Minimal Models of Foliations

叶状结构的最小模型

• 批准号: EP/X029387/1
• 负责人: Calum Spicer
• 金额: $ 52.08万
• 依托单位: King's College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX029387%2F1
• 关键词: Minimal; Models; Foliations

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.

我的研究是一个跨学科项目，重点关注纯数学两个活跃领域的交叉点：动力系统和代数几何。我对代数簇上的叶状结构特别感兴趣。叶状结构的大部分研究始于 20 世纪初，作为理解微分方程解的一种方法。通常，可能无法找到某些感兴趣的微分方程的简单封闭形式解，但是，可以对微分方程的流或轨道的一般行为进行说明。这些流将底层空间划分为不相交的浸没子流形，称为叶，这种分解称为叶状结构。因此，对微分方程的研究被定性研究所取代（例如， 几何、拓扑等）叶状结构的属性。这是一个强有力的想法，叶状结构已经出现在非常广泛的背景下，例如拓扑学、几何学、数论和数学物理。理解叶状结构越来越被理解为各个领域的一个重要研究方向。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项重大发展的核心。我的大部分研究的总体思路是从相对较新的角度理解叶状结构的定性特性：通过发展叶状结构中的技术和思想，这些技术和思想受到簇的双有理几何研究的启发（并扩展），特别是这些思想 这种跨学科融合给叶状结构研究带来的关键见解是，应该可以以受控的方式调整叶状结构，从而简化其一些几何特性，但不会改变我们最感兴趣的叶状结构的方面，例如其动力学特性。此外，通过执行这些改变，我们期望将任意叶状结构转换为可分解为的叶状结构 “原子”叶状结构。如果能够实现这种分解策略，那么人们就能够将一般叶状结构和动力学的研究减少到对这些原子叶状结构的这些特性的研究，我们期望这项研究更加可行。