Ancient Solutions to Geometric Flows

几何流是描述表面运动的演化方程，或更高维的类似物，其速度由其曲率决定。该项目主要关注的流动是平均曲率流，其特征在于它演变成表面，使其面积尽可能迅速地减小。几何流在各种物理问题中有着广泛的应用。例如，平均曲率流出现在几个多相物理模型中出现的界面演化的描述中。此外，它在材料科学中用于模拟细胞，晶粒和气泡的生长。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。这个项目将集中在古代的解决方案，平均曲率流，这些解决方案已经存在的所有时间在过去。这些解决方案中的一个大而重要的类别迄今为止一直抵制试图阐明，但一种新的方法来构建，研究和分类他们将承担。 该项目还包括培养研究生，以及组织讲习班和暑期学校，以帮助创造机会，让学生接触到几何分析的新发展。古老的解决方案在奇点的研究中发挥着重要作用;这些构成了对所有时间的流动存在的障碍，因此了解它们的几何和行为是非常感兴趣的。 正如数学中的典型情况一样，为了获得令人满意的答案，必须在一定程度上缩小焦点。因此，这个项目将集中在古代的解决方案，平均曲率流，仅限于平板地区。 这个项目将在所有维度上构建一个新的例子大家庭，对称和不对称的;该项目还将构建许多永恒的例子，这些例子不是通过翻译进化的标准类型。 此外，该项目还提出了一种新的方法来分类这些解决方案。 最后，该项目计划将结果应用于非折叠设置，以获得整个解决方案的属性。 展望未来，有强有力的证据表明，这些方法适用于更广泛的一类几何流。 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问题的讲习班的计划。 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。

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.