O-minimality and dynamical systems

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Mathematical logic has long played a role in delineating between what is formally possible and impossible. For example, Gödel's incompleteness theorem shows that no consistent formal system sophisticated enough to prove simple theorems in mathematics can demonstrate its own consistency. This type of result places a limit on the scope of what is formally knowable in mathematics. In recent years, model theorists have taken on a constructivist approach to the use of logic inside mathematics. Working below the "Gödel barrier'', they have formalized simple properties of structures that have desirable finiteness properties and which, at the same time, give rise to a rich collection of definable sets. Among the most successful of these properties is one that lives on the border of model theory and analytic geometry, namely, o-minimality. This subject has already shown its usefulness by providing key insights into the foundations of dynamical systems, hybrid systems, and learning theory through neuronal networks and has helped settle major open problems in real algebraic geometry and number theory. The motivation for my research is the investigation of dynamical systems whose solutions exhibit certain asymptotic behaviours. I am particularly interested in Hilbert's 16th problem, one of the famous list of 23 problems posed by the German mathematician David Hilbert in 1900; it remains unsolved to this day. I have developed a new approach to this problem using o-minimality, solving a very special case of it. The main lines of my research over the next five years are as follows: generalize my approach to Hilbert's 16th problem to obtain more significant cases of it; simplify our understanding of Pfaffian geometry, in order to make it more suitable for applications; and the investigation of generalizations of o-minimality relevant to the understanding of phenomena arising from dynamical systems.