# III Mini-Workshop em Geometria Simplética

# June 20-21, PUC-Rio

**Sala de reuniões do Decanato do CTC (12 º andar do prédio Cardeal Leme), PUC-Rio**

SPEAKERS

- Daniel Fadel (UNICAMP)
- Daniele Sepe (UFF)
- Misha Verbitsky (IMPA)
- Joa Weber (UNICAMP)
- Henrique Bursztyn (IMPA)
- Nicola Sansonetto (Univ. Verona)
- Paula Balseiro (UFF)
- Renato Vianna (UFRJ)
- Umberto Hryniewicz (UFRJ)
- Vinicius Ramos (IMPA)

## Talks

- Daniel Fadel (UNICAMP) -
**The Limit of Large Mass Monopoles.**

*Abstract:* Given a noncompact, oriented, Riemannian 3-manifold X, an SU(2) monopole on X is a solution to the so-called Bogomolnyi equation: a first order PDE for pairs consisting of a connection and a Higgs field on the (trivial) SU(2)-bundle over X; it describes a particular kind of critical point of the SU(2) Yang-Mills-Higgs energy functional, and can be further obtained by dimensional reduction from the (anti)-self-dual instanton equation in 4 dimensions. In this talk I will consider monopoles on an asymptotically conical 3-manifold with one end. I will explain that in this case the connected components of the moduli space of monopoles are labeled by an integer called the charge, and I will consider the problem of the limiting behavior of sequences of monopoles with fixed charge, and whose sequence of Yang-Mills-Higgs energies is unbounded. I will show that the limiting behavior of such monopoles is characterized by energy concentration along the subset of X consisting of points at which the zeros of the Higgs fields accumulate. I will end showing that this set is finite using a bubbling analysis to obtain effective bounds on its cardinality, with such bounds depending solely on the charge of the monopole. This is joint work with Gonçalo Oliveira.

- Daniele Sepe (UFF) -
**Constructing complexity one integrable systems**

*Abstract*: An integrable system on a symplectic manifold is a maximal family of Poisson commuting functions that are independent on an open, dense subset. On the one hand, these systems arise naturally in natural sciences as they can be used to model many real-life phenomena. On the other, from a mathematical standpoint, these can be seen as examples of Hamiltonian actions of the simply connected abelian Lie group. A natural (imprecise!) question is the extent to which it is possible to endow a given symplectic manifold with a "reasonable" integrable system. In general, this question is too vague (and difficult, even when made precise). The aim of this talk is to consider the following significantly simpler problem: Given a closed symplectic 4-manifold endowed with an effective Hamiltonian S^1-action, does there exist an integrable system on the given symplectic manifold one of whose components is the moment map of the given S^1-action? (The pair of such a symplectic manifold and the effective Hamiltonian S^1-action is known as a complexity one space in work of Karshon and Tolman.) We shall discuss some partial results in this direction and, time permitting, some ideas to solve the above problem completely. The partial results are in collaboration with Sonja Hohloch, Silvia Sabatini and Margaret Symington, while the ongoing extensions are in collaboration with San Vu Ngoc and Sue Tolman (in separate projects).

- Misha Verbitsky (IMPA) -
**Locality in the Fukaya category of a hyperkahler manifold**.

*Abstract:* Let (M,I,J,K) be a hyperkahler manifold, that is, a Riemannian manifold equipped with quaternion algebra action on its tangent bundle, which is Kahler with respect to the complex structures induced by quaternions. Then the complex manifold (M,I) is holomorphically symplectic, that is, equipped with a complex linear, non-degenerate,closed form. It is known that any complex Lagrangian subvariety X in (M,I) is special Lagrangian with respect to a complex structure aJ+bK, where a, b are real nubmers satisfying a^2 + b^2=1. I will prove that for all a, b except countably many, there is no holomorphic curve on a manifold (M, aJ+bK) with boundary on X. This statement should be inderstood as formality of the subcategory of Fukaya category generated by Lagrangian subvarieties of holomorphic Lagrangian type. This is a joint work with Jake Solomon, arXiv:1805.00102.

- Joa Weber (UNICAMP) -
**The shift map on Floer trajectory spaces**

*Abstract: *In this talk we explain a uniform proof, joint work with Urs Frauenfelder [arXiv:1803.03826], why the shift map on Floer homology trajectory spaces is scale smooth. This proof works for various Floer homologies, periodic, Lagrangian, Hyperkähler, elliptic or parabolic, and uses Hilbert space valued Sobolev theory. This crucially uses the linearity of the shift map in the second variable.

- Henrique Bursztyn (IMPA) -
**Deformation spaces and normal forms around transversals**

*Abstract:* Given a manifold M, any submanifold N defines a "deformation space" D(M,N), which is a manifold equipped with a submersion to the real line whose zero fiber is the normal bundle of the submanifold, and all the other fibres are equal to M. I will explain how deformation spaces can be used to study normal forms/linearization of various geometric structures that carry an underlying singular foliaton (e.g. Poisson structures, Lie or Courant algebroids...) around submanifolds transverse to the foliation. These results include local splitting theorems in these settings. Based on joint work with F Bischoff, H. Lima and E. Meinrenken.

- Nicola Sansonetto (Univ. Verona) -
**On integrable Hamiltonian systems on almost-symplectic manifolds**

*Abstract:* It is well known that nonholonomic mechanical systems with symmetry admit an almost symplectic/Poisson formulation and are often Hamiltonizable after the symmetry reduction. Motivated by this we wonder how much do systems which are Hamiltonian with respect to an almost-symplectic structure (a non-degenerate but non-closed two form) differ from standard Hamiltonian systems? If the two-form is non-closed, then Hamiltonian vector fields are not automatically symmetries of the almost-symplectic structure, and do not form a Lie algebra. However under suitable, quite natural but strong hypotheses the theory of integrable systems extends to the almost-symplectic case.

In this talk we discuss dynamical systems on almost symplectic manifolds that are integrable, investigate their geometry and possibly a relation between reduction and Hamiltonization.

[1] F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems. J.Math. Phys. 48 (2007), 092902, 13 pp.

[2] F. Fassò and N. Sansonetto, Nearly-integrable almost-symplectic Hamiltonian systems. Preprint.

[3] N. Sansonetto and D. Sepe, Twisted isotropic realizations of twisted Poisson structures. J. Geom. Mech. 5, (2014), 233-256.

[4] I. Vaisman, Hamiltonian vector fields on almost symplectic manifolds. J. Math. Phys. 54 (2013), 092902, 11 pp.

- Paula Balseiro (UFF) -
**First integrals and non-holonomic mechanics**

- Renato Vianna (UFRJ) -
**On shapes of symplectic manifolds as seen by holomorphic curves**

- Umberto Hryniewicz (UFRJ) -
**Evaluating the quadratic linking form**

*Abstract:* In this talk I will revise the definition of the quadratic linking form for flows on the three sphere. Then I would like to explain how pseudo-holomorphic curves can be used to estimate its value on many relevant invariant measures of tight Reeb flows. This talk is based on joint work with Pedro Salomao and Kris Wysocki.

- Vinicius Ramos (IMPA) -
**Symplectic embeddings, lagrangian products and integrable systems**

*Abstract*: Symplectic embedding problems are at the core of the study of symplectic topology. There are many well-known results for so-called toric domains, but very little is known about other kinds of domains. In this talk, I will mostly speak about a different kind of domain, namely a lagrangian product. These domains are of a very different nature and are related to billiards, as discovered by Artstein-Avidan and Ostrover. I will explain how to use integrable systems to see that some of these products are secretely toric domains and how to use symplectic capacities to obtain sharp obstructions to many symplectic embedding problems.

**Organizing Commitee**: Alessia Mandini and David Martinez Torres