Spinal Open Book Decompositions

Speaker: Sam Lisi, University of Mississippi.

Date: 25 nov 2019, 14h.

Place: Room 201, Bloco H, Campus Gragoatá, UFF.

Abstract: Contact manifolds admit many different descriptions, and examples can be constructed from many points of view. One natural class comes as the boundary "at infinity" of a (non-compact) symplectic manifold with a homogeneity condition -- the prototypical example is seeing the unit co-tangent bundle as the boundary of T*Q. Another description comes from an open book decomposition.

The symplectic filling problem for a contact manifold is, given a contact manifold, to determine if it arises as the boundary of a symplectic manifold, and if so, to classify all the symplectic fillings.

To address this question, at least in some cases, we introduce the notion of a spinal open book decomposition in dimension 3. Using J-holomorphic curve techniques, we obtain filling obstructions for a class of examples (using ideas originally developed by Gromov, McDuff and Eliashberg) and a complete filling classification for a smaller class of examples (using ideas from Hofer-Wysocki-Zehnder, Hutchings and Siefring).  I will give some examples of what we are able to classify, and will also illustrate some of the less technical ingredients of the proofs.


This is joint work with Jeremy Van Horn-Morris and Chris Wendl.

 

 

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.

 

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