# Program 2018

## Lectures series

**Miguel Abreu,**Instituto Superior Técnico

**A toric geometry road from Kaehler metrics to contact topology**

Toric manifolds provide interesting hands-on testing grounds for several areas of mathematics. In these lectures I will exemplify this in the context of explicit constructions of constant scalar curvature Kaehler metrics (in part jointly with Rosa Sena-Dias) and of contact structures with zero first Chern class (joint with Leonardo Macarini), and the road that connected them.

: toric symplectic/contact manifolds.

Lecture 1: Kaehler metrics on toric symplectic manifolds.

**Lecture 2**

: contact invariants of toric contact manifolds.*Lecture 3*

**Alexandru Oancea**, Institut de Mathématiques de Jussieu-Paris

**Symplectic topology and string topology**String topology is the study of algebraic structures carried by spaces of loops and paths on manifolds. Symplectic topology is the study of global properties of symplectic manifolds. The connection between the two is realized in the framework of cotangent bundles, or phase spaces, where algebraic structures from symplectic topology have counterparts in string topology, and vice versa. I intend to explore in these three lectures the faithfulness of the dictionary between these two fields, and the ways in which each of them illuminates the other.

## Lectures

**Misael Avendaño**, UNISON

**Deformations of Poisson bundles with symmetry**

Given a horizontal deformation (a 2 cocycle) of a vertical Poisson structure on Poisson fiber bundle with symmetry, we study the problem of finding a transformation under which the deformation become G- invariant. We present some normalization criteria which are based on the averaging method for Poisson connections and computing the second Poisson cohomology classes. These results have applications to the normalization procedure of a class of slow-fast Hamiltonian system and also have rise some interesting questions about the G-equivariant Poisson cohomology.

**Jasel Berra**, UASLP

**Singular representations of quantum mechanics in deformation quantization**

In this talk we study singular representations of the Weyl algebra as a distributional limits of the standard Schrodinger representation within the deformation quantization approach. These limits satisfy the Bohr’s correspondence principle and imply the appearance of a minimal length scale in the spectrum, unlike other approaches where this scale is present at the level of the Poisson structure and its formal deformations. Finally, we discuss the case of the scalar field theory by constructing a complex structure compatible with the symplectic form associated to the phase space. The singular representation in this case agrees with the formalism developed in loop quantum gravity.

**Isaac Hasse Armengol**, UNISON

The averaging method is a classical tool to construct invariant geometric structures (metric, tensor field, etc.), with respect to an action of a compact Lie group. In this talk, by using the averaging method, we describe a class of invariant Ehresmann connections on Poisson foliations with symmetries.**Invariant Poisson connections**

**Nicolás Martínez Alba**, Universidad Nacional de Colombia

The aim of this talk is to introduce a new definition of Dirac strucures of higher order that extend the condition of maximal isotropic subbundles. After some properties of such structures it will be outlooked some applications to the geometry of classical field theories and index theorems.**On higher-Dirac structures**

**Alberto Molgado**, UASLP

**Applications of the De Donder-Weyl formalism for field theories**

Conventionally, relativistic field theories have been classically analyzed in a satisfactory manner either at the Lagrangian or the Hamiltonian levels. However, implementation of a canonical formalism which considers a covariant Poisson bracket for an arbitrary field theory is still missing. In this sense, considering a few basic ingredients from variational calculus and multisymplectic geometry, and bearing in mind some physically motivated models, in this talk we will introduce the polysymplectic formalism which allow us to incorporate a concomitant Poisson structure with the covariant De Donder-Weyl equations associated to a field theory. We also briefly discuss a recent proposal under which this Poisson structure may be mapped covariantly at the quantum level. In particular, we will emphasize the application of the polysymplectic formalism for the classical description of the well-known MacDowell-Mansouri model for General Relativity. For this specific model, we will describe the behaviour of the emerging geometric constraints and we will see how the symmetry breaking

process results implemented within the polysymplectic formalism.

**Pedro Solorzano**, UNAM

**TBA**

**Eduardo Velasco-Barreras**, UNISON

**Bigraded cochain complexes and Poisson chomology**

We present an algebraic framework for the computation of the low-degree cohomology of a class of Poisson structures on fibred and foliated manifolds. We also show that, in the more general context, this framework can be applied to the study of Lie algebroid cohomology. Finally, we illustrate our results by computing the low-degree cohomology in some particular cases. Work in progress with Andrés Pedroza, and Yury Vorobiev.