Differential Geometry and Partial Differential Equations

许多物理现象都是由本质上是几何的方程来模拟的。一个最好的例子是广义相对论，它是为了使引力与狭义相对论兼容而发明的。爱因斯坦方程是描述时空几何和曲率的演化方程系统，在特定情况下出现。与许多其他几何/物理方程一样，爱因斯坦方程是非线性的，因此可以产生产生非常高曲率区域并导致数学奇异性的解，预计这些奇异性就是黑洞。奇点也出现在体积最小的曲面中，称为极小曲面。弯曲时空中的这种极小化曲面与时空的几何结构密切相关，它们的存在和性质反映了时空的性质，如引力场的能量分布，以及引力坍缩成黑洞的预言。这个项目的一个主要目标是更好地理解这种奇点的稳定性，特别是那些可能出现在时空中体积最小化表面的奇点。当我们研究弯曲的几何形状时，自然会有一些相关的数字，这些数字是这样的几何形状振动的基本频率，如果它被认为是理想化的鼓面。这个数列叫做谱。该项目的第二个目标是了解每单位体积具有最大基频的特殊几何形状。 该项目还包括对博士生和博士后研究员的重要培训。 PI还计划提供几何方面的推广讲座。这项研究的重点是微分几何，广义相对论和偏微分方程。在微分几何和广义相对论的推动下，PI将研究奇点，这可能会在高维体积最小化超曲面中形成。这个问题出现在正数量曲率流形和相关的广义相对论正质量定理的研究中。他将研究的问题是否可以通过一个小扰动的边界或周围的度量奇点扰动。他为这个问题提出了一些明确的测试案例。第二个主题是研究和比较广义相对论中出现的三种准定域质量。这些是霍金、巴特尼克和布朗-约克质量。关于这些量的相对大小提出了几个问题。布朗-约克质量和最近的多面体比较定理之间的联系将被调查与建议巩固两者之间的联系。第三个主题，他的研究几何将是研究光谱几何和相关问题的最小子流形。PI最近分析了盘的高Steklov本征值的尖锐界限，并将该研究扩展到下一个最简单的情况下的环和莫比乌斯带。最后，他计划调查的等变特征值最大化问题的surface.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Many physical phenomena are modeled by equations that are essentially geometric in nature. A prime example is the theory of General Relativity, which was invented to make gravity compatible with Special Relativity. The Einstein equations are a system of evolution equations that describe the spacetime geometry and curvature, which arise in specific situations. Like many other geometric/physical equations the Einstein equations are nonlinear and thus can produce solutions that develop very high curvature regions and lead to mathematical singularities, which are expected to be black holes. Singularities also arise in surfaces that minimize volume, so called minimal surfaces. Such minimizing surfaces in a curved spacetime are intimately connected with the geometric structure of the spacetime, and their existence and properties reflect properties of the spacetime such as the energy distribution of the gravitational field, and the prediction of gravitational collapse to a black hole. A major goal of this project is to better understand the stability properties of such singularities, especially those that can arise in volume minimizing surfaces in spacetime. When we study curved geometries, there are naturally associated numbers that are the fundamental frequencies with which such a geometry would vibrate if it were thought of as an idealized drumhead. This sequence of numbers is called the spectrum. A second goal of this project is to understand the special geometries that have the largest fundamental frequencies per unit volume. The project also includes significant training of PhD students and post-doctoral fellows. The PI also plans to deliver outreach lectures in geometry.The focus of this research is in the areas of differential geometry, general relativity, and partial differential equations. In work that is motivated both by differential geometry and general relativity the PI will study the singularities, which may form in higher dimensional volume minimizing hypersurfaces. This issue comes up in the study of manifolds of positive scalar curvature and the related positive mass theorem of general relativity. He will study the question of whether singularities can be perturbed away by a small perturbation of the boundary or the ambient metric. He poses some explicit test cases for this question. The second main theme is the study and comparison of three quasi-local masses that have arisen in general relativity. These are the Hawking, Bartnik, and Brown-York masses. Several questions are posed concerning the relative sizes of these quantities. The connections between the Brown-York mass and recent polyhedral comparison theorems will be investigated with a suggestion for solidifying the connections between the two. The third main theme of his research in geometry will be the study of spectral geometry and related questions on minimal submanifolds. The PI has recently analyzed sharp bounds on the high Steklov eigenvalues of the disk and will extend that study to the next simplest cases of the annulus and the Mobius band. Finally he plans to investigate the equivariant eigenvalue maximization problem for surfaces.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.