Minicurso

Yoshihiko Mitsumatsu
Título: Lefschetz fibrations on Milnor fibres of cusp and simple elliptic singularities
Abstract: Recently we found there exist Lefschetz fibration structures on Milnor fibres of cusp or simple elliptic singularities in three complex variables, whose regular fibre is closed 2-torus. This also implies that there exists a symplectic structure on such a Milnor fibre whose end is symplectically periodic.
We start with the notion of convexity in symplectic structures as well as the (strong) pseudo convexity in complex variables, then proceed to the topologial flexibility of the symplectic convexity. How the cohomological condition is related with the modification of convex symplectic structures into end-periodic ones is discussed.
As a motivation and as an application of such end-periodic symplectic structures, leafwise symplectic structures on Lawson's foliation on the 5-sphere (a regular Poisson structure) and on similar foliations are shown to exist.
The construction of Lefschetz fibrations is roughly sketched. Local modifications of isolated critical points are discussed on the way. Such local study might enables us to attack further global existence results in future.
As applications to 4-dim (symplectic) topology, by gluing two Milnor fibres together with Lefschetz fibrationn structures, we obtain closed symplectic 4-manifolds and in fact elliptic surfaces over CP^1. Some of them are diffeomorphic to the K3 surface. A relation of these phenomena with the singularity theory, in particular the strange duality of Arnold, is also discussed.
The results are largely form the joint work with Naohiko KASUYA, Hiroki KODAMA, and Atsyhide MORI.

Palestras

Francesco Bonechi (Università degli Studi di Firenze, Italy)
Title: Quantization of symplectic groupoids from multiplicative integrable models
Abstract: I will present a class of non trivial examples where Weinstein's dream of quantizing Poisson manifolds through the quantization of the symplectic groupoid can be concretely realized. The construction uses singular polarizations, for instance those given by integrable models that are compatible with the groupoid structure. We call such models multiplicative. The main source of examples comes from Poisson-Nijenhuis geometry. I will discuss in detail the example of Bruhat-Poisson structure on complex projective spaces.

Margaret Symington (Mercer University, USA)
Title: Integral affine surfaces: cylinders to spheres
Abstract: A key feature of an almost toric manifold (a symplectic four-manifold equipped with a singular Lagrangian fibration having singularities of elliptic and focus-focus type) is the singular integral affine structure induced on the base of the fibration. Building on an understanding of the integral affine structures induced by semitoric systems and inspired by the algebraic and combinatorial descriptions of integral affine structures on on S^2 (coming from toric degenerations and mirror symmetry), I will explain an approach to describing integral affine structures that is suitable in the more general symplectic context. I will give some examples of structures on cylinders and explain how they may help in understanding such structures on S^2.

Jethro van Ekeren (UFF, Brazil)
Title: Quantisation of Poisson Arc Spaces and Statistical Models
Abstract: Differential Poisson Algebras and Vertex Algebras provide a context in which to discuss quantisations and classical limits of 2 dimensional field theories. In this talk I will describe a homological criterion to decide freeness of classical limits of such field theories as well as recent results on classical limits of the Ising model and its generalisations and open problems. I will also discuss interesting links with modular forms and partitions and with recent work of physicists on Schur indices of 4 dimensional superconformal field theories. (Joint work with R. Heluani.)

Misha Verbitsky (IMPA, Brazil)
Title: Closed Reeb orbits on Sasakian manifolds
Abstract: Sasakian manifolds are related to contact ones in the same way as Kahler manifolds are related to symplectic. Sasakian manifold can be defined as a quotient of a Kahler manifold by R acting by non-trivial Kahler homotheties, in the same was as one defines a contact manifold as a quotient of a symplectic manifold by homotheties. For each Sasakian manifold Q there is a circle S^1 acting on Q by Sasakian automorphisms, in such a way that the quotient Q/S^1 is a projective orbifold. We prove that the number of closed Reeb orbits on Q is bounded from below by the sum of Betti numbers of Q/S^1. This implies that a Sasakian manifold of real dimension 2n+1 has at least n closed Reeb orbits. This is a joint work with Liviu Ornea. The proof is based on counting elliptic curves on a certain non-Kahler complex manifold which is naturally associated to a Sasakian manifold.

Renato Vianna (UFRJ, Brazil)
Title: Applications of almost toric fibrations
Abstract: We will survey a series of results in 4 dimensional symplectic topology in the last 6 years that rely on almost toric fibrations developed by Symington. We will present a subset of the following list of results: existence of infinitely many (Hamiltonian isotopy classes) of monotone Lagrangian tori in Del Pezzo surfaces (comments about generalisations to higher dimensions [joint with L. Diogo, D. Tonkonog, W. Wu]); computations and other results of isotopy shapes/star-shapes for (almost) toric fibres and relationship with embedding of Weinstein neighbourhoods [joint with E. Shelukhin, D. Tonkonog]; ball packing results in the complement of exotic Lagrangian tori in CP^2 [joint with W. Lee, Y-G. Oh]; recover results about volume filling embeddings of ellipsoids into the ball and other toric domains [joint with R. Cassals]; classification of almost toric fibres of CP^2 [joint with E. Shelukhin, D. Tonkonog]. We may also mention how to visualise Lagrangians fibering over tropical curves on almost toric fibrations, which was independently discovered by G. Mikhalkin / D. Matessi / J. Hicks.

Maxim Zabzine (Uppsala University, Sweden)
Title: On generalised Kahler potential
Abstract: I will discuss the problem of generalised Kahler potential. I will outline the past progress and open problems.

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