An almost existence theorem for non-contractible periodic orbits in cotangent bundles

Assume M is a closed connected smooth manifold and H:T^*M->R a smooth proper function bounded from below. Suppose the sublevel set {H<d} contains the zero section and \alpha is a non-trivial homotopy class of free loops in M. Then for almost every s>=d the level set {H=s} carries a periodic orbit z of the Hamiltonian system (T^*M,\omega_0,H) representing \alpha. Examples show that the condition that {H<d} contains M is necessary and almost existence cannot be improved to everywhere existence.Comment: 9 pages, 4 figures. v2: corrected typo

Global surfaces of section for Reeb flows in dimension three and beyond

We survey some recent developments in the quest for global surfaces of section for Reeb flows in dimension three using methods from Symplectic Topology. We focus on applications to geometry, including existence of closed geodesics and sharp systolic inequalities. Applications to topology and celestial mechanics are also presented.Comment: 33 pages, 3 figures. This is an extended version of a paper written for Proceedings of the ICM, Rio 2018; in v3 we made minor additional corrections, updated references, added a reference to work of Lu on the Conley Conjectur

The Milnor number of a hypersurface singularity in arbitrary characteristic

The Milnor number of an isolated hypersurface singularity, defined as the codimension $\mu(f)$ of the ideal generated by the partial derivatives of a power series $f$ whose zeros represent locally the hypersurface, is an important topological invariant of the singularity over the complex numbers, but its meaning changes dramatically when the base field is arbitrary. It turns out that if the ground field is of positive characteristic, this number is not even invariant under contact equivalence of the local equation $f$. In this paper we study the variation of the Milnor number in the contact class of $f$, giving necessary and sufficient conditions for its invariance. We also relate, for an isolated singularity, the finiteness of $\mu(f)$ to the smoothness of the generic fiber $f=s$. Finally, we prove that the Milnor number coincides with the conductor of a plane branch when the characteristic does not divide any of the minimal generators of its semigroup of values, showing in particular that this is a sufficient (but not necessary) condition for the invariance of the Milnor number in the whole equisingularity class of $f$.Comment: 20 page

The Milnor Number of Plane Branches With Tame Semigroup of Values

The Milnor number of an isolated hypersurface singularity, defined as the codimension $\mu(f)$ of the ideal generated by the partial derivatives of a power series $f$ that represents locally the hypersurface, is an important topological invariant of the singularity over the complex numbers. However it may loose its significance when the base field is arbitrary. It turns out that if the ground field is of positive characteristic, this number depends upon the equation $f$ representing the hypersurface, hence it is not an invariant of the hypersurface. For a plane branch represented by an irreducible convergent power series $f$ in two indeterminates over the complex numbers, it was shown by Milnor that $\mu(f)$ always coincides with the conductor $c(f)$ of the semigroup of values $S(f)$ of the branch. This is not true anymore if the characteristic of the ground field is positive. In this paper we show that, over algebraically closed fields of arbitrary characteristic, this is true, provided that the semigroup $S(f)$ is tame, that is, the characteristic of the field does not divide any of its minimal generators.Comment: arXiv admin note: substantial text overlap with arXiv:1507.0317

Três palavras para mudar a comunidade

Seminário / Webinário de homenagem a Robert L. Cooper, realizado na Universidade Aberta, em Lisboa, no dia 22 de outubro de 2013.Na evolução do domínio científico do Planeamento Linguístico tem sido feito um caminho no sentido do alargamento de diferentes objetos de estudo e intervenção, consequentemente de metodologias mais individualizadas, até pelas exigências próprias dos objetos de estudo e de intervenção. Num percurso especialmente marcado por Cooper e por Spolsky, este artigo visa expor o trabalho efetuado na comunidade da Costa da Caparica em torno de apenas 3 palavras – no que se poderia classificar de micro planeamento – que poderão ter profundas implicações na imagem identitária da comunidade.In the evolution of the scientific domain of Language Planning it is clear a path of broadening the kind of objects of study and intervention and subsequently implying more individualized methodologies, derived precisely by the characteristics inherent of the new objects of study and intervention. Specially influenced by Cooper and Spolsky, this article aims to expose to discussion the work being done in Costa da Caparica around just three words, in what could be called micro planning, but could eventually have deep consequences in the identity images of the community

Language and reality: being one with everything

International Conference "Philosophy of Science in the 21st Century – Challenges and Tasks". Lisboa, 4-6 Dezembro de 2013The first assumption of Hyperphysis is that “there is an objective Reality. This Reality is observer-independent, yet, it is understood that the observer interacts with the very same reality being able to change it and of course of being changed in a greater or lesser degree.” This principle of the existence of an objective Reality explicitly includes ideas, as J. R. Croca recently defined. Language Planning has evolved from his first steps dedicated to “nation building” to a present framework that was first enunciated by Robert L. Cooper as a tool for Social Change, and recently by Bernard Spolsky as a broader, more flexible management tool, understanding the change of, either societal or diverse sizes of communities, always including, obviously, the individuals responsible for the proposed change. Language as been the object of study of a science – Linguistics – that has difficulties accepting the inherent social nature of his object, pushing out this social nature of language to an hyphenated science: Sociolinguistics. This is far from peaceful. As A.-J. Calvet has stated, it is impossible to exclude the social nature of Language, therefore, there is no Linguistics that is not Sociolinguistics. The proposed concepts of Hyperphysis and especially of Eurhytmy can provide an important breakthrough in the understanding of the relation between speech and language, individual and social – use or change through words and languages - and also human immaterial production and Reality