A function on~$\mathbb{R}$ is continuous if and only if it is computable relative to some real parameter. This is perhaps the simplest example of the connection between computability and descriptive set theory, resulting in the field of \emph{effective descriptive set theory}. I will give a general introduction and mention some recent applications of the priority method to descriptive set theory, relying on machinery developped by Maontalbán, and independently by Debs and Saint Raymond. 

An ordered structure (M, <, ...) is called o-minimal when the definable subsets of M are finite unions of points and open intervals. That is, no matter the language, the definable sets in one free variable are precisely the sets that can be defined with the order alone. While the definition of o-minimality concerns only the definable subsets of M, there are striking consequences on all the definable sets of the structure. Remarkable applications of o-minimality have been recently found in several areas of mathematics, like algebraic geometric, number theory, probability and even outside mathematics in economics and neural networks. I will give a general introduction to the topic and then focus on some applications to group theory and ring theory. 

Quantitative models and quantitative analysis in Computer Science are receiving increased attention. The goal of this talk is to investigate quantitative automata and quantitative logics. Weighted automata on finite words have already been investigated in seminal work of Schützenberger (1961). They consist of classical finite automata in which the transitions carry weights. These weights may model, e.g., the cost, the consumption of resources, or the reliability or probability of the successful execution of the transitions. This concept soon developed a flourishing theory, as is exemplified and presented in several books by Eilenberg, Salomaa-Soittola, Kuich-Salomaa,

Berstel-Reutenauer, Sakarovitch, and the "Handbook of Weighted Automata".

We investigate weighted automata and their relationship to weighted logics. For this, we present syntax and semantics of a quantitative logic; the semantics counts ‘how often’ a formula is true in a given word. Our main result, jointly with Paul Gastin, extending classical results of Büchi, Elgot and Trakhtenbrot (1961), shows that if the weights are taken from an arbitrary semiring, then weighted automata and a syntactically defined fragment of our weighted logic are expressively equivalent. A corresponding result holds for infinite words. Moreover, this extends to quantitative automata investigated by Henzinger et al. for modeling limit average-type or discounting behaviors e.g. of power plants. Finally, we consider Fagin's seminal result (1974) characterizing NP in terms of existential second-order logic; this started the field of descriptive complexity theory. In very recent work, jointly with Guillermo Badia, Carles Noguera and Erik Paul, we obtained a weighted version of Fagin's result.