Evolution equations in geometry and related fields

几何及相关领域的演化方程

• 批准号: 2104349
• 负责人: Tobias Colding
• 金额: $ 38.54万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104349&HistoricalAwards=false
• 关键词: Evolution; equations; geometry; related; fields

Evolution equations are fundamental objects in the sciences, describing how natural phenomena change over time. The two main parts of this project concern evolution equations or results that should lead to applications to evolution equations. The first part is about mean curvature flow which has been used and studied in materials science for almost a century to model things like cell, grain, and bubble growth. The second is related to the new methods for dealing with the collections of transformations of (the gauge group) for non-compact spaces with applications to Ricci flow. The mean curvature flow’s computational and theoretical study and its applications as well as that of other similar flows have had enormous impact in diverse areas of pure and applied mathematics, theoretical physics,material science, and engineering. For broader impacts the PI will continue mentoring and advising graduate students and post-doctoral scholars, write a second graduate level book on “Heat equation in Analysis, Geometry and Probability", and disseminate their work via lectures, workshops and conferences.In mean curvature flow (MCF) the project will study questions about stable structures; both for hypersurfaces and submanifolds of higher codimension. In the second project a new way of dealing with the diffeomorphism group will be investigated that should be useful in many settings. Indeed, in many problems across a wide swath of areas singularities occur. Understanding them and the set where they form becomes a question of great importance. When the objects sit inside some canonical space, one can use ``extrinsic'' reference points to make progress on important questions like these. However, many key problems are defined intrinsically and there are no canonical reference points. In those problems, the infinite dimensional diffeomorphism group (the gauge group) becomes a major issue and dealing with it, and understanding it, becomes a major obstacle. Ricci flow is such an example. The PI expects that the projects on MCF, Ricci flow and gauge groups will have implications to a number of other fields of mathematics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

演化方程是科学中的基本对象，描述自然现象如何随时间变化。 这个项目的两个主要部分涉及发展方程或结果，应导致应用到发展方程。 第一部分是关于平均曲率流，它在材料科学中已经被使用和研究了将近一个世纪，用来模拟细胞、晶粒和气泡的生长。 第二部分是关于处理非紧空间（规范群）的变换集合的新方法及其在Ricci流中的应用。 平均曲率流的计算和理论研究及其应用，以及其他类似的流动，在纯数学和应用数学，理论物理，材料科学和工程的各个领域产生了巨大的影响。 为了更广泛的影响，PI将继续指导和建议研究生和博士后学者，撰写第二本研究生水平的书《分析，几何和概率中的热方程》，并通过讲座，研讨会和会议传播他们的工作。在平均曲率流（MCF）中，该项目将研究关于稳定结构的问题;无论是超曲面还是更高余维的子流形。 在第二个项目中，将研究一种新的方法来处理同构群，这在许多情况下都是有用的。 事实上，在许多问题中，在很宽的范围内都会出现奇点。 了解它们以及它们形成的集合成为一个非常重要的问题。 当对象位于某个规范空间内时，人们可以使用“外部”参考点来在这些重要问题上取得进展。 然而，许多关键问题是内在定义的，没有规范的参考点。 在这些问题中，无限维规范群（规范群）成为一个主要问题，处理和理解它成为一个主要障碍。 利玛窦流就是这样一个例子。 PI期望MCF、Ricci流和规范群的项目将对其他数学领域产生影响。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。