Countable Choice and Compactness

We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p greater or equal to 1 (resp. . p = 0), and some closed subset F of [0, 1]^I which is a bounded subset of l^p(I), we show that AC(N) (resp. DC, the axiom of Dependent Choices) implies the compactness of F

Generalized gradient flow structure of internal energy driven phase field systems

In this paper we introduce a general abstract formulation of a variational thermomechanical model, by means of a unified derivation via a generalization of the principle of virtual powers for all the variables of the system, including the thermal one. In particular, choosing as thermal variable the entropy of the system, and as driving functional the internal energy, we get a gradient flow structure (in a suitable abstract setting) for the whole nonlinear PDE system. We prove a global in time existence of (weak) solutions result for the Cauchy problem associated to the abstract PDE system as well as uniqueness in case of suitable smoothness assumptions on the functionals

Higher algebraic structures in Hamiltonian Floer theory I

This is the first of two papers devoted to showing how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures on the symplectic cohomology of open symplectic manifolds. Using the SFT of Hamiltonian mapping tori we show how to define a homotopy extension of the well-known Lie bracket on symplectic cohomology. Apart from discussing applications to the existence of closed Reeb orbits, we outline how the $L_{\infty}$-structure is conjecturally related via mirror symmetry to the extended deformation theory of complex structures.Comment: Results of arXiv:1310.6014 got merged into arXiv:1412.2682, now entitled "Higher algebraic structures in Hamiltonian Floer theory" and published in Advances in Geometry (DOI: 10.1515/advgeom-2019-0017). Extensions of other announced results have been turned into an ongoing PhD thesis projec