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.

 

Hamiltonian S1-spaces with large equivariant pseudo-index.

Speaker: Isabelle Charton, Universität zu Köln.

Date: 18 oct 2019, 11h. Cancelado! :(

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

Abstract: Let (M,\omega) be a compact symplectic manifold of dimension 2n endowed with a Hamiltonian circle action with only isolated fixed points. Whenever M admits a toric 1-skeleton S, which is a special collection of embedded 2-spheres in M, we define the notion of equivariant pseudo-index of S: this is the minimum of the evaluation of the first Chern class c1 on the spheres of S. This can be seen as the analog in this category of the notion of pseudo-index for complex Fano varieties. In this talk we discuss upper bounds for the equivariant pseudo-index. In particular, when the even Betti numbers of M are unimodal, we prove that it is at most n+1 . Moreover, when it is exactly n+1, M must be homotopically equivalent to CPn.

 

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).

 

Special Lagrangian spheres

Speaker: Gonçalo Oliveira, UFF.

Date: 27 sep 2019, 11h. CANCELADO (Palestrante vai receber o Prêmio Jovem Cientista do Nosso Estado. Parabéns!)

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

Abstract: A conjecture of Richard Thomas gives a stability condition supposed to control the existence of a special Lagrangian sphere in a Hamiltonian isotopy class. In this talk, I will describe joint work with Jason Lotay where we prove a version of this conjecture for real 4-dimensional examples arising from the Gibbons-Hawking ansatz. In these same examples, I will also give a description of Seidel's symplectically knotted Lagrangian spheres and, if time permits, I will give a thorough study of minimal submanifolds of this class of examples.

 

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.

 

Integrating Lie 2-algebras

Speaker: Camilo Angulo, UFF.

Date: 06 sep 2019, 11h.

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

Abstract: A Lie 2-algebra is a groupoid object in the category of Lie algebras. These can naturally be seen as an infinitesimal version of Lie 2-groups which are groupoids in the category of Lie groups. Lie 2-algebras are known to be integrable in this sense. To understand this integration process from a cohomological point of view, we present appropriate notions of representations for both Lie 2-groups and Lie 2-algebras and the corresponding complexes whose cohomologies classify extensions. Finally, we discuss a van Est type theorem.

 

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.

 

Symplectic Picard-Lefschetz theory

Speaker: Hossein Movasati, IMPA.

Date: 05 jul 2019, 15h30.

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

Abstract: Is it possible to classify all homological cycles of a given symplectic manifold supported in Lagrangian spheres? The question in this generality might be ambiguous and too difficult. However, for complex projective varieties endowed with the Fubini-Study metric, Lefschetz vanishing cycles turn out to be supported in Lagrangian spheres and the monodromy action on them gives us a big class of such homological cycles. In this talk, I will report on a partial result in this direction for a family of Calabi-Yau threefolds called mirror quintic. The talk is partially based on my book 'A course in Hodge Theory: With Emphasis on multiple integrals' and Daniel Lopes Ph.D. thesis.

 

The Cohomology of Courant algebroids and their characteristic classes

Speaker: Miquel Cueca Ten, IMPA.

Date: 05 jul 2019, 14h.

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

Abstract: Courant algebroids originated over 20 years ago motivated by constrained mechanics but now play an important role in Poisson geometry and related areas. Courant algebroids have an associated cohomology, which is hard to describe concretely. Building on work of Keller and Waldmann, I will show an explicit description of the complex of a Courant algebroid where the differential satisfies a Cartan-type formula. This leads to a new viewpoint on connections and representations of Courant algebroids and allows us to define new invariants as secondary charcateristic classes, analogous to what Crainic and Fernandes did for Lie algebroids.

This is joint work with R. Mehta.

 

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