The parameterization method in dynamical systems This paper presents an overview of the parameterization method to study invariant manifolds in dynamical systems. Our purpose is to present the key ideas in a digestible way to the reader not familiar with the topic. First, we will state the general problem of finding invariant manifolds, and then we will discuss specific problems: manifolds associated with fixed points or equilibria; general normally hyperbolic invariant manifolds and invariant

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... ori with quasi-periodic dynamics. The latter problem is included in the so-called KAM theory. Keywords: parameterization methods, invariant manifolds, KAM theory, small divisors and not so small ones. MSC2010 Subject Classification: 37C15, 37C20, 34C45, 34D05. Elitza Maneva P versus NP: the star problem of the mathematics of computation The problem "P versus NP" is one of the seven Millennium Prize Problems of the Clay Mathematics Institute, whose solution would be awarded one million dollars. In this article we present in an informal manner the problem and its origin, giving along the way examples of computational problems of diferent levels of hardness, some non-trivial algorithms, the definition of Turing machine-the mathematical model of a computer-and the concept of polynomial reduction between problems. The most advanced part of the