Minimal Models of Foliations

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My research is an interdisciplinary project focused at the interface of two active fields of pure mathematics: dynamical systems and algebraic geometry.I am particularly interested in the case of foliations on algebraic varieties.Much of the study of foliations began in the early 20th century as a way to understandthe solutions of differential equations. Often it may not be possible to finda simple closed form solution to some differential equation of interest, however,it may be possible to say something about the general behaviour of the flowsor orbits of the differential equation.These flows partition the underlying space into disjointimmersed submanifolds, called leaves, and this decomposition is referred to as a foliation.The study of the differential equation is therefore replaced by the studyof the qualitative (e.g., geometric, topological, etc.) properties of the foliation. This is a powerful idea and foliations have arisenin a very wide range of contexts, for instance,topology, geometry, number theory, and mathematical physics.Understanding foliations is increasingly being understood as an essential research directionin a wide range of fields. My particular interests are in foliations in the context of algebraic geometrywhere, in the past few years, the study of foliations has been at the heart of several recentmajor developments.The general idea of much of my research is to understand the qualitative propertiesof foliations from a relatively recent perspective: by developing techniques and ideas in foliation theoryinspired by (and extending) the study of the birational geometry of varieties, in particular, the ideas of the Minimal Model Program.The key insight that this interdisciplinary fusion brings to the study of foliations isthat it should be possible to tweak a foliation in a controlled way which simplifies some of its geometric properties,but which does not alter the aspects of the foliation we are most interested in, for instance its dynamical properties.Moreover, by performing these alterations we expect to transform an arbitrary foliation into one which decomposes into ``atomic"foliations. If one could realize this decomposition strategy then one be able to reduce the study of foliation geometry and dynamics in generalto the study of these properties on these atomic foliations, where we expect this study to be much more feasible.