Minimal Models of Foliations

我的研究是一个跨学科的项目，专注于两个活跃的纯数学领域：动力系统和代数几何。我对代数族上的叶的情况特别感兴趣。对叶的研究大多始于20世纪初，作为理解微分方程解的一种方式。通常不可能找到一些感兴趣的微分方程的简单的闭合形式的解，然而，也许可以描述一些关于微分方程的流动或轨道的一般行为。这些流动将下面的空间划分成不相交的、被称为叶的子流形，这种分解被称为叶。因此，对微分方程的研究被定性的(例如，几何的、拓扑的等)研究所取代。叶理的性质。这是一个强大的想法，叶在非常广泛的背景下出现，例如，拓扑学、几何学、数论和数学物理。理解叶正在越来越多地被理解为广泛领域中的一个基本研究方向。我特别感兴趣的是在代数几何的背景下的叶状结构。在过去的几年里，对叶状结构的研究一直是最近几个主要发展的核心。我的许多研究的总体想法是从相对较新的角度来理解叶状结构的定性性质：通过发展叶状结构理论中的技术和思想来启发(和扩展)变种的二元几何，特别是最小模型程序的思想。这种跨学科的融合给叶状结构的研究带来的关键见解是，应该可以通过可控的方式来调整叶状结构，以简化它的一些几何性质，但这并不改变我们最感兴趣的叶理的方面，例如它的动力学性质。此外，通过执行这些改变，我们希望将任意的叶理转变为分解成`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层几何和动力学的研究，而转向研究这些原子叶层的这些性质，我们预计这项研究将更加可行。

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.