Topological string theories in dimension less than one are known to be governed by integrable hierarchies. The most famous example is intersection theory on the moduli space of Riemann surfaces, which according to Witten and Kontsevich is governed by the KdV hierarchy. In this talk I will present a general framework to understand the integrability of topological strings, as well as some new results along this direction for topological strings on noncompact Calabi-Yau threefolds.

Let M be a symplectic manifold, and let G be a connected compact rank 1 Lie group acting on M in a Hamiltonian fashion. We show that as fundamental groups of topological spaces, the fundamental group of the symplectic reduction of M is the same as the one of M.

Using ternary trees we build operads and use them to define a family of ideals in the (non-commutative) algebra generated by pointed trees. These constructions have several applications to iterations of polynomial maps and conjectures in algebraic geometry.

The purpose of this course is to introduce a technique that has proven useful, especially in the theory of algebraic surfaces, to construct interesting varieties and families of varieties and to test general conjectures.

CONTENTS:

Double covers. Finite group actions on complex manifolds, quotient singularities. Structure theorem for abelian covers. Natural deformations of abelian covers. Applications and examples.

Etale topology, functor of points of a scheme, algebraic spaces; moduli problems, fine and coarse moduli spaces. Groupoids, groupoids in a category, groupoid fibrations, stacks.

Definition of algebraic stack. Examples: quotient stacks, moduli stacks. Morphisms to an algebraic stack, algebraic stacks as a $2$-category; fibered products. Representable morphisms, charts. (Quasi)coherent sheaves on a stack. properties of stacks and moprphisms (smoothness, separatedness, properness, etc).

More examples: modular forms, moduli spaces of stable curves and maps, Witten and Gromov-Witten invariants. The special case of orbifolds.

The purpose of this course is to introduce a technique that has proven useful, especially in the theory of algebraic surfaces, to construct interesting varieties and families of varieties and to test general conjectures.

CONTENTS:

Double covers. Finite group actions on complex manifolds, quotient singularities. Structure theorem for abelian covers. Natural deformations of abelian covers. Applications and examples.

Etale topology, functor of points of a scheme, algebraic spaces; moduli problems, fine and coarse moduli spaces. Groupoids, groupoids in a category, groupoid fibrations, stacks.

Definition of algebraic stack. Examples: quotient stacks, moduli stacks. Morphisms to an algebraic stack, algebraic stacks as a $2$-category; fibered products. Representable morphisms, charts. (Quasi)coherent sheaves on a stack. properties of stacks and moprphisms (smoothness, separatedness, properness, etc).

More examples: modular forms, moduli spaces of stable curves and maps, Witten and Gromov-Witten invariants. The special case of orbifolds.

The purpose of this course is to introduce a technique that has proven useful, especially in the theory of algebraic surfaces, to construct interesting varieties and families of varieties and to test general conjectures.

CONTENTS:

Double covers. Finite group actions on complex manifolds, quotient singularities. Structure theorem for abelian covers. Natural deformations of abelian covers. Applications and examples.

Etale topology, functor of points of a scheme, algebraic spaces; moduli problems, fine and coarse moduli spaces. Groupoids, groupoids in a category, groupoid fibrations, stacks.

Definition of algebraic stack. Examples: quotient stacks, moduli stacks. Morphisms to an algebraic stack, algebraic stacks as a $2$-category; fibered products. Representable morphisms, charts. (Quasi)coherent sheaves on a stack. properties of stacks and moprphisms (smoothness, separatedness, properness, etc).

More examples: modular forms, moduli spaces of stable curves and maps, Witten and Gromov-Witten invariants. The special case of orbifolds.

We will describe the coherent state transform for Lie groups. Then, we will explore some geometric applications to the quantization of cotangent bundles of Lie groups and to the theory of theta functions.

We show that any nonsingular regular surface of general type with $p_g=6$, $K^2=13$ and whose canonical image is not contained in a pencil of quadrics is a complete intersection in a generalized weighted Grassmannian.

In this talk I will explain how a general moment map formal set-up, developed by Donaldson for the action of a symplectomorphism group on the space of compatible complex structures, can be used to interpret and extend known results on the topology of symplectomorphism groups of rational ruled surfaces. The actual proofs are based on a result of independent interest: the space of integrable complex structures compatible with a fixed (but arbitrary) symplectic form on a rational ruled surface is contractible. This is joint work with Gustavo Granja and Nitu Kitchloo.

Wrinkling. Application of the wrinkling theory: new proofs of Thurston theorem on foliations and new proof of Eliashberg's theorem on mappings with prescribed singularities.

We construct an equivariant pro-object that represents Poincare Duality for the infinite dimensional manifold LM given by the space of loops on a smooth manifold M. By applying a suitably completed K-theory to this object, we get the structure of a TQFT (or Frobenius algebra) which has the value on the surface of genus zero given by the Witten genus of M.

A complex manifold $X$ is hyperbolic if there are no nonconstant holomorphic maps $f:\mathbb{C} \to X$. A projective variety $X$ is quasi-hyperbolic if it has only finitely many rational and elliptic curves. The Kobayashi conjecture states that: The generic hypersurface of degree $d\ge 5$ in $\mathbb{P}^3$ is hyperbolic (hence also quasi-hyperbolic). Clemens proved quasi-hyperbolicity for very generic hypersurfaces. We show that nodal hypersurfaces with a sufficiently large number of nodes are also quasi-hyperbolic. An ingredient of the proof of this counter intuitive result is Bogomolov's theory of symmetric differentials.

By the work of Delzant, we know that all Hamiltonian toric $4$-manifolds are obtained by equivariant blow-ups of either $CP^2$ or an Hirzebruch surface. On the other end, given such a manifold, the classification of all distinct toric structures one can put on it is still unknown. In this talk, we will explain how one can use $J$-holomorphic curves techniques to prove a) that there is only finitely many such structures on a given $4$-manifold; b) that many symplectic blow-ups of $CP^2$ don't admit any Hamiltonian toric action. The ideas naturaly extend to Hamiltonian $S^1$-actions under mild hypothesis and give rise to some interesting questions.

Semistable degenerations of curves and surfaces. Normal crossings, triple point formulas, components with multiplicity; examples of degenerations of rational and $K3$ surfaces.

CAMGSD, Project "Classification of complex algebraic surfaces ", (POCTI/MAT/44068/2002)

In the first two and a half lectures I will thouch these themes:

generalities on divisors and linear systems on a surface;

(birational) classification of movable linear systems whose general curve is irreducible of low geometric genus (say ≤ 2);

generalities on adjoint linear systems, concentrating on conditions for their nefness. If I have time I may say the little I know about their base point freeness,birationality etc.

I will indicate the proof of a classical theorem of Castelnuovo-Enriques which bounds (under suitable conditions) the dimensions of a linear system of curves in terms of their geometric genus. I will talk about extensions of this theorem both classical (Castelnuovo himself) and more recent (Reid). If time will be left, I will discuss possible extensions to threefolds.

CAMGSD, Project "Classification of complex algebraic surfaces ", (POCTI/MAT/44068/2002)

Embedded degenerations in projective space with special incidence on rational surfaces, Veronese surfaces, line bundles on degenerations, degenerations of surfaces to unions of planes, formulas for plurigenera and $K^2$ .

In the first two and a half lectures I will thouch these themes:

generalities on divisors and linear systems on a surface;

(birational) classification of movable linear systems whose general curve is irreducible of low geometric genus (say ≤ 2);

generalities on adjoint linear systems, concentrating on conditions for their nefness. If I have time I may say the little I know about their base point freeness,birationality etc.

I will indicate the proof of a classical theorem of Castelnuovo-Enriques which bounds (under suitable conditions) the dimensions of a linear system of curves in terms of their geometric genus. I will talk about extensions of this theorem both classical (Castelnuovo himself) and more recent (Reid). If time will be left, I will discuss possible extensions to threefolds.

Embedded degenerations in projective space with special incidence on rational surfaces, Veronese surfaces, line bundles on degenerations, degenerations of surfaces to unions of planes, formulas for plurigenera and $K^2$ .

In the first three lectures I will thouch these themes:

generalities on divisors and linear systems on a surface;

(birational) classification of movable linear systems whose general curve is irreducible of low geometric genus (say ≤ 2);

generalities on adjoint linear systems, concentrating on conditions for their nefness. If I have time I may say the little I know about their base point freeness,birationality etc.

I will indicate the proof of a classical theorem of Castelnuovo-Enriques which bounds (under suitable conditions) the dimensions of a linear system of curves in terms of their geometric genus. I will talk about extensions of this theorem both classical (Castelnuovo himself) and more recent (Reid). If time will be left, I will discuss possible extensions to threefolds.

Applications to interpolation theory. Matching formulas for a degenerating family of embedded curves. Reduction lemmas for linear systems of plane curves with base fat points; recent results on linear systems via degeneration methods.

Applications to interpolation theory. Matching formulas for a degenerating family of embedded curves. Reduction lemmas for linear systems of plane curves with base fat points; recent results on linear systems via degeneration methods.

There are two central theorems in algebraic geometry due to Hironaka: resolution of singularities, and Log-principalization of ideals. The second theorem relates to the elimination of base points of a linear system, and hence to the study of morphisms among smooth projective varieties. We intend to give some indications of an alternative simplified proof of both theorems, which has lead to various computer implementations in recent years.

Given a parallel $m$-calibration $\Omega$ on a Riemannian manifold $N$, we define for each $m$-submanifold $M$ an $\Omega$-angle that measures the deviation of $M$ to a $\Omega$-calibrated one. If $M$ is minimal but not $\Omega$-calibrated we expect that the $\Omega$-calibrated points can be expressed as a residue on a polinomial formula of invariants of $M$, the normal bundle $NM$, and $N$. This residue should be expressed on a PDE (possibly with singularities) on the $\Omega$-angle, via Chern-Weil theory. In this way, we propose to establish a link between two major theories of Harvey and Lawson: The theory of Calibrations and the one of Geometric Residues. We also propose the classification of minimal submanifolds with constant $\Omega$-angle. We study in detail the particular cases:

Complex and Lagrangian points of Cayley 4-submanifolds of Calabi-Yau (real 8)-manifolds.

Totally complex and quaternionic points of complex 4-submanifolds of Quaternionic-Kahler (QK) and Hyper-Kahler (HK) 8-manifolds.

Moreover, we give a description of complex submanifolds with constant quaternionic angle. We obtain some rigidity theorems for submanifolds of QK manifolds.

Rigidity and flexibility phenomena are well-known in symplectic geometry. In this talk, I will argue that similar phenomena appear in Poisson geometry, but are even harder to study. On a positive note, I will give criteria for stability of symplectic leaves of Poisson manifolds, which are analogous to criteria for stability of leaves of foliations and stability of orbits of group actions.

We define an invariant of embedded graphs in 3-manifolds making use of the Turaev-Viro state sum invariant, defining natural observables for 3d quantum Gravity. We relate this invariant with the coloured Jones Polynomial of links. We make a similar construction for the 4 dimensional Crane-Yetter invariant.

In spite of interesting representations of noncompact real Lie groups be typically infinite dimensional, it is hopeless to attempt to classify all of them. In fact, a given representation can be modified in countless ways by simply changing the topology on the representation space, and there are pathological representations, badly behaved, which are unlikely to fit into any reasonable classification scheme. Consequently, the study of the representations of a real semisimple Lie group $G$ is usually restricted to some class, containing the representations which are known to arise in interesting problems and for which there are some hope of a reasonable classification scheme. Harish-Chandra introduced such a class – the class of admissible representations. Harish-Chandra’s strategy to study admissible representations of $G$ was by studying their Harish-Chandra modules. In this talk, we present the classification of irreducible Harish-Chandra modules developed by Beilinson and Bernstein, which is defined by an equivalence of categories between the category of Harish-Chandra modules and the category of coherent sheaves of D-modules on the flag variety of the complexification of the Lie algebra of $G$.

If $G$ is a reductive group and $X$ a (normal) affine $G$-variety, we call $X$ spherical if every irreducible $G$-module occurs at most once in the coordinate ring $C[X]$ of $X$. Inspired by a conjecture of Friedrich Knop, which states that any such variety is determined by the module structure of $C[X]$, we classified the smooth affine spherical varieties up to coverings, central tori, and $C^*$-fibrations. We give an outline of the classification and briefly sketch the conjecture's origins in symplectic geometry: it is equivalent to Delzant's conjecture that compact multiplicity free Hamiltonian $K$-manifolds ($K$ a compact connected Lie group) are uniquely determined by their generic isotropy group and the image of the moment map.