Seminário SimpléticoConjunto no Rio

O Seminário Simplético Conjunto é uma iniciativa de pesquisadores do IMPA, PUC-RJ, UFF e UFRJ, com encontros mensais rotativos. Cada encontro conta com duas palestras, ministradas por pesquisadores locais ou convidados, em temas relacionados à geometria simplética (num sentido amplo). Alunos são particularmente encorajados a participar.

The Joint Symplectic Seminar is organized by IMPA, PUC-RJ, UFF and UFRJ, and consists of monthly meetings that alternate among these institutions. Each meeting features two talks on topics related to symplectic geometry (in a broad sense). Students are particularly encouraged to attend the seminars.

Encontros 2015
# Terça 01/dez @ UFF, Auditorio Bloco G, Campus Gragoata, Niteroi.
14:30. Michael Hutchings (U.C. Berkeley) Mean action and the Calabi invariant.
Given an area-preserving diffeomorphism of the closed unit disk which is a rotation near the boundary, one can naturally define an “action” function on the disk which agrees with the rotation number on the boundary. The Calabi invariant of the diffeomorphism is the average of the action function over the disk. We show that if the Calabi invariant is less than the boundary rotation number, then the infimum over periodic orbits of the average of the action over the periodic orbit is less than or equal to the Calabi invariant. The proof uses a new filtration on embedded contact homology determined by a transverse knot.
16:00. Tudor Ratiu (EPFL) The $U(n)$ free rigid body: integrability and stability analysis of the equilibria.
I will present joint work with Daisuke Tarama on the unitary free rigid body. The dynamics is described by the Euler equation on the Lie algebra $\mathfrak{U}(n)$. This system is bi-Hamilronian and it can be reduced onto the adjoint orbits, as in the case of the $SO(n)$-free rigid body. The complete integrability and the stability of the isolated equilibria on the generic orbits are considered by using the method of Bolsinov and Oshemkov. In particular, it is shown that all the isolated equilibria on generic orbits are Lyapunov stable, in stark contrast to the $SO(n)$-case.
# Quinta 29/out @ PUC , sala 856L.
14:30. Jaap Eldering (PUC-Rio) Symmetry reduction of fluid(-like) dynamics to jetlet particles.
I will present a particle model for a smoothed version of incompressible fluid dynamics in R^n. We follow Arnold's idea of viewing fluid dynamics as geodesics on the group of diffeomorphisms. The symmetry reduction is obtained through a dual pair of momentum maps and removes subgroups of the diffeomorphism group that fix (the jets of) a set of points. This leads to a finite-dimensional system of "jetlet" particles that are special, but exact solutions of the original system. I will outline the reduction procedure, present the resulting jetlet dynamics (which is a Hamiltonian system) and possibly say something about the feature of "particle merging" and show some numerical simulations. This is joint work with Colin Cotter, Darryl Holm, Henry Jacobs and David Meier.
16:00. Marco Castrillón (UCM, Madrid) Hamilton equations of gauge invariant problems.
Within the formulation of the multisymplectic formalism and the Hamilton-Cartan equations in Field Theory, those problems defined by a gauge invariant variational principle are non-regular. Without this regularity, the bridge between Lagrangian and Hamiltonian formalism becomes more complicated. However, a weaker condition of regularity (satisfied by most of the field theoretical models existing in the literature) can be defined and proves to be enough to analyse the structure of the set of solution of the Hamilton equations. This structure will be studied in the talk as well as the subsequent structures for the moduli spaces and the Jacobi fields.
# Quinta 1/out @ UFRJ , bloco C do CT, sala C116.
14:30. Pedro Frejlich (PUC-Rio) A new cobordism (joint with I. Marcut).
We discuss a new approach to cutting-and-pasting in the Poisson category, which employs a new normal form for Hamiltonian spaces as a key ingredient. We also extend the Poisson Transversal model in order to realize cut-and-paste as a funny cobordism in the Poisson category.
16:00. Jingzhi Yan (UFF) Existence of periodic points near an isolated fixed point with Lefschetz index 1 and zero rotation for area preserving surface homeomorphisms.
Let f be an orientation and area preserving diffeomorphism of an oriented surface M with an isolated degenerate fixed point z with Lefschetz index one. Le Roux conjectured that z is accumulated by periodic orbits. We can approach Le Roux's conjecture by proving that if f is isotopic to the identity by an isotopy fixing z and if the area of M is finite, then z is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at z is the limit in weak-star topology of a sequence of invariant probability measures supported on periodic orbits. Moreover, our proof is purely topological. It works for homeomorphisms and is related to the notion of local rotation set.
# Segunda 31/ago @ IMPA , sala 236.
15:30. Dan Cristofaro-Gardiner (Harvard University) Higher dimensional symplectic embeddings and the Fibonacci staircase.
McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a four-dimensional ball, and found that when the ellipsoid is close to round, the answer is given by an infinite "staircase" determined by the odd-index Fibonacci numbers. We show that a similar result holds in all even dimensions. This is joint work with Richard Hind.
17:00. Pedram Hekmati (IMPA) Basic gerbal representations of compact Lie groups.
Representations of compact Lie groups on complex vector spaces are well-understood and their classification dates back to the fundamental work of Cartan and Weyl. In this talk, I will discuss a notion of categorical representations introduced by Frenkel and Zhu in 2008. For compact simple Lie groups I will first recall how to construct the monoidal Lie groupoid associated to its natural 2-plectic structure defined by the Cartan 3-form, and then provide an explicit example of how such a categorical representation can be constructed.
# Segunda 15/jun @ UFF , posgraduação no campus do Gragoatá (Bloco H).
14:30. Matias del Hoyo (IMPA) Metrics on fibred Lie groupoids.
Fibred categories were introduced in descent theory by A. Grothendieck. In this talk I will discuss fibred Lie groupoids, an incarnation of that formalism in differential geometry, studied by K. Mackenzie, among others. I will explain how we construct, in a joint work with R. Fernandes, suitable metrics on fibred Lie groupoids, and present two major applications. The first one on deformation of Lie groupoids, extending a rigidity result from equivariant geometry by R. Palais, an alternative approach to that of M. Crainic et al. The second one, the Morita invariance of groupoid metrics, opens up the possibility to perform Riemannian geometry over stacks.
16:00. Leonardo Macarini (UFRJ) Multiplicity of periodic orbits for dynamically convex contact forms.
The contact Conley conjecture (CCC) establishes that the Reeb flow of any contact form has infinitely many simple closed orbits whenever the contact manifold meets some conditions. In a joint work with V. Ginzburg and B. Gurel we proved, under some mild extra assumptions, that an index admissible contact form on a prequantization of an aspherical symplectic manifold possesses infinitely many simple closed orbits, furnishing a partial positive answer to the CCC. When the basis of the prequantization is not aspherical, it is easy to see that the CCC can fail. In this talk, I will present some recent results on the multiplicity of periodic orbits for Reeb flows when the basis is not aspherical, assuming a sort of convexity of the contact form. This is joint work with Miguel Abreu.
# Segunda 27/abr @ PUC-RJ, Sala 856L.
14:30. Paula Balseiro (UFF) Using Poisson geometry to understand Nonholonomic mechanics.
In this talk I will give an overview of nonholonomic systems -- mechanical systems with constraints defined by nonintegrable distributions. We will discuss geometric features of such systems and, in particular, we will see how Poisson geometry helps in understanding how far these systems are from being hamiltonian. More precisely, I will show how the failure of the Jacobi identity of the bracket describing the dynamics characterizes the nonholonomic character of a mechanical system. Afterwards we will analyze its behaviour after a reduction by a group of symmetries and we will show how twisted Poisson brackets and gauge transformations may appear in the framework of nonholonomic mechanics.
16:00. Ugo Bruzzo (SISSA) Nonabelian Lie algebroid extensions.
We classify nonabelian extensions of Lie algebroids in the holomorphic or algebraic category, and introduce and study a spectral sequence that one can attach to any such extension and generalizes the Hochschild-Serre spectral sequence associated to an ideal in a Lie algebra.
# Terça 31/mar @ UFRJ, Fundão, Sala C116.
13:30. Renato Viana (U. Cambridge) Infinitely many monotone Lagrangian Tori in $\mathbb{C}P^2$.
In previous work, we constructed an exotic monotone Lagrangian torus in $\mathbb{C}P^2$ (not Hamiltonian isotopic to the known Clifford and Chekanov tori) using techniques motivated by mirror symmetry. We named it $T(1,4,25)$ because, when following a degeneration of $\mathbb{C}P^2$ to the weighted projective space $\mathbb{C}P(1,4,25)$, it degenerates to the central fiber of the moment map for the standard torus action on $\mathbb{C}P(1,4,25)$. Related to each degeneration from $\mathbb{C}P^2$ to $\mathbb{C}P(a^2,b^2,c^2)$, for $(a,b,c)$ a Markov triple - $a^2 + b^2 + c^2 = 3abc$ - there is a monotone Lagrangian torus, which we call $T(a^2,b^2,c^2)$. We employ techniques from symplectic field theory to prove that no two of them are Hamiltonian isotopic to each other.
15:00. Ionut Marcut (U. Nijmegen) Geometric structures on the space of Poisson transversals.
The role of Poisson transversals in Poisson geometry is analogous to the one played by symplectic submanifolds in symplectic geometry, and by transverse submanifolds in foliation theory. They are defined as submanifolds that intersect the symplectic leaves transversally and symplectically. I will talk about the geometry and topology of the infinite dimensional manifold of compact pointed Poisson transversals. In particular, this space carries a canonical weakly nondegenerate Vorobjev triple, and has a complete Poisson map to the original manifold. I will also discuss some relations between its topology and the topology of the group of Hamiltonian automorphisms. This is joint work with Pedro Frejlich.