**P rincipal Investigators: **

**José Ignacio Burgos & Francisco Presas Mata**

*Research Members**:*

Indranil Biswas

Iván Blanco Chacón

Roger Casals Gutiérrez

Viktor Ginzburg

Tomás Luis Gómez Quiroga

Marina Logares Jiménez

Daniel Macías Castillo

Alberto Navarro Garmendia

Ignacio Sols

María Jesús Vázquez Gallo

Alfonso Zamora Saiz

__Summary__:

The present project is the continuation of other research projects funded by the MTM program (MTM2004-0709-C03-02, MTM2007-63582, MTM2010-17389, MTM2013-42135P). As in the previous projects, the main objects of study are the symmetries (group actions) existent in symplectic topology, arithmetic geometry and moduli spaces theory. The origin of the group was the research work of Ignacio Sols in the 90s. Around 15 years ago, Ignacio Sols started a group to study symplectic geometry using techniques inspired in algebraic geometry, following the work of Donaldson and Gromov. The application of algebro-geometric tools in conjunction with different types of h-principles is the identity signature of Francisco Presas and his group. They are currently studying different geometric structures in the groups of symmetries of contact and symplectic manifolds. Another main subject of the group is the study of moduli spaces. The techniques we use come from algebraic geometry, andalso from differential geometry. We study different properties of moduli spaces: computation of the automorphismgroups stability of the Picard and Poincaré bundles, Torelli type theorems, etc, More recently the group has incorporated experts in arithmetic geometry, more concretely in Arakelov theory and special values of Lfunctions. We are particularly interested in the relations between this field and the other main topics of the project. For instance our group has developed an arithmetic intersection theory specially suited for the study of moduli spaces and our classification of toric arithmetic varieties uses ideas coming from symplectic geometry. The main goals of the project are the following:

1.1 To develop the approximately holomorphic techniques introduced by Donaldson in the context of symplecticcobordism; In particular study suitable decompositions of Weinstein domains.

1.2 To study Engel structures in 4 manifolds. The expectation is that they will become a meningful research area in low dimensional differential topology.

1.3 To study dinamical and geometric properties of the group of contact and or symplectic symmetries.

2.1 The search for new Torelli-type theorems that tell us that the isomorphism class of a moduli space allow us to recover the data that define the space.

2.2 To study the algebro-geometric properties of certain moduli spaces, like the moduli space of parabolic bundles.

2.3 To study the singularities of the invariant metrics of natural line bundles on moduli spaces and relate them with the geometry of Bdivisors.

3.1 Continue our development of Arakelov geometry and in particular our study of toric varieties, developing tools to deal with higher rank vector bundles.

3.2 To work on the equivariantTamagawa number conjecture, obtaining a proof for certain Tate motives and, in relation with Birch and Swinnerton Dyer conjecture, to produce global cohomology classes and relate the fact that such classes are crystalline with the vanishing of a p-adic L-function.