Jesus Alvarez Lopez (Universidade Santiago de Compostela, Spain)
Zeta invariants of Morse forms.

Given any closed 1-forma real η on a Riemannian manifold, we have the Witten's perturbation of the de Rham differential operator, depending on a complex parameter z, which induces perturbations of the codiferencial and Laplacian operators on differential forms. Using these operators, we construct a zeta function, ζ(s,z), which is meromorphic on the variable s in the complex plain. For a class of Morse forms η, we show that ζ(s,z) is regular at s=1 if |Re(z)| is large enough, that ζ(1,z) converges to some real number as Re(z) goes to infinity, uniformly on Im(z), and we describe this limit. Any real cohomology class of degree one has some representative η satisfying the hipotheses we use, realizing any prescribed real number as the limit.

Alexandre Pombo (Czech Academy of Sciences, Czech Republic)
Virial identities in General Relativity

Virial (a.k.a. scaling) identities are integral identities that are useful for a variety of purposes in non-linear field theories, including establishing no-go theorems for solitonic and black holes solutions as well as checking the accuracy of numerical solutions. In this presentation, we provide an algorithm for the derivation of such integral identities. We show that a complete treatment of virial identities in the relativistic gravity must take into account the appropriate boundary term. For General Relativity this is the Gibbons-Hawking-York boundary term. There is, however, a particular "gauge" choice i.e. a choice of coordinates and parametrizing the metric functions, that simplifies the computation of the virial identities in General Relativity, making both the Einstein-Hilbert action and the Gibbons-Hawking-York boundary term non-contributing. Under this choice, the virial identity results exclusively from the matter action. For generic "gauge" choices, however, this is not the case.

Ana Cristina Casimiro (Universidade Nova de Lisboa, Portugal)
Reductive algebraic monoids.

The theory of linear algebraic monoids has its origin around 1980, its systematic study began by Putcha and Renner independently. They are affine algebraic varieties endowed with an associative composition law which is a morphism of varieties and an unity. The theory is a mixture of topics from semigroups, algebraic geometry and algebraic groups. In this talk we intend to introduce these objects, giving some background about affine varieties and algebraic groups. Our goal is to study reductive algebraic monoids. They are to the theory of linear algebraic monoids what reductive groups are to linear algebraic groups. Namely, they have a nice structure and are relevant in the representation theory. More concretely, we want to study the homomorphims from a free monoid into a reductive algebraic monoid, and the conjugation action by the unit group of the monoid, which is a reductive algebraic group.

Ana Cristina Ferreira (CMAT, Universidade do Minho)

Locally conformal SKT structures

It is well known that a Hermitian metric on a complex manifold is called SKT(strong Kähler with torsion) if the Bismut torsion form H is such that dH = 0. In this talk, as a conformal generalization of the SKT condition, we will introduce a new type of Hermitian structure, called locally conformal SKT. Precisely, a Hermitian structure (J, g) is said to be locally conformal SKT if there exists a closed 1-form α such that dH = α ∧ H . We will discuss the existence of such structures on Lie groups and their compact quotients by lattices. This is joint work with Anna Fino and Zineb Larbi.

Vítor Bessa (CMAT, Universidade do Minho)
Dynamical Systems in General Relativity

The scope of this work is the analysis of cosmological models arising from Einstein's theory of General Relativity. Motivated by cosmological models of the early universe we focus on matter models such as nonlinear scalar and vector fields in co-evolution with perfect-fluids with linear equations of state in spatially homogeneous spacetimes. We consider three different scenarios: Massless and massive Yang-Mills fields with perfect-fluids in flat Robertson-Walker spacetimes; Monomial scalar-field potentials interacting with perfect fluids in flat Robertson-Walker spacetimes with a friction-like interaction term; Monomial scalar field potentials in Bianchi type I spacetimes. The analysis rely on the introduction of new regular dynamical systems formulation of the Einstein field equations on compact (or future invariant) state spaces, and the use of dynamical systems tools such as monotone functions, quasi-homogeneous blow-ups, and averaging methods involving a timedependent perturbation parameter. This allow us to give proofs concerning the global dynamics of the models, and their past and future asymptotics. In particular we discuss the issues of asymptotic self-similarity and self-similarity breaking as well as asymptotic source dominance, i.e., if the model is scalar/vector field dominated or fluid dominated towards the asymptotic regimes.

Catarina Faustino (CMAT, Universidade do Minho)
On the homology graph of a precubical set.

Precubical sets are combinatorial-topological objects that may be used to model concurrent systems. In this talk, I will discuss a concept of directed homology for precubical sets, called the homology graph. I will be particularly interested in the behavior of the homology graph with respect to the tensor product of precubical sets, which corresponds to both the cartesian product of topological spaces and the parallel composition of independent concurrent systems.

Lucile Vandembroucq (CMAT, Universidade do Minho)
On the topological complexity of manifolds with abelian fundamental group

The topological complexity (TC) is a homotopy invariant which was introduced by M. Farber in order to give a topological measure of the complexity of the motion planning problem. We will give sufficient conditions for the topological complexity of a closed manifold M with abelian fundamental group to be nonmaximal, that is to satisfy TC(M)<2dim(M). This generalizes for manifolds some results of Costa and Farber on the topological complexity of spaces with small fundamental group. We will also 1see through examples that our conditions are sharp. This is a joint work with Daniel Cohen.

José Manuel Oliveira (CMAT, Universidade do Minho)
Triviality of transitive Lie algebroids on contractible bases

It is well known that a vector bundle over a contractible manifold is trivial. In the context of Lie algebroids, a transitive Lie algebroid defined on a contractible smooth manifold is also trivial. The classical way to show this is made by using integrability theory. In this seminar, we show that the triviality of transitive Lie algebroids on contractible bases can also be obtained by using the classical theory of extensions.