Ancient Solutions to Geometric Flows

几何流是描述表面运动的演化方程，或更高维度的类似物，其速度由其曲率决定。这个项目主要关注的流是平均曲率流，其特点是它演变一个表面，使其面积尽可能快地减少。几何流在各种物理问题中有着广泛的应用。例如，平均曲率流动出现在几种多相物理模型中对界面演化的描述中。此外，它还被用于材料科学中模拟细胞、颗粒和气泡的生长。几何流也被证明有非常有用的几何应用，特别是分类定理和几何不等式。这个项目将专注于平均曲率流的古老解决方案，这些解决方案在过去的所有时间都存在。迄今为止，这些解法中有一大类很重要的解法还没有得到阐明，但是一种新的构造、研究和分类它们的方法将会出现。该项目还包括对研究生的培训以及研讨会和暑期学校的组织，以帮助学生创造接触几何分析新发展的机会。古代解在奇点研究中具有重要作用；这些构成了气流在任何时候存在的障碍，因此了解它们的几何形状和行为是主要的兴趣。就像数学中典型的情况一样，为了得到令人满意的答案，注意力必须在某种程度上缩小。因此，该项目将重点关注限于平板区域的平均曲率流的古老解决方案。该项目将在所有维度上构建一个大型的新示例家族，包括对称的和不对称的；该项目还将构建许多永恒的例子，这些例子不是通过翻译演变的标准类型。此外，该项目还提出了一种对此类解决方案进行分类的新方法。最后，项目计划将结果应用于非坍缩环境，以获得整个解的性质。展望未来，有强有力的证据表明，这些方法适用于更广泛的几何流类别。该项目包括举办研讨会、培训和指导学生以及组织关于流动主题的讲习班的计划。倡议还旨在通过各种活动扩大代表性不足群体的参与。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。

Geometric flows are evolution equations that describe motions of surfaces, or higher dimensional analogues, with speeds determined by their curvatures. A flow of principal concern for this project is the Mean Curvature Flow, characterized by the property that it evolves a surface so that its area decreases as rapidly as possible. Geometric flows have extensive applications to various physical problems. For instance, mean curvature flow occurs in the description of the evolution of interfaces arising in several multiphase physical models. Moreover, it is used in material science to model cell, grain and bubble growth. Geometric flows have also proven to have extremely useful geometric applications, specifically classification theorems and geometric inequalities. This project will focus on ancient solutions to mean curvature flow, these being solutions that have existed for all times in the past. A large and important class of these solutions has hitherto resisted attempts at elucidation, but a new method of constructing, studying and classifying them will be brought to bear. The project also includes training of graduate students as well as the organization of workshops and summer schools to help create opportunities for students to be exposed to new developments in geometric analysis.Ancient solutions have an important role in the study of singularities; these constitute an obstruction to the existence for all times of the flow, and it is therefore of major interest to understand their geometry and behavior. As is typically the case in mathematics, focus has to be narrowed to some extent in order to obtain satisfying answers. Therefore this project will focus on ancient solutions to mean curvature flows that are confined to slab regions. This project will construct, in all dimensions, a large family of new examples, both symmetric and asymmetric; the project will also construct many eternal examples which are not of the standard kind that evolve by translation. Additionally, the project proposes a novel way of classifying such solutions. Finally, the project plans to apply results to the non-collapsed setting to obtain properties of entire solutions. Looking further ahead, there is strong evidence that these methods apply to a much wider class of geometric flows. The project includes plans to run seminars, train and mentor students and organize workshops on the subject of flows. The PI also aims to broaden participation of under-represented groups through various activities.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.