Singularities of Minimal Hypersurfaces and Lagrangian Mean Curvature Flow

奇点自然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。本研究的重点是极小子流形和拉格朗日平均曲率流的奇异性。最小子流形是最小曲面的高维推广，它模拟了肥皂膜。他们的研究非常经典，但关于它们在奇点附近的行为，一些基本问题仍未得到解答。特殊拉格朗日子流形是一种特殊的极小子流形，由于其在弦理论中的出现，近年来受到了极大的关注。拉格朗日平均曲率流是一个自然演化过程，我们可以通过它来尝试找到特殊的拉格朗日子流形，然而奇点的出现再次成为基本的困难。本研究项目旨在了解家庭中出现的奇点的一些共同特征，而不是以往大多数研究集中在孤立的奇点上。这个问题的进展将应用于几何学和分析学的许多其他相关问题。该项目还包括一些教育活动，旨在提高各级STEM领域的兴趣和成功。具体而言，教育活动包括：针对三至五年级学生为期一周的暑期数学圈；支持本科生的研究工作；几何学和拓扑学暑期本科生研讨会的延续；继续为刚开始读研究生的学生提供一个过渡项目，以方便他们过渡到研究生院。在许多几何问题中，从奇点可以得到的最基本的信息是它的切锥或切流。这种“切线对象”的唯一性问题是几何分析中最基本的问题之一，只有在奇点被孤立的情况下才能很好地理解。PI将研究两种相关设置中的非孤立奇点：最小超曲面和拉格朗日平均曲率流。在这些情况下，为理解切锥和切线流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最小超曲面的分类；极小超曲面的一般光滑性；以及在拉格朗日平均曲率流的奇点处进行手术的可能性，这与托马斯-丘猜想有关。除了这些特定的应用之外，PI期望与上述问题相关的新方法将在几何分析的其他领域得到应用。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。

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.