Singularities of Minimal Hypersurfaces and Lagrangian Mean Curvature Flow

奇点自然而然地出现在科学和数学的许多领域。有时它们是需要克服的技术障碍，而在其他情况下，它们编码了正在考虑的问题的基本特征。本文主要研究极小子流形的奇异性和拉格朗日平均曲率流的奇异性。极小子流形是极小曲面的高维推广，用于模拟肥皂膜。他们的研究非常经典，但关于他们在奇点附近的行为，一些基本问题仍然没有得到回答。特殊拉格朗日子流形是一类特殊的极小子流形，由于它在弦理论中的出现，近年来引起了人们的极大关注。拉格朗日平均曲率流是一个自然演化过程，我们可以通过它来寻找特殊的拉格朗日子流形，然而奇点的出现又是一个基本的困难。这项研究项目旨在了解家庭中出现的奇点的一些共同特征，而不是以往大多数专注于孤立奇点的研究。这一问题的进展将适用于几何和分析中的许多其他相关问题。该项目还包括几项教育活动，目的是在各级提高人们对科学、技术和经济管理领域的兴趣并取得成功。具体地说，教育活动包括：针对三至五年级学生的为期一周的暑期数学圈；支持本科生研究；继续暑期本科生几何学和拓扑学研讨会；继续为研究生提供过渡到研究生的桥梁计划。从许多几何问题中的奇点可以获得的最基本信息是它的切锥或切线流。这种“相切物体”的唯一性问题是几何分析中最基本的问题之一，只有当奇点是孤立的时才能很好地理解它。PI将在两个相关的环境中研究非孤立奇点：极小超曲面和拉格朗日平均曲率流。为了解切锥和流在这些设置下的唯一性而开发的工具也将应用于重要的几何问题：在无穷远处具有规定行为的极小超曲面的分类；极小超曲面的一般光滑性；以及与Thomas-Yau猜想有关的拉格朗日平均曲率流在奇点的手术的可能性。除了这些具体的应用，PI预计与上述问题相关的新方法将应用于几何分析的其他领域。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Singularities arise naturally in many areas of science and mathematics. Sometimes they are technical obstacles to overcome, while in other cases they encode essential features of the problem being considered. In this research the focus is on singularities of minimal submanifolds and of the Lagrangian mean curvature flow. Minimal submanifolds are higher dimensional generalizations of minimal surfaces, which model soap films. Their study is very classical, but some basic questions remain unanswered about their behavior near singularities. Special Lagrangian submanifolds are a particular kind of minimal submanifolds, which have received a great deal of attention recently due to their appearance in string theory. The Lagrangian mean curvature flow is a natural evolution process by which we can attempt to find special Lagrangian submanifolds, however once again the appearance of singularities is the basic difficulty. This research project aims to understand some common features of singularities which appear in families, in contrast with most previous research that focused on isolated singularities. Progress on this problem will have applications to many other related questions in geometry and analysis. The project also includes several educational activities aimed at increasing interest and success in STEM fields at all levels. Specifically, the educational activities include: a week long summer math circle aimed at students in grades three to five; support for undergraduate research; the continuation of a summer undergraduate workshop in geometry and topology; and the continuation of a bridge program for beginning graduate students to ease transitioning to graduate school.The most basic information that can be obtained from a singularity in many geometric problems is its tangent cone or tangent flow. The question of the uniqueness of such "tangent objects" is one of the most basic problems in geometric analysis and it is only well understood when singularities are isolated. The PI will study non-isolated singularities in two related settings: minimal hypersurfaces and the Lagrangian mean curvature flow. The tools developed for understanding the uniqueness of tangent cones and flows in these settings will also have applications to important geometric problems: the classification of minimal hypersurfaces with prescribed behavior at infinity; the generic smoothness of minimal hypersurfaces; and the possibility of surgeries at singularities along the Lagrangian mean curvature flow in connection with the Thomas-Yau conjecture. In addition to these specific applications, the PI expects that the new methods introduced in connection with the above problems will have applications in other areas of geometric analysis.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.