###
09/07/2002, 14:30 — 15:30 — Amphitheatre Ea2, North Tower, IST

Alexander Varchenko, *North Carolina*

```
```###
Arrangements, Hypergeometric Functions, and KZ-Type Equations - III

Conformal blocks in conformal field theory and their
quantizations are solutions of Knizhnik-Zamolodchikov differential
and difference equations. The geometric version of these objects is
multidimensional hypergeometric functions and hypergeometric
equaitons. The interrelations of these subjects, the theory of
solvable models in statistical mechanics, and representation theory
will be discussed.

The course will be elementary and accessible to graduate
students and advanced undergraduate students.

###
09/07/2002, 11:30 — 12:30 — Amphitheatre Ea2, North Tower, IST

Ieke Moerdijk, *Utrecht*

```
```###
Non-abelian Cohomology and Gerbes - II

The purpose of these lectures is to introduce the audience to
the topological concepts of stack, gerbe and bundle gerbe, and the
non-abelian cohomology classes to which they give rise.

#### Lecture 2

Introduction to stacks and gerbes.

#### References

- L. Breen, On the classification of 2-gerbes and 2-stacks,
Asterisque 225, (1994).
- J. Giraud, Cohomologie nonabelienne, Springer-Verlag,
1971.
- I. Moerdijk, On the classification of regular Lie groupoids,
preprint math.DG/0203099.
- M. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996).

###
09/07/2002, 09:30 — 10:30 — Amphitheatre Ea2, North Tower, IST

Alexander Varchenko, *North Carolina*

```
```###
Arrangements, Hypergeometric Functions, and KZ-Type Equations - II

Conformal blocks in conformal field theory and their
quantizations are solutions of Knizhnik-Zamolodchikov differential
and difference equations. The geometric version of these objects is
multidimensional hypergeometric functions and hypergeometric
equaitons. The interrelations of these subjects, the theory of
solvable models in statistical mechanics, and representation theory
will be discussed.

The course will be elementary and accessible to graduate
students and advanced undergraduate students.

###
08/07/2002, 14:00 — 15:00 — Room P3.10, Mathematics Building

Viktor Ginzburg, *Santa Cruz*

```
```###
Periodic Orbits and Symplectic Topology

Introduction: Hamiltonian flows, the Arnold and Weinstein
conjectures, examples: convex Hamiltonians and flows on twisted
cotangent bundles, review of results.

Floer homology: review of Morse theory, Floer homology and
symplectic homology, application: the Arnold conjecture.

Almost existence theorems for periodic orbits: symplectic
capacities almost existence and the Hofer-Zehnder capacity,
application: Viterbo's proof of Weinstein conjecture and almost
existence in twisted cotangent bundles.

The Hamiltonian Seifert conjecture: the Seifert conjecture and
counterexamples, Hamiltonian dynamical systems without periodic
orbits.

###
08/07/2002, 11:00 — 12:00 — Room P3.10, Mathematics Building

Alexander Varchenko, *North Carolina*

```
```###
Arrangements, Hypergeometric Functions, and KZ-Type Equations - I

Conformal blocks in conformal field theory and their
quantizations are solutions of Knizhnik-Zamolodchikov differential
and difference equations. The geometric version of these objects is
multidimensional hypergeometric functions and hypergeometric
equaitons. The interrelations of these subjects, the theory of
solvable models in statistical mechanics, and representation theory
will be discussed.

The course will be elementary and accessible to graduate
students and advanced undergraduate students.

###
08/07/2002, 09:00 — 10:00 — Amphitheatre Ea2, North Tower, IST

Ieke Moerdijk, *Utrecht*

```
```###
Non-abelian Cohomology and Gerbes - I

The purpose of these lectures is to introduce the audience to
the topological concepts of stack, gerbe and bundle gerbe, and the
non-abelian cohomology classes to which they give rise.

#### Lecture 1

Sheaves, torsors and non-abelian cohomology in degree 1.

#### References

- L. Breen, On the classification of 2-gerbes and 2-stacks,
Asterisque 225, (1994).
- J. Giraud, Cohomologie nonabelienne, Springer-Verlag,
1971.
- I. Moerdijk, On the classification of regular Lie groupoids,
preprint math.DG/0203099.
- M. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996).

###
28/06/2001, 16:30 — 17:30 — 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.

###
28/06/2001, 15:00 — 16:00 — 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.

###
27/06/2001, 16:30 — 17:30 — 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.

###
26/06/2001, 16:30 — 17:30 — 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.

###
22/06/2001, 16:00 — 17:00 — 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.

###
22/06/2001, 13:30 — 14:30 — 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.

###
22/06/2001, 09:30 — 10:30 — Amphitheatre Pa2, Mathematics Building

François Lalonde, *Université de Montréal*

```
```###
Symplectic fibrations and quantum homology - IV

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.

###
21/06/2001, 13:00 — 14:00 — 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.

###
21/06/2001, 09:30 — 10:30 — 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.

###
20/06/2001, 11:30 — 12:30 — 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.

###
20/06/2001, 09:30 — 10:30 — 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.

###
19/06/2001, 11:00 — 12:00 — 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.

###
18/06/2001, 11:00 — 12:00 — 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.