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11/06/2013, 15:00 — 16:00 — Room P3.10, Mathematics Building

Daniele Sepe, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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From semi-toric systems to Hamiltonian ${S}^{1}$-spaces

Semi-toric integrable systems on closed four dimensional
manifolds, introduced by Vu Ngoc, lie at the intersection of
integrable Hamiltonian systems and Hamiltonian torus actions. In
particular, they are integrable Hamiltonian systems which have one
integral that generates an effective Hamiltonian ${S}^{1}$-action. Vu
Ngoc showed that these systems share an important property with
symplectic toric manifolds, i.e. it is possible to associate a
family of convex polygons to each of them. On the other hand,
considering the manifold only with the Hamiltonian ${S}^{1}$-action,
they give rise to examples of Hamiltonian ${S}^{1}$-spaces, which have
been classified by Karshon. The aim of this talk is to illustrate
how, given a semi-toric system, Karshon's invariants of the
underlying Hamiltonian ${S}^{1}$-space can be reconstructed from one
(and hence any) convex polygon associated to the system. Time
permitting, we will consider how to construct examples of these
systems starting from symplectic toric manifolds.

This is joint work with Sonja Hohloch and Silvia Sabatini.

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06/06/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building

Daniele Sepe, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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Singular integral affine structures and integrable Hamiltonian
systems

An important outstanding question in the theory of integrable
Hamiltonian systems is their classification, i.e. the construction
of some (hopefully computable!) invariants which completely
determine these systems up to a suitable notion of equivalence. In
this talk, a weaker (but still considerably hard) problem is going
to be introduced, namely that of determining when two integrable
Hamiltonian systems are equivalent. In the absence of
singularities, i.e. equilibria, this problem can be solved using
integral affine geometry, which studies symmetries of Euclidean
space fixing the standard integral lattice therein; this was first
observed by Duistermaat and Dazord and Delzant. One way to extend
this result to include singularities is to define an appropriate
notion of singular integral affine geometry, which is the aim of
this talk. Time permitting, some low dimensional explicit examples
will be discussed. This is joint ongoing work with Rui Loja
Fernandes.

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04/06/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building

Daniele Sepe, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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On complete isotropic realisations of Poisson manifolds

Poisson geometry can be viewed as the study of those manifolds
which admit a (possibly singular) foliation by symplectic manifolds
(e.g. \(\mathbb{R}^3\) foliated by spheres centred at the origin of
increasing radius and the radius). A complete symplectic
realisation of a Poisson manifold is a desingularisation, in the
sense that it consists of a symplectic manifold together with a
submersion onto the original Poisson manifold which reflects its
Poisson structure. A beautiful theorem of Crainic and Fernandes
proves that existence of such a desingularisation is equivalent to
the Poisson manifold (viewed as an infinitesimal object, the
analogue of a Lie algebra) admitting a global integration (the
analogue of a Lie group). In this talk, we consider complete
isotropic realisations of Poisson manifolds, which are
desingularisations as above of minimal dimension; these are related
to non-commutative integrable Hamiltonian systems. The fundamental
driving question behind this talk is the following: what can be
said about Poisson manifolds that admit such desingularisations?
Some results in this direction, both old and new, will be
presented. This is ongoing joint work with Ioan Marcut.

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10/04/2012, 10:30 — 11:30 — Room P4.35, Mathematics Building

Daniele Sepe, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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Lecture VII - Lagrangian fibrations with elliptic singularities

The Eliasson-Miranda-Zung linearisation theorem provides a
symplectic model in a neighbourhood of a compact non-degenerate
orbit of a completely integrable Hamiltonian system. A natural
question to ask is whether such a model exists for singular fibres
of Lagrangian fibrations (which, generally, consist of several
disjoint orbits), the aim being a symplectic classification of
Lagrangian fibrations with singularities. In this lecture the
simplest case of purely elliptic singularities is studied in order
to illustrate some of the difficulties that arise in developing
such a classification theory.

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03/04/2012, 10:30 — 11:30 — Room P4.35, Mathematics Building

Daniele Sepe, *CAMGSD/IST*

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Lecture VI - The Eliasson-Miranda-Zung linearisation theorem

The definition of non-degenerate singular orbits of completely
integrable Hamiltonian systems gives local linear models for the
underlying Hamiltonian ${\mathbb{R}}^{n}$-action. A natural question is
whether there exists a symplectomorphism from an open neighbourhood
of a fixed singular orbit to the local linear model which preserves
the respective Lagrangian fibrations. The Eliasson-Miranda-Zung
theorem answers the above questions affirmatively when the orbit is
compact. In this lecture this theorem is stated and a strategy for
its proof is outlined, so as to explain the main underlying
geometric ideas.

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27/03/2012, 10:30 — 11:30 — Room P4.35, Mathematics Building

Daniele Sepe, *CAMGSD/IST*

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Lecture V - Non-degenerate singularities

Having established several structural results for Lagrangian
fibrations which are submersions in the first four sessions, this
lecture aims to study a more general class of fibrations which
allows for topologically well-behaved (i.e. Morse-Bott in some
sense) singularities. These naturally arise in the theory of
completely integrable Hamiltonian systems (e.g. if the phase space
is compact, they must exist) and in mirror symmetry. The notion of
non-degenerate singularities is introduced and illustrated with
several examples. Time permitting, the Eliasson-Miranda
linearization theorem for non-degenerate singularities is going to
be studied in some detail (but probably without proof).

#### See also

http://www.math.ist.utl.pt/~jmourao/inves/D_Sepe_Lagrangian_Fibrations.pdf

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20/03/2012, 10:30 — 11:30 — Room P4.35, Mathematics Building

Daniele Sepe, *CAMGSD/IST*

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Lecture IV - (Integral) affine manifolds and Lagrangian fibrations
II

This lecture continues with the study of (integral) affine
manifolds. First, some important invariants associated to these
manifolds (the affine and linear holonomies and the radiance
obstruction) are constructed. Particular attention is devoted to
the radiance obstruction, a cohomology class constructed by Goldman
and Hirsch which contains important information about the given
(integral) affine structure. Secondly, it is proved that a manifold
is the base of any Lagrangian fibre bundle with compact and
connected fibres if and only if it is an integral affine manifold,
which allows to study the problem of constructing all Lagrangian
fibre bundles with compact and connected fibres over a fixed
integral affine manifold. As usual, the theory developed is
illustrated by means of examples.

#### See also

http://www.math.ist.utl.pt/~jmourao/inves/D_Sepe_Lagrangian_Fibrations.pdf

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13/03/2012, 10:30 — 11:30 — Room P4.35, Mathematics Building

Daniele Sepe, *CAMGSD/IST*

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Lecture III - (Integral) affine manifolds and Lagrangian fibrations
I

A theorem due to Weinstein states that the leaves of any Lagrangian
foliation admit a flat, torsion-free connection. A manifold
admitting such a connection is called affine; these have been
extensively studied since the '50s as a generalisation of flat
Riemannian manifolds. This lecture first proves the above result
directly for Lagrangian fibrations and then introduces more
formally (integral) affine manifolds, illustrating the theory with
examples.

#### See also

http://www.math.ist.utl.pt/~jmourao/inves/D_Sepe_Lagrangian_Fibrations.pdf

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06/03/2012, 10:30 — 11:30 — Room P4.35, Mathematics Building

Daniele Sepe, *CAMGSD/IST*

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Lecture II - Topological and symplectic classification

A theorem due to Liouville, Mineur and Arnol'd states that if a
Lagrangian fibration admits a compact and connected fibre $F$, then
the fibre is diffeomorphic to a torus, nearby fibres are also tori
and there exists a symplectomorphism between a neighbourhood of $F$
and the zero section of the cotangent bundle to the torus which
preserves the fibrations. In this lecture, a generalisation of this
theorem for complete Lagrangian fibrations is proved, using the
natural fibrewise action of the cotangent bundle to the base on the
total space of the fibration. This construction allows to develop a
topological (in fact, smooth) and symplectic classification theory
for such fibrations, which yields two topological invariants, the
period net and Chern class, and one symplectic characteristic
class, the Lagrangian Chern class.

#### See also

http://www.math.ist.utl.pt/~jmourao/inves/D_Sepe_Lagrangian_Fibrations.pdf

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28/02/2012, 10:30 — 11:30 — Room P4.35, Mathematics Building

Daniele Sepe, *CAMGSD/IST*

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Lecture I - An introduction to Lagrangian fibrations

This lecture presents an overview on the whole series, introducing
some fundamental concepts in the study of Lagrangian fibrations by
means of some examples. Special attention is dedicated to
completely integrable Hamiltonian systems, which are a type of
mechanical systems that are intimately connected to Lagrangian
fibrations. The structure of the cotangent bundle as a Lagrangian
fibration (in fact a fibre bundle) is studied, thus illustrating
the ideas that lead to the topological and symplectic
classification of Lagrangian fibrations.

#### See also

http://www.math.ist.utl.pt/~jmourao/inves/D_Sepe_Lagrangian_Fibrations.pdf

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14/02/2012, 10:30 — 11:30 — Room P4.35, Mathematics Building

Rosa Sena Dias, *CAMGSD/IST*

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Hyperkähler manifolds - II

Hypertoric manifolds are $4n$ dimensional hyperkähler manifolds
admiting a tri-hamiltonian ${\mathbb{T}}^{n}$ action. We describe a
construction of such manifolds as quotients of ${\u2102}^{2d}$
by subtori of ${\mathbb{T}}^{d}$. This construction mimics Delzant's
construction of toric Kahler manifolds as quotients of
${\u2102}^{d}$ by subtori of ${\mathbb{T}}^{d}$. We will discuss
relations between the two constructions and explain how to use the
hyperkahler version to give a description of the hyperkähler
metric in hyper action-angle coordinates.

Main reference: R. Bielawski and
A. Dancer, *The geometry and topology of toric hyperkähler
manifolds*

#### See also

http://www.math.ist.utl.pt/~jmourao/ws/HK.pdf

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07/02/2012, 10:30 — 11:30 — Room P4.35, Mathematics Building

José Mourão, *CAMGSD/IST*

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Hyperkahler manifolds - I

##### Motivational introduction

- Q1
- Why would a differential/symplectic/algebraic geometer care
about hyperkähler manifolds?
- Q2
- Why would a
*quantum* geometer care?

##### Technical introduction

Definition and basic properties of hyperkähler manifolds.
Relation with the holomorphic symplectic point of view.
Hyperkähler quotient/holomorphic symplectic quotient as a tool to
construct interesting examples.

#### See also

http://www.math.ist.utl.pt/~jmourao/ws/HK.pdf

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21/06/2011, 10:30 — 11:30 — Room P4.35, Mathematics Building

Luís Diogo, *Stanford University*

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Symplectic quasi-states and intersections - III

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14/06/2011, 10:30 — 11:30 — Room P4.35, Mathematics Building

Luís Diogo, *Stanford University*

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Symplectic quasi-states and intersections - II

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07/06/2011, 10:30 — 11:30 — Room P4.35, Mathematics Building

Luís Diogo, *Stanford University*

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Symplectic quasi-states and intersections - I