Lifting Lagrangians from symplectic divisors

Speaker: Luís Diogo, UFF.

Date: 01 nov 2019, 11h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: We prove that there are infinitely many non-symplectomorphic monotone Lagrangian tori in complex projective spaces, quadrics and cubics of complex dimension at least 3. This is a consequence of the following: if Y is a codimension 2 symplectic submanifold  of a closed symplectic manifold X, then we can explicitly relate the superpotential of a monotone Lagrangian L in Y with the superpotential of a monotone Lagrangian lift of L in X. This sometimes involves relative Gromov-Witten invariants of the pair (X,Y). We start by defining the superpotential, which is a count of pseudoholomorphic disks with boundary on a Lagrangian, and which plays an important role in Floer theory and mirror symmetry.

This is joint work with D. Tonkonog, R. Vianna and W. Wu.

Leaf topology of minimal foliations on 3-manifolds

Speaker: Carlos Meniño, UFF.

Date: 11 oct 2019, 15h30m.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: We show that every oriented and noncompact surface is homeomorphic to a leaf of a minimal hyperbolic foliation of a closed 3-manifold. The example is a suspension of a suitable circle group action over the bitorus. Moreover, every prescribed countable family of noncompact oriented surfaces can be simultaneusly realized as leaves of the same minimal hyperbolic foliation. The interest of this example relies in the fact that there were no examples of minimal hyperbolic foliatons with leaves with leaves with finitely and infinitely generated groups coexisting in the same foliation (and in the first case, only foliations with leaves homeomorphic to planes and cylinders were described!). Our example cannot be smoothed to transverse regularity C2, this suggests possible obstructions on the leaf topology of minimal hyperbolic foliations in that regularity. This is a joint work with P. Gusmão (UFF).

Coadjoint orbits of the group of contact diffeomorphisms

Speaker: Cornelia Vizman, West University of Timisoara.

Date: 20 sep 2019, 11h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: (Joint work with Stefan Haller from the University of Vienna) Generalizing the canonical symplectization of contact manifolds, we construct an infinite dimensional manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with equivariant moment maps, respectively, giving rise to a dual pair, called the EPContact dual pair. Via symplectic reduction, it provides a conceptual identification of nonlinear Grassmannians of weighted isotropic submanifolds of the contact manifold with certain coadjoint orbits of the contact group. For the projectivized cotangent bundle, the EPContact dual pair is closely related to the EPDiff dual pair due to Holm and Marsden, and leads to a geometric description of some coadjoint orbits of the full diffeomorphism group.

Classification of Hamiltonian Circle Actions on Compact Symplectic Orbifolds of Dimension Four

Speaker: Grace Mwakyoma, IST Lisboa - IMPA.

Date: 16 aug 2019, 11h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: Circle actions have attracted much recent attention in geometry and topology. In the terminology of dynamical systems, they are regarded as periodic flows and their fixed points correspond to equilibrium points.

A complete classification of Hamiltonian circle actions on compact manifolds of dimension four was obtained by Y. Karshon, following the work of M. Audin and K. Ahara and A. Hattori. In particular, it was shown that all these spaces are Kahler, that every example can be obtained from a simple model by a sequence of symplectic blowups and, if the fixed points are isolated, the circle actions extend to toric actions. In higher dimensions much less is known. There are however some partial classification results.

The present research aims at completely classifying Hamiltonian circle actions on compact orbifolds of dimension 4 when the fixed points are isolated. These spaces appear, for example, as reduced spaces of Hamiltonian torus actions at regular level sets of the moment map where the action is not free. L. Godinho shows in her classification of semifree Hamiltonian circle actions on compact 4-orbifolds that the situation is much different from the manifold case. For example, these actions can have any number of fixed points while, in the manifold case, they have exactly four fixed points.