemática
Jonathan Weitsman, University of California at Santa Cruz
Measures on Banach manifolds and supersymmetric quantum field theory (IV)
16/06/2005, 12:00 — 13:00 — Sala P12, Pavilhão de Matemática
Jonathan Weitsman, University of California at Santa Cruz
Measures on Banach manifolds and supersymmetric quantum field theory (III)