Independent Partitions in Boolean Algebras

This dissertation introduces a generalization of the cardinal invariant independence for Boolean algebras, suggested by the proof of the Balcar-Franek Theorem. The objects of study are independent sets of partitions under this new notion of independence. Generalizations of several known results regarding large and small independence are formulated and proved, and counterexamples provided for others. Notably the Balcar-Franek theorem itself is generalized

Logic Meets Algebra: the Case of Regular Languages

The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoid. We also explain the impact of these connections on circuit complexity theory.Comment: 37 page

Free Probability Theory

The workhop brought together leading experts, as well as promising young researchers, in areas related to recent developments in free probability theory. Some particular emphasis was on the relation of free probability with random matrix theory

Sheaves on T-topologies

The aim of this paper is to give a unifying description of various constructions (subanalytic, semialgebraic, o-minimal site) using the notion of T-topology. We then study the category of T-sheaves.Comment: 31 pages, uses xy-pic, revised versio