Nonlinear Geometric Flows: Ancient Solutions, Non-Compact Surfaces, and Regularity

数学和物理学中一些最重要的问题与对奇点的理解有关。这些是物理量行为中的异常，用于测量这些量的数学表达式失效了。这可能与对黑洞或湍流对癌细胞积累的理解有关。这些物理现象通常用涉及时间和空间的微分方程来描述。研究这类方程解的定性行为往往有助于更好地理解相关的物理问题。为了捕获一个奇点，我们使用了一个放大程序，它允许我们将焦点集中在潜在的奇点附近，并利用所涉及的微分方程的缩放特性。由于空间和时间尺度的变化，爆炸后的新解决方案是针对所有空间和时间定义的——换句话说，它是一个“全局解决方案”。这些解的分类，在可能的情况下，为奇点分析和相关的物理问题提供了新的见解。本课题研究非线性几何椭圆型和抛物型偏微分方程整体解的存在性、唯一性和定性行为问题。重点是古代解的分类和奇点的研究。分析技术和几何技术之间的相互作用对于解决拟议的研究活动至关重要。该项目涉及广泛的数学活跃领域，特别是非线性偏微分方程、几何和经典分析。PI打算研究数学问题在其他学科的应用，如量子场论和图像处理。研究结果将通过各种会议和研究论文的出版向研究界传播。将特别强调培养博士生和鼓励少数民族和妇女在数学领域取得成功。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。

Some of the most important problems in mathematics and physics are related to the understanding of singularities. These are anomalies in the behavior of a physical quantity where the mathematical expressions that are used to measure such quantities break down. It may be related to the understanding of black holes or turbulence to the accumulation of cancer cells. These physical phenomena are often described via a differential equation which involves time and space. Studying the qualitative behavior of the solutions of such equations often results in a better understanding of the related physical problem. To capture a singularity one uses a blow up procedure which allows one to focus near the potential singularity and use the scaling properties of the differential equation involved. Because of the change in the scaling of space and time, the new solution after the blow up is defined for all space and time -- in other words it is a "global solution". The classification of such solutions, when possible, sheds new insight into the singularity analysis and the related physical problem.This project addresses the questions of existence, uniqueness and qualitative behavior of global solutions to nonlinear geometric elliptic and parabolic partial differential equations. Emphasis is given to the classification of ancient solutions and the study of singularities. The interplay between analytical and geometric techniques will be crucial for the resolution of the proposed research activities. The project links a wide range of active fields of mathematics, in particular nonlinear partial differential equations, geometry and classical analysis. The PI intends to study the applications of the mathematical problems to other disciplines such as quantum field theory and image processing. Results will be disseminated to the research community at various meetings and by publication of research articles. Special emphasis will be given to the training of PhD students and the encouragement of minorities and women to pursue a successful career in mathematics.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.