If one would like to study representation theory of groupoids, some examples would be useful. We shall discuss a method to construct such examples generalizing the well-known procedure of geometric quantization. Starting point of this procedure is a coadjoint orbit. For groupoids such a coadjoint orbit has the structure of a symplectic fibration. A central role in the talk will be played by a new notion of momentum map, suitable for groupoid actions on symplectic fibrations. If one would not like to study representation theory of groupoids, one might still be interested symplectic fibrations an sich. Constructing a coupling form for symplectic fibrations, a momentum map of the above kind comes for free.

Wonderful varieties are algebraic varieties, equipped with an action of a linear semisimple group $G$ and satisfying axioms inspired by the well known compactifications of symmetric spaces of De Concini and Procesi. They are a sort of generalization of complete $G$-homogenous spaces (i.e. partial flag varieties, if $G$ is the special linear group $SL(n)$). In the talk we will discuss some of their basic properties, and many examples. In particular, we will focus on recent results about their full automorphism groups, generalizing the results of Demazure (1974) about automorphism groups of complete $G$-homogeneous spaces.

We present a geometric method in Calculus of Variations that, in the presence of a constraint given by partial differential equations, allows to determine, for any Lagrangian density, a 'constrained' Cartan form which satisfies an analogous of the classical Lepage congruences. The problematic of defining a second variation from the Lagrangian density can be solved with this new object, from which we derive Euler-Lagrange and Cartan equations for constrained critical sections, a Hessian bilinear form and a notion of Jacobi vector fields. We compare our results with those arising in the classical Lagrange multiplier approach, showing that certain special care should be taken when working with this classical framework. An example of immersions of surfaces in the euclidean space with prescribed area element illustrates the theory.

Every complex projective algebraic surface $S$ satisfies the inequality \[9\chi({\cal O}_S)\geq c_1^2\geq 2\chi({\cal O}_S)-6.\] This talk will focus on results (recent and less recent) about the algebraic fundamental group of surfaces of general type with $c_1^2$ “small” with respect to $\chi({\cal O}_S)$. In particular some results obtained in colaboration with R. Pardini and C. Ciliberto will be discussed.

We will present canonical open books decomposition carrying large degree Boothby-Wang contact forms. These are obtained using holomorphic sections.

We study the quotients for the diagonal action of $\operatorname{SL}_3(C)$ on the product of $n$-fold of $\mathbb{P}^2(C)$: we are interested in describing how the quotient changes when we vary the polarization (i.e. the choice of an ample linearized line bundle). We illustrate the different techniques for the construction of a quotient, in particular the numerical criterion for semi-stability and the elementary transformations which are resolutions of precisely described singularities (case $n=6$).

Harmonic morphisms are mappings of Riemannian manifolds which preserve solutions of Laplace's equation; elementary examples are conformal transformations of the complex plane. The concept can be traced back to work of Jacobi in 1848. For probabilists working in stochastic processes, harmonic morphisms are Brownian path-preserving transformations. Harmonic morphisms can be characterized as harmonic maps which satisfy an additional condition which is dual to the condition of weak conformality.

We shall give an overview of the subject, mentioning especially twistorial constructions, and finishing with some recent work to understand these using the bicomplex numbers.

I will explain how to view the construction of moduli spaces of semistable sheaves on a projective variety as a functorial embedding into more basic projective varieties, namely moduli spaces of semistable Kronecker modules. This sheds new light on how to construct 'theta functions', i.e. natural projective coordinates on these moduli spaces. This is joint work with Alastair King (Invent. Math. 168 (2007) 613-666).

The study of Higgs pairs has had many unexpected applications. One of the most spectacular, has been the understanding of the fundamental group from the algebraic geometric point of view, and identifying an action of $C^*$, which mimics the Hodge structure on cohomology. The lecture will seek to explain the construction of this new gadget and give one or two applications

It is well known that analytic invariants totally determine the topology of complex curves. Hodge theory gives a process to obtain topological information on Kähler manifolds from the the spaces of holomorphic differential forms. We in this talk analyse if there are topological implications to be derived from the spaces of symmetric differentials on complex surfaces. The dimensions of the spaces of symmetric differentials do not encode topological information, but we will show that there is a special type of symmetric differentials, which we call closed, which reflect topological properties.

Morse theory is a geometric way to understand the homology of manifolds. Orbifolds are spaces that locally look like the quotient of a manifold by a finite group. I will explain how Morse theory generalizes to orbifolds, giving methods to compute several different notions of "the homology of an orbifold", including the orbifold cohomology of Chen and Ruan.

We consider the set $\Sigma_0$ of smooth connected closed hypersufaces of contact type in $T^*N$ whose intersection with each fiber is either empty or bounds a compact region that contains the zero point. We show that if $N$ has dimension bigger than one and $H_1(N) = 0$ then there exists a subset $\widehat\Sigma_0 \subset \Sigma_0$ of symplectically non-convex hypersurfaces which is $C^1$-dense and open in the Hausdorff topology. The proof involves the Maslov index of periodic orbits and symplectic homology.

I will discuss the reduction procedure for Courant algebroids and Dirac/generalized complex structures developed in joint work with Cavalcanti and Gualtieri. After illustrating this construction, I will show how it has a natural interpretation in super-geometric terms: viewing Courant algebroids as symplectic graded manifolds, the reduction procedure can be seen as a symplectic (Marsden-Weinstein) quotient.

Godeaux surfaces are surfaces of general type with the lowest numerical examples possible. Since the first example was given by Lucien Godeaux in 1930 there is an extensive search for more such surfaces and for a classification. In this talk we consider Godeaux surfaces over fields of positive characteristic with non-vanishing $h^1$, i.e., non-reduced Picard scheme. The existence of these surfaces was already observed by Rick Miranda in 1984. We prove that such surfaces can exist over fields of characteristic at most $5$ and give a complete classification in characteristic $5$.

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.

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.

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.

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.

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.

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.

We introduce the theory of $G$-Higgs bundles over a compact Riemann surface $X$ for any real semisimple Lie group $G$ and show their relation to the representations of the fundamental group of $X$ in $G$. We then focus on the case in which $G$ is the isometry group of a non-compact Hermitian symmetric space.

In this talk we show how the theory of Higgs bundles can be applied to the study of the following geometric problem: When can a maximal $Sp(4,R)$-representation of a surface group be deformed to a representation which factors through a proper reductive subgroup of $Sp(4,R)$?

A very long-standing problem in Algebraic Geometry is to determine the stability of exceptional bundles on smooth projective varieties. We first show that if $X\subset P^n$, $n\geq 3$, is a smooth and irreducible hypersurface of degree $1\leq d\leq n-1$ and $E$ is an exceptional bundle on $X$ given by \[ 0 \rightarrow \mathcal{O}_{X}(-1)^{r} \rightarrow \mathcal{O}_{X}^{s} \rightarrow E \rightarrow 0, \] for some $s,t\geq 1$, then $E$ is stable. We then prove that any exceptional bundle on a smooth complete intersection $3$-fold $Y\subset P^n$ of type $(d_1,\ldots,d_{n-3})$ with $d_1+\cdots+ d_{n-3}\leq n$ and $n\geq 4$, is stable. (joint work with Rosa M. Miró-Roig)

Considering toric deformations of complex structures, described by the symplectic potentials of Abreu and Guillemin, we study degenerate limits of holomorphic polarizations corresponding to the toric Lagrangian fibration. We use these toric metric degenerations to study the (Gromov-Hausdorff) limit of hypersurfaces and show that in Legendre transformed variables they are described by compact pieces of tropical amoebas in the sense of Gelfand-Kapranov-Zelevinsky and Mikhalkin. This talk will be based on the preprint arXiv:0806.0606 (joint work with C. Florentino, J. M. Mourão and J. P. Nunes).

Let a torus $T$ act on a compact symplectic manifold $M$ in a Hamiltonian way, with discrete fixed point set $M^T$ . Then the moment map can be used to construct a basis for the $T$-equivariant cohomology of $M$ (as a module over the equivariant cohomology of a point). In recent work, R. Goldin and S. Tolman used the symplectic structure of $M$ to compute the restriction of the elements of this basis to the fixed points of the action, which gives all the information about the equivariant cohomology ring of $M$, since in this case the restriction map to the equivariant cohomology of the fixed point set is injective. I will first explain recent results obtained jointly with S. Tolman on how it is possible to simplify their formula using special equivariant symplectic fibrations, and get in some nice cases positive integral formulas. Then I’ll focus on generic coadjoint orbits, where positive integral formulas are already known, and explain how to use “GKM fiber bundles” to get positive integral formulas, which are not equivalent to the ones found previously.

In this lecture, we will talk about Hamiltonian spaces associated with pairs $(E,A)$, where $E$ is a Courant algebroid and $A\subset E$ is a Dirac structure. These spaces are defined in terms of morphisms of Courant algebroids with suitable compatibility conditions. Several of their properties will be discussed, including a reduction procedure. This set-up encompasses familiar moment map theories, such as group-valued moment maps, and it provides an intrinsic approach from which different geometrical descriptions of moment maps can be naturally derived.

In the last few years the definition of overtwisted structure has been generalized in contact geometry as a tool to bulid non-fillable contact structures in any dimension. We introduce a generalization of this definition, GPS structures, that still captures the non-fillability properties and has some unexpected connections with non-squeezing results in contact geometry. In particular, we will prove some non-squeezing results of embeddings of overtiwted 3-folds.

Complex algebraic surfaces of general type with $p_g=q=1$ are still not completely understood. Until recently only a few examples were known. In this talk I will give some results about surfaces $S$ with $p_g=q=1$ having an involution (automorphism of order $2$).

If the bicanonical map $\phi_2$ of $S$ is of degree 2 onto a non-ruled surface, then $\phi_2(S)$ is birational to a $K3$ surface. We will use the Computational Algebra System Magma to construct such a surface with $K_S^2=6,$ as a double cover of a Kummer quartic surface (quartic in $P^3$ with 16 ordinary double points).

We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group ${\bf U}(n)$ in any dimension, and so all harmonic maps from the $2$-sphere, in terms of freely chosen meromorphic functions on the surface and their derivatives, using only combinations of projections and avoiding the usual $\overline{\partial}$-problems or loop group factorizations.

A general theme is Quantum Mechanics is "What is the asymptotics of the so-called quantum observables?". An analogue of this question in Riemannian geometry is "What is the asymptotics of the eigenfunctions of the Laplacian?". A recent result of Burns, Guillemin and Uribe gives a method to approach the first question on toric varieties endowed with a special metric, the so called reduced metric . In this talk we will explain their result and how it generalizes to all toric metrics. We will also speak about an unexpected connection between this result and an important problem in Kähler geometry namely that of approximating Kähler metrics by algebraic ones.