# Programme

## GenĂ¨ve, May 15^{th}

## Villa Batelle

** 10:00 Claude Viterbo ** (Paris)

On the topology of fillings of contact manifolds

We address the following
question: given a contact manifold (Σ ,ξ), what can be said
about the
symplectically aspherical manifolds (W ,ω) bounded by (Σ,ξ) ?
We first extend a theorem of Eliashberg, Floer and McDuff to prove that,
under suitable assumptions, the map from H_{*}(Σ) to H_{*}(W) induced by inclusion is surjective.
We apply this method in the case of contact manifolds admitting a
contact embedding in ℝ^{2n} or in a subcritical Stein
manifold. We prove in many cases that the homology of the fillings is
uniquely determined by the boundary.
This is joint work with A. Oancea

**11:30 Sergei Tabachnikov** (Pennsylvania State)

Pentagram Map, twenty years after

Introduced by R. Schwartz about 20 years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. The pentagram map is a discrete completely integrable system whose continuous limit is the Boussinesq equation, a completely integrable PDE of soliton type. In this talk I shall survey recent work on the pentagram map and its generalizations, emphasizing its close ties with the theory of cluster algebras, a new and rapidly developing area with numerous connections to diverse fields of mathematics. In particular, I shall describe a higher-dimensional version of the pentagram map and, somewhat counter-intuitively, its 1-dimensional version.

**14:30 Conan Leung** (Hong Kong)

SYZ transformation for coisotropic A-branes

Kapustin-Li observed that Lagrangian cycles alone is not enough for mirror symmetry for Calabi-Yau manifolds away from LCSL and they introduced the notion of coistropic A-branes. In the semiflat case, Yi Zhang and I showed recently that the SYZ transformation of B-branes are precisely given by coisotropic A-branes. In this talk, I will explain this work. **16:00 Gabi Farkas** (Berlin)

Parametrizations of moduli spaces of abelian varieties.

I will discuss various approaches to describe the geometry of the moduli spaces of abelian varieties respectively Prym varieties of relatively small dimension, in concrete terms using curves and their correspondences as well as special K3 surfaces.The workshop is supported by the TROPGEO project of the European Research Council.