This talk is about turn based two player infinite duration zero sum games. Such objects have been originally introduced for logical purposes in descriptive set theory, and progressively became a key tool in automata theory and verification. The purpose of the first part of the talk will be to convey the meaning of such objects and justify their importance. A prominent problem in the computer-science context is the question of the complexity of knowing the winner of finite parity games. This problem has the interesting complexity status to be in NP and coNP wile not being known be in P! Though the problem is still open, there have been significant progress during the last two years. The second part of this talk concern a collaboration with Nathanaël Fijalkow that explain these new methods in an elementary way, and show at the same time their limitation.

I am a senior researcher of the Cnrs. I am mainly interested in logic, automata theory, games and categories related to verification. I have defended my PhD thesis in 2004, and my habilitation thesis in 2013 (the document in french). More precisely, I am interested in model theory, finite model theory, monadic-second order logic, databases, language theory, semigroups, automata theory and in particular quantitative forms of automata (weighted automata, regular cost functions, nominal automata), as well as games for verification and model-checking. I also started to investigate the connections between all these subjects and category theory.