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11/09/2008, 11:00 — 12:00 — Amphitheatre Pa1, Mathematics Building
, 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 — Amphitheatre Pa1, Mathematics Building
, 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 — Amphitheatre Pa1, Mathematics Building
, 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 — Room P3.10, Mathematics Building
, 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 — Room P3.10, Mathematics Building
, 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 — Room P3.10, Mathematics Building
, 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 — Room P3.10, Mathematics Building
, 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 — Room P3.10, Mathematics Building
, 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 — Room P8, Mathematics Building, IST
, University of California at Santa Cruz, USA

Equivariant Morse Theory, Old and New IV

28/06/2006, 13:30 — 14:30 — Room P8, Mathematics Building, IST
, University of California at Santa Cruz, USA

Equivariant Morse Theory, Old and New III

27/06/2006, 14:30 — 15:30 — Room P8, Mathematics Building, IST
, University of California at Santa Cruz, USA

Equivariant Morse Theory, Old and New II

26/06/2006, 14:30 — 15:30 — Room P8, Mathematics Building, IST
, University of California at Santa Cruz, USA

Equivariant Morse Theory, Old and New I

17/06/2005, 12:00 — 13:00 — Room P12, Mathematics Building
, 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 — Room P12, Mathematics Building
, University of California at Santa Cruz

Measures on Banach manifolds and supersymmetric quantum field theory - V

17/06/2005, 09:30 — 10:30 — Room P12, Mathematics Building
, University of California at Santa Cruz

Measures on Banach manifolds and supersymmetric quantum field theory - IV

16/06/2005, 12:00 — 13:00 — Room P12, Mathematics Building
, University of California at Santa Cruz

Measures on Banach manifolds and supersymmetric quantum field theory - III

16/06/2005, 11:00 — 12:00 — Room P12, Mathematics Building
, University of Maryland

Projective geometry on manifolds - IV

Poisson geometry on Fricke spaces

  • Symplectic geometry
  • Fricke's theorem on rank two free groups
  • Fenchel-Nielsen coordinates and their generalizations
  • Action of the mapping class group
  • Penner-Fock coordinates

16/06/2005, 09:30 — 10:30 — Room P12, Mathematics Building
, University of Maryland

Projective Geometry on Manifolds - III

Deformation spaces: geometries on the space of geometries

  • Markings
  • Definition of the deformation space
  • The Ehresmann-Thurston holonomy theorem
  • Representation spaces

15/06/2005, 15:30 — 16:30 — Room P12, Mathematics Building
, University of California at Santa Cruz

Measures on Banach manifolds and supersymmetric quantum field theory - II

15/06/2005, 14:00 — 15:00 — Room P12, Mathematics Building
, University of California at Santa Cruz

Measures on Banach manifolds and supersymmetric quantum field theory - I

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