###
28/06/2011, 14:00 — 15:00 — Room P3.10, Mathematics Building

Mark Behrens, *MIT*

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Topological Automorphic Forms

#### Topological Automorphic Forms I: definition.

I will review the definition of certain moduli spaces of abelian
varieties (Shimura varieties) which generalize the role that the
moduli space of elliptic curves plays in number theory. Associated
to these Shimura varieties are cohomology theories of Topological
Automorphic Forms (TAF) which generalize the manner in which
Topological Modular Forms are associated to the moduli space of
elliptic curves. These cohomology theories arise as a result of a
theorem of Jacob Lurie.

#### References

- Mark Behrens, Notes
on the construction of TMF (2007).
- Mark Behrens and Tyler Lawson,
Topological Automorphic Forms, Memoirs of the AMS 958
(2010).
- Paul Goerss,
Topological modular forms (after Hopkins, Miller and Lurie),
Séminaire Bourbaki, 2009.
- Mike Hopkins,
Topological modular forms, the Witten genus and the Theorem of the
cube, Proceedings of the 1994 ICM.
- Mike Hopkins,
Algebraic Topology and Modular Forms, Proceedings of the 2002
ICM.
- Tyler Lawson, An overview of
abelian varieties in homotopy theory (2008).

Doug Ravenel's web page for a seminar on topological
automorphic forms contains a comprehensive list of references.

#### See also

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/Behrenstalk2.pdf

###
27/06/2011, 14:00 — 15:00 — Room P3.10, Mathematics Building

Mark Behrens, *MIT*

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Topological Automorphic Forms

#### Modular forms and topology

In this survey talk I will describe how modular forms give
invariants of manifolds, and how these invariants detect elements
of the homotopy groups of spheres. These invariants pass through a
cohomology theory of Topological Modular Forms (TMF). I will review
the role that K-theory plays in detecting periodic families of
elements in the homotopy groups of spheres (the image of the J
homomorphism) in terms of denominators of Bernoulli numbers. I will
then describe how certain higher families of elements (the divided
beta family) are detected by certain congruences between
q-expansions of modular forms.

#### References

- Mark Behrens, Notes
on the construction of TMF (2007).
- Mark Behrens and Tyler Lawson,
Topological Automorphic Forms, Memoirs of the AMS 958
(2010).
- Paul Goerss,
Topological modular forms (after Hopkins, Miller and Lurie),
Séminaire Bourbaki, 2009.
- Mike Hopkins,
Topological modular forms, the Witten genus and the Theorem of the
cube, Proceedings of the 1994 ICM.
- Mike Hopkins,
Algebraic Topology and Modular Forms, Proceedings of the 2002
ICM.
- Tyler Lawson, An overview of
abelian varieties in homotopy theory (2008).

Doug Ravenel's web page for a seminar on topological
automorphic forms contains a comprehensive list of references.

#### See also

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/Behrenstalk1.pdf

###
25/06/2010, 11:00 — 15:45 — Room P3.10, Mathematics Building

Vicente Muñoz, *Universidad Complutense de Madrid*

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Moduli spaces of pairs: Hodge-Deligne polynomials

We shall study moduli spaces of vector bundles over a complex
curve, and moduli spaces of pairs formed by a vector bundle and a
global section. There is a concept of stability for pairs which
depends on a real parameter. These moduli spaces suffer a
birational transformation when the parameter crosses certain
critical values. We shall see how properties of the algebraic
structure of the mentioned moduli spaces (such as Hodge numbers,
Hodge structures, Torelli theorems) can be deduced from a careful
study of these birational transformations.

We give the computation of Hodge-Deligne polynomials of the
moduli spaces of pairs for small rank. This gives in particular the
Poincaré polynomials of the moduli spaces. An analogous procedure
allows to determine the K-theory class of these spaces.

###
24/06/2010, 15:30 — 15:45 — Room P3.10, Mathematics Building

Peter Newstead, *University of Liverpool*

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Coherent systems on algebraic curves: existence results for stable
coherent systems

A coherent system on an an algebraic curve (or Riemann surface)
is a pair consisting of a vector bundle (algebraic or holomorphic)
and a subspace of its space of sections. These are direct
generalisations of the classical linear systems studied since the
mid 19th century. Moduli spaces for coherent systems exist using a
concept of stability dependent on a real parameter. The purpose of
these lectures is to introduce coherent systems and describe the
current state of knowledge.

###
24/06/2010, 14:00 — 15:45 — Room P3.10, Mathematics Building

Vicente Muñoz, *Universidad Complutense de Madrid*

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```###
Moduli spaces of pairs: Hodge structures

We shall study moduli spaces of vector bundles over a complex
curve, and moduli spaces of pairs formed by a vector bundle and a
global section. There is a concept of stability for pairs which
depends on a real parameter. These moduli spaces suffer a
birational transformation when the parameter crosses certain
critical values. We shall see how properties of the algebraic
structure of the mentioned moduli spaces (such as Hodge numbers,
Hodge structures, Torelli theorems) can be deduced from a careful
study of these birational transformations.

We study more specific properties of the algebraic structure of
the moduli spaces, like the (mixed) Hodge structures.

###
24/06/2010, 11:00 — 15:45 — Room P3.10, Mathematics Building

Peter Newstead, *University of Liverpool*

```
```###
Coherent systems on algebraic curves: existence results for stable
coherent systems

A coherent system on an an algebraic curve (or Riemann surface)
is a pair consisting of a vector bundle (algebraic or holomorphic)
and a subspace of its space of sections. These are direct
generalisations of the classical linear systems studied since the
mid 19th century. Moduli spaces for coherent systems exist using a
concept of stability dependent on a real parameter. The purpose of
these lectures is to introduce coherent systems and describe the
current state of knowledge.

###
23/06/2010, 15:30 — 15:45 — Room P3.10, Mathematics Building

Vicente Muñoz, *Universidad Complutense de Madrid*

```
```###
Moduli spaces of pairs: Torelli theorem

We shall study moduli spaces of vector bundles over a complex
curve, and moduli spaces of pairs formed by a vector bundle and a
global section. There is a concept of stability for pairs which
depends on a real parameter. These moduli spaces suffer a
birational transformation when the parameter crosses certain
critical values. We shall see how properties of the algebraic
structure of the mentioned moduli spaces (such as Hodge numbers,
Hodge structures, Torelli theorems) can be deduced from a careful
study of these birational transformations.

We use an inductive argument on the rank involving the moduli
spaces of bundles and the moduli of pairs to get topological and
geometrical properties like: irreducibility, Picard groups, or
"Torelli theorems" which say that the moduli space determines the
curve.

###
23/06/2010, 14:00 — 15:45 — Room P3.10, Mathematics Building

Peter Newstead, *University of Liverpool*

```
```###
Coherent systems on algebraic curves; the classical case

A coherent system on an an algebraic curve (or Riemann surface)
is a pair consisting of a vector bundle (algebraic or holomorphic)
and a subspace of its space of sections. These are direct
generalisations of the classical linear systems studied since the
mid 19th century. Moduli spaces for coherent systems exist using a
concept of stability dependent on a real parameter. The purpose of
these lectures is to introduce coherent systems and describe the
current state of knowledge.

###
23/06/2010, 11:00 — 15:45 — Room P3.10, Mathematics Building

Vicente Muñoz, *Universidad Complutense de Madrid*

```
```###
Moduli spaces of pairs: moduli spaces of pairs and of bundles

We shall study moduli spaces of vector bundles over a complex
curve, and moduli spaces of pairs formed by a vector bundle and a
global section. There is a concept of stability for pairs which
depends on a real parameter. These moduli spaces suffer a
birational transformation when the parameter crosses certain
critical values. We shall see how properties of the algebraic
structure of the mentioned moduli spaces (such as Hodge numbers,
Hodge structures, Torelli theorems) can be deduced from a careful
study of these birational transformations.

We shall describe the moduli spaces of pairs, its relationship
with the moduli of bundles, with emphasis on the role of the
stability parameter and the birational transformations which happen
when varying it, which are called flips.

###
22/06/2010, 11:00 — 15:45 — Room P3.10, Mathematics Building

Peter Newstead, *University of Liverpool*

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```###
Coherent systems on algebraic curves: moduli spaces of vector
bundles and of coherent systems; Brill-Noether loci; basic
properties.

A coherent system on an an algebraic curve (or Riemann surface)
is a pair consisting of a vector bundle (algebraic or holomorphic)
and a subspace of its space of sections. These are direct
generalisations of the classical linear systems studied since the
mid 19th century. Moduli spaces for coherent systems exist using a
concept of stability dependent on a real parameter. The purpose of
these lectures is to introduce coherent systems and describe the
current state of knowledge.

###
02/07/2009, 14:00 — 15:00 — Room P3.10, Mathematics Building

Kai Behrend, *University of British Columbia*

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Moduli Spaces via differential graded Lie algebras

I will explain how many interesting moduli spaces in algebraic
geometry can be constructed as the solution set of the
Maurer-Cartan equation in a differential graded Lie algebra, modulo
the action of the gauge group. The advantage of this approach is
that it gives directly the higher derived structure on the moduli
space in question. We will focus on the case of sheaves on
projective varieties. We will examine the case of the Hilbert
scheme of points on a Calabi-Yau threefold in particular
detail.

#### Referências

- Deformation
theory via differential graded Lie algebras - Marco
Manetti
- Lectures on
deformations of complex manifolds - Marco Manetti
- Injective
resolutions of BG and derived moduli spaces of local systems -
M. Kapranov
- A functorial
construction of moduli of sheaves - Luis Álvarez-Cónsul,
Alastair King

###
02/07/2009, 11:00 — 12:00 — Room P3.10, Mathematics Building

Dietmar Salamon, *ETH Zurich*

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```###
Floer Homology

Floer homology groups in hyperkaehler geometry

###
01/07/2009, 14:00 — 15:00 — Room P3.10, Mathematics Building

Kai Behrend, *University of British Columbia*

```
```###
Moduli Spaces via differential graded Lie algebras

I will explain how many interesting moduli spaces in algebraic
geometry can be constructed as the solution set of the
Maurer-Cartan equation in a differential graded Lie algebra, modulo
the action of the gauge group. The advantage of this approach is
that it gives directly the higher derived structure on the moduli
space in question. We will focus on the case of sheaves on
projective varieties. We will examine the case of the Hilbert
scheme of points on a Calabi-Yau threefold in particular
detail.

#### Referências

- Deformation
theory via differential graded Lie algebras - Marco
Manetti
- Lectures on
deformations of complex manifolds - Marco Manetti
- Injective
resolutions of BG and derived moduli spaces of local systems -
M. Kapranov
- A functorial
construction of moduli of sheaves - Luis Álvarez-Cónsul,
Alastair King

###
01/07/2009, 11:00 — 12:00 — Room P3.10, Mathematics Building

Dietmar Salamon, *ETH Zurich*

```
```###
Floer Homology

The Atiyah-Floer conjecture and 3-manifolds with boundary

###
30/06/2009, 11:00 — 12:00 — Room P3.10, Mathematics Building

Dietmar Salamon, *ETH Zurich*

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```###
Floer Homology

Floer homology groups in symplectic topology II

###
29/06/2009, 14:00 — 15:00 — Room P3.10, Mathematics Building

Kai Behrend, *University of British Columbia*

```
```###
Moduli Spaces via differential graded Lie algebras

I will explain how many interesting moduli spaces in algebraic
geometry can be constructed as the solution set of the
Maurer-Cartan equation in a differential graded Lie algebra, modulo
the action of the gauge group. The advantage of this approach is
that it gives directly the higher derived structure on the moduli
space in question. We will focus on the case of sheaves on
projective varieties. We will examine the case of the Hilbert
scheme of points on a Calabi-Yau threefold in particular
detail.

#### Referências

- Deformation
theory via differential graded Lie algebras - Marco
Manetti
- Lectures on
deformations of complex manifolds - Marco Manetti
- Injective
resolutions of BG and derived moduli spaces of local systems -
M. Kapranov
- A functorial
construction of moduli of sheaves - Luis Álvarez-Cónsul,
Alastair King

###
29/06/2009, 11:00 — 12:00 — Room P3.10, Mathematics Building

Dietmar Salamon, *ETH Zurich*

```
```###
Floer Homology

Floer homology groups in symplectic topology I

###
12/09/2008, 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 - 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 — Amphitheatre Pa1, Mathematics Building

Anton Kapustin, *California Institute of Technology*

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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 — Amphitheatre Pa1, Mathematics Building

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.