Seminário Simplético do Rio - 2021

Abstract: In nonholonomic mechanics, the presence of constraints in the velocities breaks the well-understood link between symmetries and first integrals of holonomic systems, expressed in Noether's Theorem.

In this talk, we show that, under certain conditions on the symmetry Lie group, the (nonholonomic) momentum map is conserved along the nonholonomic dynamics, thus extending Noether Theorem to the nonholonomic framework. Our analysis leads to a constructive method, with fundamental consequences to the integrability of some nonholonomic systems as well as their hamiltonization.

Abstract: A Lagrangian field on a symplectic manifold $M$ is a family $\Lambda=\{\Lambda_x|x \in M\}$ of pointed Lagrangian submanifolds of $M$. This notion is a generalization of a real Lagrangian polarization for which each $\Lambda_x$ is the leaf containing $x$. Two Lagrangian fields $\Lambda$ and $\tilde \Lambda$ are called transversal if $\Lambda_x$ intersects $\tilde\Lambda_x$ transversally at $x$ for every $x$. Two transversal Lagrangian fields determine an almost para-K\"ahler structure on $M$.

We construct a local symplectic groupoid on a neighborhood of the zero section of $T^\ast M$ from two transversal Lagrangian fields on $M$. The Lagrangian manifold of $n$-cycles of this groupoid in $(T^\ast M)^n$ has a generating function whose germ around the diagonal of $M^n$ is given by the $n$-point cyclic Calabi function of a closed (1,1)-form on a neighborhood of the diagonal of $M^2$ obtained from the symplectic form on~$M$.

Abstract: Lie bialgebras were introduced by Drinfeld in the 80s while studying quantum groups. They are Lie algebras coupled with a compatible Poisson structure and they integrate to Poisson groups. Their theory has a natural extension to groupoids and algebroids, developed by Weinstein and Mackenzie, among others. Jointly with H. Bursztyn and A. Cabrera, and building over previous work on Poisson double structures, we recently obtained a (new) approach to the categorification of Lie bialgebras. In this talk I will first overview the fundamentals on Lie bialgebras, then discuss the categorification of vector spaces and Lie algebras, and finally present our contributions to the subject.

Abstract: The question of whether a Symplectic manifold embeds into another is central in Symplectic topology. Since Gromov nonsqueezing theorem, it is known that it is a different problem from volume preserving embeddings. Embedded contact homology has been shown to be very useful to obtain obstructions for Symplectic embeddings in dimension 4. In this talk we will see some results in dimension 4 due to Mcduff-Schlenk Hutchings, Frenkel-Müller, Cristofaro-Gardiner, and Ramos. Furthermore, we will explain new results about the disk cotangent bundle of the two dimensional round sphere. This is joint work with Vinicius Ramos.

Abstract: I will report on recent joint work (https://arxiv.org/abs/2102.12641) with Stéphane Druel, Brent Pym, Frédéric Touzet where we prove that when a compact Kähler Poisson manifold has a symplectic leaf with finite fundamental group, then after passing to a finite étale cover, it decomposes as the product of the universal cover of the leaf and some other Poisson manifold.

Abstract: A foundational rigidity result in contact geometry is Gray stability: up to contactomorphism, the only deformations of closed contact manifolds are constant ones. More recently, an analogous rigidity result for Lie groupoids was obtained independently (using different techniques) by Crainic, Mestre and Struchiner, and by del Hoyo and Fernandes: up to isomorphism, the only deformations of compact Lie groupoids are constant ones. It is natural to ask whether the two rigidity results can be achieved simultaneously for compact contact groupoids. In this talk we will show that this is possible (hence the title of the talk), and we will illustrate some of the consequences of this result to the study of rigidity of Poisson and Jacobi structures. No previous knowledge of either contact structures or Lie groupoids will be assumed. This is ongoing joint work with Camilo Angulo and María Amelia Salazar.

Abstract: The topological entropy of a flow on a compact manifold is a measure of complexity related to many other notions of growth. By celebrated works of Katok and Besson-Courtois-Gallot, the topological entropy of geodesic flows of Riemannian metrics with a fixed volume on a manifold M that carries a metric of negative curvature is uniformly bounded from below by a positive constant depending only on M. We show that this result persists for all (possibly irreversible) Finsler flows, but that on every closed contact manifold there exists a Reeb flow of fixed volume and arbitrarily small entropy. This is joint work with Alberto Abbondandolo, Murat Saglam and Felix Schlenk.

Abstract: Using tools from Dirac geometry, we show through an explicit construction that any Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which relates, for a principal bundle M→B, integrations of a Poisson structure on B to integrations of its pullback Dirac structure on M by pre-symplectic groupoids (joint work with H. Bursztyn and J.H. Lu).

Abstract: Hutchings uses embedded contact homology to show the following for area-preserving disc diffeomorphisms that are a rotation near the boundary of the disc: If the asymptotic mean action on the boundary is bigger than the Calabi invariant, then the infimum of the mean action of the periodic points is less than or equal to the Calabi invariant. In this talk, we extend this to all area-preserving disc diffeomorphisms. Our strategy is to extend the diffeomorphism to a larger disc with nice properties and apply Hutchings' theorem. As an application, we show that the Calabi invariant of a smooth pseudo-rotation is equal to its rotation number.

Abstract: In this talk, we will study some aspects of Dirac-Nijenhuis structures, which encompasses the notions of Poisson-Nijenhuis and presymplectic-Nijenhuis structures introduced by Magri and Morosi in the context of integrable systems. The main goal is to discuss the integration of Dirac-Nijenhuis structures to presymplectic-Nijenhuis groupoids, and present the important case when the Nijenhuis tensor is a holomorphic structure. The talk is based on joint work with H. Bursztyn and T. Drummond.

Abstract: In this talk, we will discuss some geometric structures over Lie groupoids and differentiable stacks. In the first half of the talk, after recalling some background, we will see two notions of a gerbe over a stack and show their correspondence. In the second half of the talk, we see the notion of the principal bundle over a Lie groupoid and associate a Chern-Weil map to this structure over Lie groupoid.
This talk is based on the works https://www.sciencedirect.com/science/article/pii/S0007449720300567 and https://arxiv.org/abs/2012.08447

Abstract: First I review N-vortex equations on surfaces - in terms of the Green and Robin functions (and why not, Batman's) associated to the Laplace-Beltrami operator. I mention physical motivations, and show some numerical simulations, work with SB (//doi.org/10.1063/1.3146241) and with Cesar Castilho and Adriano Regis (doi: 10.3934/jgm.2018007, http://mi.mathnet.ru/eng/rcd389). These simulations confirm the conjecture of Y. Kimura in 1999, that vortex dipoles are "curvature checkers", in the sense that they move along geodesics (//doi.org/10.1098/rspa.1999.0311). The outline for a possible proof was given in Stefanella Boatto and JK (arXiv:0802.4313). In the second part of the talk I present some results from ongoing work with Umberto Hryniewicz and Alejandro Cabrera. In particular, we claim that close by vortex pairs are also "topology checkers": the dynamics of a vortex pair should be a good way to probe not only the local geometry but also the topology in the large, very much like geodesics do - only more so. In principle, all the ingredients for surfaces of genus 0 and 1 are already available. Time permitting, I will show some preliminary estimates by Anilatmaja Aryasomayajula for genus \geq 2, for both Robin and Batman's functions, in terms of the injectivity radius and first eigenvalue of the hyperbolic Laplacian.

Abstract: Symplectic groupoids are geometric objects that function as global counterparts to Poisson manifolds, in the same way that Lie groups are global counterparts to Lie algebras. In this talk I will first give an idea of what these objects are and of how that analogy works, and I will then present the construction of the deformation cohomology controlling deformations of symplectic groupoids. I will then compute this cohomology in some examples, explain how to use it in a Moser path argument, and relate it to the deformation theory of the corresponding Poisson manifolds. The talk is based on joint work with Cristian Cárdenas (UFF) and Ivan Struchiner (USP).

Abstract: This talk discusses aspects of the theory of submanifolds and submersions in Poisson geometry. In the first part we present the general picture concerning manifolds which inherit a Poisson structure from an ambient Poisson manifold, and among those, we select a class (coregular submanifolds) which have particularly nice functorial properties. The second part is devoted to Poisson submersions with coregular fibers. Coregular submersions restrict nicely over symplectic leaves in the base (coupling property), and we determine when they split into commuting vertical and horizontal Poisson structures. In the last part we present instances in which such coregular Poisson submersions appear. Our illustrations all revolve around Poisson actions of Poisson-Lie groups. This is joint work with L. Brambila and P. Frejlich.

Abstract: We will show sharp dynamical implications of convexity on symmetric spheres that do not follow from dynamical convexity. It allows us to furnish new examples of dynamically convex contact forms that are not equivalent to convex ones via contactomorphisms that preserve the symmetry in any dimension. Moreover, these examples are $C^1$-stable in the sense that they are actually not equivalent to convex ones via contactomorphisms that are $C^1$-close to those preserving the symmetry. We also show the multiplicity of symmetric non-hyperbolic closed Reeb orbits under suitable pinching conditions and the existence of symmetric elliptic periodic Reeb orbits. This is ongoing joint work with Miguel Abreu.

Abstract: In 2002 Polterovich notably showed that Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must be trivial on closed symplectically aspherical manifolds. We study the existence of Hamiltonian torsion and its metric rigidity properties in more general situations. First, we extend Polterovich's result to closed symplectically Calabi-Yau and closed negative monotone manifolds. Second, going beyond topological constraints, we describe how Hamiltonian torsion is related to the existence of pseudo-holomorphic spheres and answer a close variant of Problem 24 from the introductory monograph of McDuff-Salamon. Finally, we prove an analogue of Newman’s 1931 theorem for Hofer’s metric and Viterbo’s spectral metric on the Hamiltonian group of monotone symplecitc manifolds: a sufficiently small ball around the identity contains no torsion. During the talk, I shall discuss the results above and some of the key ingredients of their proofs. This talk is based on joint work with Egor Shelukhin.