Abstract
Differential equations are a natural way to model phenomena in mathematics, physics, biology, chemistry and many other areas. When considered under the ect of evolution in time, one speaks of `dynamical systems'. This proposal focuses on systems on three and four dimensional manifolds whose flow induces an S^1 x S^1- or an S^1 x R-action. These cases are called 'toric' and 'semitoric', respectively. We are interested in these systems in the context of symplectic and contact geometry. During the last 3 decades, these topics grew into a very active and influential field with ramifications to many areas in mathematics and physics. Symplectic geometry only exists in even dimensions and is the natural setting for Hamiltonian dynamics. Contact geometry only exists in odd dimension and provides the setting for Reeb dynamics. `Symplectization' and `contactization' admit in certain cases to pass from one to the other. When toric systems on compact symplectic manifolds were classified in the 1980s, the classification of toric systems in contact geometry followed not long after. But toric systems are very rare such that the search for less restrictive classes of systems began. About 10 years ago, semitoric systems were classified in the symplectic Hamiltonian context. The main goal of this proposal is to define and classify semitoric systems in contact geometry and study the interaction of toric and semitoric systems in contact geometry with semitoric systems in symplectic geometry.
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