Evolution equations in geometry and related fields

进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进化方程或应该导致进化方程应用的结果。第一部分是关于平均曲率流，它已经在材料科学中被使用和研究了将近一个世纪，用来模拟细胞、颗粒和气泡的生长。第二部分涉及处理非紧空间（规范群）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与其他类似流的研究一样，在纯数学和应用数学、理论物理、材料科学和工程等各个领域产生了巨大的影响。为了产生更广泛的影响，PI将继续为研究生和博士后学者提供指导和建议，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，该项目将研究稳定结构的问题；对于高余维的超曲面和子流形。在第二个项目中，将研究一种在许多情况下应该有用的处理差分同构群的新方法。事实上，在许多领域的许多问题中都出现了奇点。理解它们以及它们形成的环境是一个非常重要的问题。当对象处于某个规范空间中时，人们可以使用“外在”参考点来解决诸如此类的重要问题。然而，许多关键问题是固有的，没有规范的参考点。在这些问题中，无限维微分同构群（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。里奇流就是这样一个例子。PI期望MCF、Ricci流量和测量组的项目将对许多其他数学领域产生影响。该奖项反映了美国国家科学基金会的法定使命，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。

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.