Singularities of Minimal Hypersurfaces and Lagrangian Mean Curvature Flow

奇点在科学和数学的许多领域中自然出现。有时它们是需要克服的技术障碍，而在另一些情况下，它们包含了正在考虑的问题的基本特征。在这项研究中，重点是奇异性的极小子流形和拉格朗日平均曲率流。极小子流形是极小曲面的高维推广，它模拟肥皂膜。他们的研究是非常经典的，但关于他们在奇点附近的行为，一些基本问题仍然没有答案。特殊拉格朗日子流形是一类特殊的极小子流形，由于其在弦理论中的出现，近年来受到了广泛的关注。拉格朗日平均曲率流是一个自然的演化过程，通过它我们可以试图找到特殊的拉格朗日子流形，然而奇点的出现再次是基本的困难。这个研究项目的目的是了解一些共同的特点，出现在家庭中的奇点，在以往的研究，主要集中在孤立的奇点。这个问题的进展将应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，旨在提高各级对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的为期一周的暑期数学圈;支持本科生研究;继续举办几何和拓扑学暑期本科生讲习班;和继续一个桥梁计划，为开始研究生，以减轻过渡到研究生院。最基本的信息，可以从一个奇点在许多几何问题是它的切向锥或切向流。问题的唯一性，这样的“切线对象”是一个最基本的问题，在几何分析，它是只有很好地理解时，奇点孤立。PI将研究两个相关设置中的非孤立奇点：极小超曲面和拉格朗日平均曲率流。为理解切锥的唯一性而开发的工具和在这些设置中的流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;以及与丘-游猜想有关的在沿着拉格朗日平均曲率流的奇点处进行手术的可能性。除了这些具体的应用外，PI还希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。

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.