###
11/09/2008, 11:00 — 12:00 — Amphitheatre Pa1, Mathematics Building

James Sparks, *Mathematical Institute, Oxford*

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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

James Sparks, *Mathematical Institute, Oxford*

```
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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

Anton Kapustin, *California Institute of Technology*

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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

Hansjörg Geiges, *Mathematical Institute of the University of Cologne, Germany*

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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

Hansjörg Geiges, *Mathematical Institute of the University of Cologne, Germany*

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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

Hansjörg Geiges, *Mathematical Institute of the University of Cologne, Germany*

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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

Hansjörg Geiges, *Mathematical Institute of the University of Cologne, Germany*

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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

Hansjörg Geiges, *Mathematical Institute of the University of Cologne, Germany*

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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

Jonathan Weitsman, *University of California at Santa Cruz, USA*

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Equivariant Morse Theory, Old and New IV

###
28/06/2006, 13:30 — 14:30 — Room P8, Mathematics Building, IST

Jonathan Weitsman, *University of California at Santa Cruz, USA*

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Equivariant Morse Theory, Old and New III

###
27/06/2006, 14:30 — 15:30 — Room P8, Mathematics Building, IST

Jonathan Weitsman, *University of California at Santa Cruz, USA*

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Equivariant Morse Theory, Old and New II

###
26/06/2006, 14:30 — 15:30 — Room P8, Mathematics Building, IST

Jonathan Weitsman, *University of California at Santa Cruz, USA*

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Equivariant Morse Theory, Old and New I

###
17/06/2005, 12:00 — 13:00 — Room P12, Mathematics Building

William Goldman, *University of Maryland*

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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

Jonathan Weitsman, *University of California at Santa Cruz*

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Measures on Banach manifolds and supersymmetric quantum field
theory - V

###
17/06/2005, 09:30 — 10:30 — Room P12, Mathematics Building

Jonathan Weitsman, *University of California at Santa Cruz*

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Measures on Banach manifolds and supersymmetric quantum field
theory - IV

###
16/06/2005, 12:00 — 13:00 — Room P12, Mathematics Building

Jonathan Weitsman, *University of California at Santa Cruz*

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Measures on Banach manifolds and supersymmetric quantum field
theory - III

###
16/06/2005, 11:00 — 12:00 — Room P12, Mathematics Building

William Goldman, *University of Maryland*

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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

William Goldman, *University of Maryland*

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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

Jonathan Weitsman, *University of California at Santa Cruz*

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Measures on Banach manifolds and supersymmetric quantum field
theory - II

###
15/06/2005, 14:00 — 15:00 — Room P12, Mathematics Building

Jonathan Weitsman, *University of California at Santa Cruz*

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Measures on Banach manifolds and supersymmetric quantum field
theory - I