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Advances in Mean Curvature Flow: Theory and Applications

平均曲率流的进展：理论与应用

基本信息

批准号：
EP/S012907/1
负责人：
Reto Buzano
金额：
$ 78.14万
依托单位：
Queen Mary University of London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2019
资助国家：
英国
起止时间：
2019 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FS012907%2F1
关键词：
Advances Mean Curvature Flow Theory

项目摘要

This project aims to develop the theoretical framework of the singularity formation of the Mean Curvature Flow. The Mean Curvature Flow is a geometric flow that describes the motion of a surface. It was introduced by Mullins as a model for the formation of grain boundaries in annealing metals. It also appears as the flow to equilibrium of soap films, the motion of embedded branes in approximations of the renormalisation group flow in theoretical physics, boundaries of Ginzburg-Landau equations of simplified superconductivity and as a method of denoising in image processing. The results of this project, as well as the methods pioneered, will enable the next generation of applications.This proposal lies in the intersection of the EPSRC research areas Mathematical Analysis and Geometry and Topology with applications to Mathematical Physics and Algebra. It has underpinning relevance ranging from fundamental problems in theoretical physics to current issues in engineering. At the heart of these problems is a single system of geometric Partial Differential Equations (PDE). Such equations have had a tremendous impact in mathematics: they have been extremely successful with applications over diverse areas such as topology (Poincaré Conjecture, Geometrisation conjecture), Kähler geometry (minimal model problem), gravitation (Penrose inequality), image processing and material science (Martensite, nonlinear plate models).Geometric PDE are inherently nonlinear and therefore singularities are expected to occur. In fact, these singularities turn out to be useful - they tell us something about the underlying geometry of our object. This project will develop our understanding of singularities of systems of geometric PDE, an extremely important area in geometry, analysis and PDE theory which is relatively poorly understood. In principle, the singularities of systems of nonlinear partial differential equations may be unstructured, but due to their geometric origins, the singularities of systems of geometric PDE display a surprising order. This project seeks to characterise the mechanisms of the formation of singularities, obtain classifications of the singularity models, and to develop a geometric hierarchy of singularity models and quantitatively analyse their stability. Modern applications of geometric flows require a detailed understanding of singularities using an integrated approach combining algebra, analysis, geometry and topology.More precisely, the proposed research consists of the following themes: understanding the singularity formation of the Mean Curvature Flow in high codimension and in curved background spaces, developing new concentration compactness results to analyse the singularities and surgery procedures to geometrically undo the singularity formation, and finally exploring applications of the new theory to various fields of mathematics. The results pioneered in this project will have a direct and significant impact on geometry, analysis, and topology. Furthermore, the methodologies and techniques developed in this project can also be applied to a number of outstanding problems in physics, biology, engineering and computer imaging.

本项目旨在发展平均曲率流奇异性形成的理论框架。平均曲率流是一种描述曲面运动的几何流。它是由Mullins引入的，作为退火金属中晶界形成的模型。它还表现为肥皂膜的流动到平衡，嵌入的薄膜在理论物理中重正化群流动的近似下的运动，简化超导的金兹堡-朗道方程的边界，以及图像处理中的一种去噪方法。这个项目的结果，以及所开创的方法，将使下一代应用成为可能。这项建议位于EPSRC的研究领域数学分析和几何与拓扑学与应用于数学物理和代数的交叉点。它具有从理论物理的基本问题到工程中的当前问题的基础相关性。这些问题的核心是一个几何偏微分方程组(PDE)。这些方程在数学上产生了巨大的影响：它们在不同领域的应用中都非常成功，如拓扑学(Poincaré猜想、几何猜想)、Kähler几何学(最小模型问题)、引力(彭罗斯不等式)、图像处理和材料科学(马氏体、非线性板材模型)。几何偏微分方程天生是非线性的，因此预计会出现奇点。事实上，这些奇点被证明是有用的--它们告诉我们一些关于我们物体的基本几何结构的信息。这个项目将发展我们对几何偏微分方程组的奇点的理解，几何偏微分方程组是几何、分析和偏微分方程组理论中一个非常重要的领域，人们对它的了解相对较少。原则上，非线性偏微分方程组的奇点可能是无结构的，但由于其几何起源，几何偏微分方程组的奇点显示出令人惊讶的顺序。该项目旨在描述奇点形成的机制，获得奇点模型的分类，并建立奇点模型的几何层次并定量分析它们的稳定性。几何流的现代应用需要用代数、分析、几何和拓扑学相结合的方法来详细理解奇点，更确切地说，该研究包括以下主题：理解高余维和弯曲背景空间中平均曲率流的奇点形成，发展新的集中紧致性结果来分析奇点和几何地消除奇点形成的手术步骤，最后探索新理论在数学的各个领域的应用。该项目首创的结果将对几何图形、分析和拓扑产生直接而重大的影响。此外，该项目中开发的方法和技术还可以应用于物理学、生物学、工程学和计算机成像中的一些突出问题。