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Room P3.10, Mathematics Building

Parametrizações de Espaços Analíticos Singulares: aplicações às equações diferenciais não lineares

Geometria em Lisboa Seminar

Room P3.10, Mathematics Building

Fuensanta Aroca, CMAF-UL

Parametrizações de Espaços Analíticos Singulares: aplicações às equações diferenciais não lineares

Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Holger Spielberg, CAMGSD-IST

Quantum cohomology and symplectic cuts

The quantum cohomology ring of a symplectic manifold is a deformation of the usual cohomology ring. Since its structure constants are given by Gromov-Witten invariants, the quantum ring, too, is an invariant of the symplectic topology of the manifold. In this talk I will report on recent progress (by Li/Ruan, Ionel/Parker, and others) on the question how this quantum ring changes under symplectic cuts (or gluing, which is the inverse operation). In the case of Kähler manifolds, these operations include for example the blow-up along complex submanifolds.

Room P3.10, Mathematics Building

Yvette Kosmann-Schwarzbach, École Polytechnique

We first introduce quasi-Poisson actions and their momentum maps, which take values in homogeneous spaces. We then show that Poisson actions of Poisson Lie groups (when the Manin pair under consideration is in fact a Manin triple) and the Hamiltonian actions with group-valued momentum maps (when the Manin pair has vanishing cobracket) are special cases of this construction.

Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Leonor Godinho, Instituto Superior Técnico

Room P3.10, Mathematics Building

António Araújo, CMAF, Universidade de Lisboa

Moduli Spaces of germs of Legendrian Curves

We construct the moduli spaces of germs of conic singular Lagrangean subvarieties of a symplectic manifold of dimension four. The projectivisation of a Lagrangean variety is a Legendrian subvariety of a contact manifold.

Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Sílvia Anjos, Instituto Superior Técnico

Topology of the symplectomorphism groups of $S^2\times S^2$

In this talk we discuss the topology of the symplectomorphism group of a product of two $2$-dimensional spheres where the ratio of the areas of the spheres is bigger than $1$. We compute the homotopy type of this symplectic manifold and we how that the group of symplectomorphisms contains two finite dimensional Lie groups generating the homotopy.

Amphitheatre Pa2, Mathematics Building

François Lalonde, Université de Montréal

Symplectic fibrations and quantum homology I

By the very definition of a symplectic form, every symplectic manifold is locally fibered. Donaldson's theorem on Lefschetz pencils shows that a somehow similar statement (with singularities) is actually true at a global level. This means that symplectic fibrations play a fundamental role in symplectic topology and geometry. The goal of these lectures is to introduce to the theory of non- singular symplectic fibrations in arbitrary dimensions. Here, by “symplectic fibration”, we mean a symplectic manifold which is fibered by symplectic submanifolds. I will show that this is essentially equivalent to a topological fibration \((M,\omega) \to P \to B\) with structure group equal to the group of Hamiltonian diffeomorphisms of \(M\) (at least when \(B\) is a simply connected symplectic manifold). I will discuss the topology of such fibrations, showing in particular that their rational cohomology splits in a many interesting cases (this is a strong generalisation of works of Kirwan and Atiyah-Bott). This gives hard obstructions to the construction of new symplectic manifolds by twisted products of two symplectic manifolds. I will sketch the proof, based on a geometric interpretation of the Seidel map in quantum homology. If time permits, I will mention the consequences of this on the discovery of new rigidity phenomenon in Hamiltonian dynamics. Most of this work is the result of a collaboration with McDuff and Polterovich.

Amphitheatre Pa2, Mathematics Building

François Lalonde, Université de Montréal

Symplectic fibrations and quantum homology II

By the very definition of a symplectic form, every symplectic manifold is locally fibered. Donaldson's theorem on Lefschetz pencils shows that a somehow similar statement (with singularities) is actually true at a global level. This means that symplectic fibrations play a fundamental role in symplectic topology and geometry. The goal of these lectures is to introduce to the theory of non- singular symplectic fibrations in arbitrary dimensions. Here, by “symplectic fibration”, we mean a symplectic manifold which is fibered by symplectic submanifolds. I will show that this is essentially equivalent to a topological fibration \((M,\omega) \to P \to B\) with structure group equal to the group of Hamiltonian diffeomorphisms of \(M\) (at least when \(B\) is a simply connected symplectic manifold). I will discuss the topology of such fibrations, showing in particular that their rational cohomology splits in a many interesting cases (this is a strong generalisation of works of Kirwan and Atiyah-Bott). This gives hard obstructions to the construction of new symplectic manifolds by twisted products of two symplectic manifolds. I will sketch the proof, based on a geometric interpretation of the Seidel map in quantum homology. If time permits, I will mention the consequences of this on the discovery of new rigidity phenomenon in Hamiltonian dynamics. Most of this work is the result of a collaboration with McDuff and Polterovich.

Amphitheatre Pa2, Mathematics Building

Charles Epstein, University of Pennsylvania

Survey of the geometric and analytic results in contact structures I

Several complex variables. After a quick introduction to complex structures and holomorphic functions of several variables we turn to the special features of higher dimensions: the Hartogs phenomenon, CR-structures, pseudoconvexity and the Lewy extension theorem. We define the \(d\)-barb complex and the Kohn-Rossi cohomology, discussing its connection to interior singularities. The lecture concludes with an introduction to 3-dimensional CR-manifolds.

Amphitheatre Pa2, Mathematics Building

Bruno de Oliveira, University of Pennsylvania

Introduction to the Hartshorne Conjecture I

In the early 60's Hartshorne studied the subvarieties that are the generalization of ample divisors for higher codimensions. Motivated by his study Hartshorne proposed the following conjecture: Let \(X\) be a smooth projective variety, \(A\) and \(B\) be two smooth subvarieties of \(X\) with ample normal bundle and such that \(\dim A + \dim B \geq \dim X\). Then \(A\) intersects \(B\). We will use this problem to illustrate the interplay of complex, differential and algebraic geometry. We will always target a diverse audience. To that effect we review notions of algebraic geometry : line bundles, vector bundles, \(P^n\)-bundles and the ampleness property. From complex differential geometry: Kahler manifolds, Hermitean metrics, connections, curvature, the positivity of vector bundles and vanishing theorems. From several complex variables: strongly q-convex spaces, the finiteness of cohomology groups and cycle spaces of complex manifolds. The notions and results mentioned above will then be applied to explain the reason of the conjecture, why the conjecture might not be true and to prove special cases. In particular we will do the case where the ambient variety \(X\) is a hypersurface in \(P^n\) (done by Barlet), a \(P^ 2\)-bundle over a surface (done by Barlet, Schneider and Peternel) and a \(P^1\)-bundle over a threefold.

Amphitheatre Pa2, Mathematics Building

François Lalonde, Université de Montréal

Symplectic fibrations and quantum homology III

By the very definition of a symplectic form, every symplectic manifold is locally fibered. Donaldson's theorem on Lefschetz pencils shows that a somehow similar statement (with singularities) is actually true at a global level. This means that symplectic fibrations play a fundamental role in symplectic topology and geometry. The goal of these lectures is to introduce to the theory of non- singular symplectic fibrations in arbitrary dimensions. Here, by “symplectic fibration”, we mean a symplectic manifold which is fibered by symplectic submanifolds. I will show that this is essentially equivalent to a topological fibration \((M,\omega) \to P \to B\) with structure group equal to the group of Hamiltonian diffeomorphisms of \(M\) (at least when \(B\) is a simply connected symplectic manifold). I will discuss the topology of such fibrations, showing in particular that their rational cohomology splits in a many interesting cases (this is a strong generalisation of works of Kirwan and Atiyah-Bott). This gives hard obstructions to the construction of new symplectic manifolds by twisted products of two symplectic manifolds. I will sketch the proof, based on a geometric interpretation of the Seidel map in quantum homology. If time permits, I will mention the consequences of this on the discovery of new rigidity phenomenon in Hamiltonian dynamics. Most of this work is the result of a collaboration with McDuff and Polterovich.

Amphitheatre Ea2, North Tower, IST

Charles Epstein, University of Pennsylvania

Survey of the geometric and analytic results in contact structures II

Filling three dimensional CR-manifolds. Three dimensional CR-manifolds have, in some sense, too many deformations because most cannot be realized as the boundaries of Stein spaces. This problem is related to that of finding symplectic fillings for contact manifolds. We give examples and consider the general features of this pathology. Lempert's algebraic approximation theorem gives a way to address this problem. We introduce this approach and consider the case of hypersurfaces in lines bundles over \(P^1\) in detail. We define the relative index which provides a measure of the change in the algebra of CR-functions under a deformation of the CR-structure.

Room P3.10, Mathematics Building

Rita Gaio, Universidade do Porto

J-holomorphic curves, moment maps and adiabatic limits

We study pseudoholomorphic curves in symplectic quotients as adiabatic limits of solutions of a system of nonlinear first order elliptic partial differential equations in the ambient symplectic manifold. The symplectic manifold carries a Hamiltonian group action. The equations involve the Cauchy-Riemann operator over a Riemann surface, twisted by a connection, and couple the curvature of the connection with the moment map. Our work should prove a conjecture that the genus zero invariants of Hamiltonian group actions defined by these equations agree with the genus zero Gromov-Witten invariants of the symplectic quotient in the monotone case.

Amphitheatre Pa2, Mathematics Building

François Lalonde, Université de Montréal

Symplectic fibrations and quantum homology IV

Amphitheatre Pa2, Mathematics Building

Charles Epstein, University of Pennsylvania

Survey of the geometric and analytic results in contact structures III

The relative index. The theory of relative index is presented in detail. Using this concept we give an analytic proof that the set of fillable deformations of the CR-structure on certain three manifolds is a closed set. If time permits we will discuss connections between the relative index and the contact mapping class group.

Amphitheatre Pa2, Mathematics Building

Bruno de Oliveira, University of Pennsylvania

Introduction to the Hartshorne Conjecture II

In the early 60's Hartshorne studied the subvarieties that are the generalization of ample divisors for higher codimensions. Motivated by his study Hartshorne proposed the following conjecture: Let \(X\) be a smooth projective variety, \(A\) and \(B\) be two smooth subvarieties of \(X\) with ample normal bundle and such that \(\dim A + \dim B \geq \dim X\). Then \(A\) intersects \(B\). We will use this problem to illustrate the interplay of complex, differential and algebraic geometry. We will always target a diverse audience. To that effect we review notions of algebraic geometry : line bundles, vector bundles, \(P^n\)-bundles and the ampleness property. From complex differential geometry: Kahler manifolds, Hermitean metrics, connections, curvature, the positivity of vector bundles and vanishing theorems. From several complex variables: strongly q-convex spaces, the finiteness of cohomology groups and cycle spaces of complex manifolds. The notions and results mentioned above will then be applied to explain the reason of the conjecture, why the conjecture might not be true and to prove special cases. In particular we will do the case where the ambient variety \(X\) is a hypersurface in \(P^n\) (done by Barlet), a \(P^ 2\)-bundle over a surface (done by Barlet, Schneider and Peternel) and a \(P^1\)-bundle over a threefold.

Room P3.10, Mathematics Building

Yum-Tong Siu, Harvard University

Introduction to the application of \(\overline{\partial}\) estimates to complex geometry I

The series of three lectures will discuss the recent applications of \(L^2\) estimates of to geometric problems. The main technique of these applications is the use of multiplier ideal sheaves. The use of \(\overline{\partial}\) multiplier ideal sheaves is a completely new way of deriving a priori estimates of partial differential equations. So far the technique has been developed only for the equation but should be adaptable to other systems of partial differential equations arising from geometric problems.

When a priori estimates cannot be readily derived by the usual methods of integration by parts, one multiplies the quantity to be estimated by a function to make the a priori estimate hold. The set of all such multipliers form an ideal sheaf. Global geometric conditions are studied which can force the ideal to be the unit ideal, thereby making the desired a priori estimate automatically hold. This method is a powerful tool for many geometric problems.

Without the assumption of any pre-requisites, this series of lectures starts with the derivation of \(L^2\) estimates of \(\overline{\partial}\). Then the two kinds of multiplier ideal sheaves, Kohn's and Nadel's, are introduced, along with the problems and motivations from which they originate.

As examples of the geometric application of multiplier ideal sheaves, the following kinds of problems in algebraic geometry are discussed:

- Fujita's conjecture which says that if L is a positive holomorphic line bundle over a compact complex manifold \(X\) of complex dimension n with canonical line bundle \(K_X\), then \(m L+K_X\) is generated by global holomorphic sections when \(m\geq n+1\) and is very ample when \(m\geq n+2\) (in the sense that any basis of the space of global holomorphic sections define a holomorphic embedding of \(X\) into a complex projective space).
- Effective Matsusaka's Theorem which says that if \(L\) is a positive holomorphic line bundle over a compact complex manifold \(X\) of complex dimension \(n\), then \(m L\) is very ample when \(m\) is greater than some explicit effective function of the Chern numbers \(L^n\) and \(L^{n-1} K_X\).
- Deformational invariance of plurigenera which says that \(m\)-genus of a compact complex projective algebraic manifold \(X\) ( i.e., the dimension of the space of global holomorphic sections of \(m K_X\) over \(X\)) is unchanged when \(X\) is holomorphically deformed.

Room P3.10, Mathematics Building

Yum-Tong Siu, Harvard University

Introduction to the application of \(\overline{\partial}\) estimates to complex geometry II

The series of three lectures will discuss the recent applications of \(L^2\) estimates of to geometric problems. The main technique of these applications is the use of multiplier ideal sheaves. The use of \(\overline{\partial}\) multiplier ideal sheaves is a completely new way of deriving a priori estimates of partial differential equations. So far the technique has been developed only for the equation but should be adaptable to other systems of partial differential equations arising from geometric problems.

When a priori estimates cannot be readily derived by the usual methods of integration by parts, one multiplies the quantity to be estimated by a function to make the a priori estimate hold. The set of all such multipliers form an ideal sheaf. Global geometric conditions are studied which can force the ideal to be the unit ideal, thereby making the desired a priori estimate automatically hold. This method is a powerful tool for many geometric problems.

Without the assumption of any pre-requisites, this series of lectures starts with the derivation of \(L^2\) estimates of \(\overline{\partial}\). Then the two kinds of multiplier ideal sheaves, Kohn's and Nadel's, are introduced, along with the problems and motivations from which they originate.

As examples of the geometric application of multiplier ideal sheaves, the following kinds of problems in algebraic geometry are discussed:

- Fujita's conjecture which says that if L is a positive holomorphic line bundle over a compact complex manifold \(X\) of complex dimension n with canonical line bundle \(K_X\), then \(m L+K_X\) is generated by global holomorphic sections when \(m\geq n+1\) and is very ample when \(m\geq n+2\) (in the sense that any basis of the space of global holomorphic sections define a holomorphic embedding of \(X\) into a complex projective space).
- Effective Matsusaka's Theorem which says that if \(L\) is a positive holomorphic line bundle over a compact complex manifold \(X\) of complex dimension \(n\), then \(m L\) is very ample when \(m\) is greater than some explicit effective function of the Chern numbers \(L^n\) and \(L^{n-1} K_X\).
- Deformational invariance of plurigenera which says that \(m\)-genus of a compact complex projective algebraic manifold \(X\) ( i.e., the dimension of the space of global holomorphic sections of \(m K_X\) over \(X\)) is unchanged when \(X\) is holomorphically deformed.

Room P3.10, Mathematics Building

Bruno de Oliveira, University of Pennsylvania

Introduction to the Hartshorne Conjecture III

In the early 60's Hartshorne studied the subvarieties that are the generalization of ample divisors for higher codimensions. Motivated by his study Hartshorne proposed the following conjecture: Let \(X\) be a smooth projective variety, \(A\) and \(B\) be two smooth subvarieties of \(X\) with ample normal bundle and such that \(\dim A + \dim B \geq \dim X\). Then \(A\) intersects \(B\). We will use this problem to illustrate the interplay of complex, differential and algebraic geometry. We will always target a diverse audience. To that effect we review notions of algebraic geometry : line bundles, vector bundles, \(P^n\)-bundles and the ampleness property. From complex differential geometry: Kahler manifolds, Hermitean metrics, connections, curvature, the positivity of vector bundles and vanishing theorems. From several complex variables: strongly q-convex spaces, the finiteness of cohomology groups and cycle spaces of complex manifolds. The notions and results mentioned above will then be applied to explain the reason of the conjecture, why the conjecture might not be true and to prove special cases. In particular we will do the case where the ambient variety \(X\) is a hypersurface in \(P^n\) (done by Barlet), a \(P^ 2\)-bundle over a surface (done by Barlet, Schneider and Peternel) and a \(P^1\)-bundle over a threefold.

Room P3.10, Mathematics Building

Yum-Tong Siu, Harvard University

Introduction to the application of \(\overline{\partial}\) estimates to complex geometry III

The series of three lectures will discuss the recent applications of \(L^2\) estimates of to geometric problems. The main technique of these applications is the use of multiplier ideal sheaves. The use of \(\overline{\partial}\) multiplier ideal sheaves is a completely new way of deriving a priori estimates of partial differential equations. So far the technique has been developed only for the equation but should be adaptable to other systems of partial differential equations arising from geometric problems.

When a priori estimates cannot be readily derived by the usual methods of integration by parts, one multiplies the quantity to be estimated by a function to make the a priori estimate hold. The set of all such multipliers form an ideal sheaf. Global geometric conditions are studied which can force the ideal to be the unit ideal, thereby making the desired a priori estimate automatically hold. This method is a powerful tool for many geometric problems.

Without the assumption of any pre-requisites, this series of lectures starts with the derivation of \(L^2\) estimates of \(\overline{\partial}\). Then the two kinds of multiplier ideal sheaves, Kohn's and Nadel's, are introduced, along with the problems and motivations from which they originate.

As examples of the geometric application of multiplier ideal sheaves, the following kinds of problems in algebraic geometry are discussed:

- Fujita's conjecture which says that if L is a positive holomorphic line bundle over a compact complex manifold \(X\) of complex dimension n with canonical line bundle \(K_X\), then \(m L+K_X\) is generated by global holomorphic sections when \(m\geq n+1\) and is very ample when \(m\geq n+2\) (in the sense that any basis of the space of global holomorphic sections define a holomorphic embedding of \(X\) into a complex projective space).
- Effective Matsusaka's Theorem which says that if \(L\) is a positive holomorphic line bundle over a compact complex manifold \(X\) of complex dimension \(n\), then \(m L\) is very ample when \(m\) is greater than some explicit effective function of the Chern numbers \(L^n\) and \(L^{n-1} K_X\).
- Deformational invariance of plurigenera which says that \(m\)-genus of a compact complex projective algebraic manifold \(X\) ( i.e., the dimension of the space of global holomorphic sections of \(m K_X\) over \(X\)) is unchanged when \(X\) is holomorphically deformed.

Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Le Dung-Trang, Université de Provence

Resolution of surfaces by normalized blow-ups

A celebrated theorem of O. Zariski asserts that if $S$ is an algebraic surface over an algebraically closed field of characteristic zero one may resolve the singularities of $S$ by a finite sequence of point blow-ups followed by normalizations. We present a new approach to the proof of this theorem.

Room P3.10, Mathematics Building

Pedro Cristiano, CMAF, Universidade de Lisboa

On a Class of Local Systems Associated to Irreducible Plane Curves

We study a class of local systems on the complementary of an irreducible plane curve. We exhibit families of such local systems which by microlocal Riemann-Hilbert correspondence,give rise to regular holonomic $D$-modules with characteristic variety equal to the union of the zero section with the conormal of the curve.

Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Miguel Abreu, Instituto Superior Técnico

Kahler metrics on toric orbifolds

In this talk I will describe how a symplectic approach to Kahler geometry on toric manifolds, presented in this seminar on July 2000, can be extended to the more general context of toric orbifolds. The main result is again an effective parametrization of all invariant Kahler metrics on a toric orbifold, via smooth functions on the corresponding moment polytope.

As an application I will give a simple explicit description of recent work of R. Bryant, producing interesting new families of extremal Kahler metrics. In particular, one obtains an extremal Kahler metric on any weighted projective space.

Room P3.10, Mathematics Building

José Basto Gonçalves, Universidade do Porto

Linearization of resonant vector fields

Classical linearization results hold in the non-resonant case. For resonant vector fields, we show how the non-linear part either gives simple, geometric, conditions for linearization, or else provides information on which resonant terms are relevant for its normal form.

Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Iakovos Androulidakis, Instituto Superior Técnico

Geometric quantization and integrability of Lie algebroids

A classical theorem of Kostant gives the conditions for a symplectic form to be the curvature form of a Hermitian connection on a Hermitian line bundle.

We consider a generalization of this result and explain how it relates to the integrability of Lie algebroids and the classification of extensions of Lie groupoids. Also, we clarify the nature of the obstructions using the notion of a crossed module.

Room P3.10, Mathematics Building

Orlando Neto, CMAF, Universidade de Lisboa

On the de Rham complex of an holonomic $\cal D$-module

An holonomic $\mathcal{D}$-module $\mathcal{M}$ is a system of differential equations on a complex manifold $X$ (module over the ring of differential equations on $X$) that equals a flat connection $(E,\nabla )$ on the complementary of an hypersurface $Y$ of $X$. If $\mathcal{M}$ verifies a reasonable condition the de Rham complex of $\mathcal{M}$ only depends on the the Rham complex of $(E,\nabla )$. We obtain in this way a dictionary between a certain class of linear representations of the fundamental group of $X\setminus Y$ and a certain class of holonomic $\mathcal{D}$-modules.

Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Paul Norbury, Adelaide University

Boundary algebras of hyperbolic monopoles

We prove the conjecture that a monopole in hyperbolic space can be completely determined by its “holographic image” on the conformal boundary two-sphere. The main tool used is an $n$-point function defined for a given monopole and any ordered collection of points on the conformal boundary two-sphere. The n-point function can be used to define structure coefficients of an interesting associative algebra abstractly generated by points of the conformal boundary two-sphere.