Minimal Models of Foliations

我的研究是一个跨学科的项目，集中在两个活跃的纯数学领域的接口：动力系统和代数几何。我对代数变种上的叶的情况特别感兴趣。叶化的研究大多始于20世纪初，当时是为了理解微分方程的解。对于某些感兴趣的微分方程，通常不可能找到简单的闭形式解，然而，可能有可能说明微分方程的流或轨道的一般行为。这些流动将底层空间划分为不相交的子流形，称为叶，这种分解被称为叶状。因此，对微分方程的研究被对叶理的定性（如几何、拓扑等）性质的研究所取代。这是一个强大的思想，叶状运算在非常广泛的背景下出现，例如，拓扑，几何，数论和数学物理。了解叶状结构越来越被认为是一个重要的研究方向，在广泛的领域。我特别感兴趣的是代数几何背景下的叶状结构，在过去的几年里，叶状结构的研究一直是最近几个主要发展的核心。我的大部分研究的总体思路是从一个相对较新的角度来理解叶理的定性性质：通过发展叶理理论中的技术和思想，这些技术和思想受到（并扩展）品种的双几何研究的启发，特别是最小模型程序的思想。这种跨学科的融合为叶理研究带来的关键见解是，应该有可能以一种可控的方式调整叶理，从而简化其一些几何性质，但不会改变我们最感兴趣的叶理方面，例如其动力学性质。此外，通过进行这些改变，我们期望将任意的叶理转变为分解为“原子”叶理的叶理。如果人们能够实现这种分解策略，那么人们就能够减少对叶理几何和动力学的研究，从而研究这些原子叶理的性质，我们希望这项研究在这些方面更加可行。

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.