Differential Equations and the Geometry of Manifolds

在更多的技术术语，该奖项的研究的主要组成部分是研究临界点的曲率泛函黎曼流形。一个重要的问题是理解模空间的紧性和临界度量的存在性，如反自对偶和极值Kaehler度量。在某些几何非坍缩假设下，适当的模空间可以通过添加具有类轨道奇点的度量来紧致化。一个长期的目标是将紧性定理扩展到包括崩溃的可能性，并找到其他应用到四维流形的微分拓扑。在四维中，Weyl能量的临界点被称为Bach平坦度量，它包含了反自对偶度量。这种度量有非常有趣的性质，可以用扭量理论来研究。在更高的维度，PI将研究二次曲率泛函，以及它们的变分性质，如临界点的稳定性和刚性。这有应用体积比较定理和胶合结果。 PI还对临界度量的非紧例子感兴趣，例如渐近局部欧几里得临界度量，并获得此类空间的最佳衰减率。这对理解模空间的结构和去除奇点有应用。PI研究的一个重要动机是理解空间的几何和拓扑之间的关系。后者，拓扑学，是研究空间的性质，这些性质在空间的连续拉伸或弯曲下是不变的，而前者，几何学，涉及理解距离，并且更加严格。例如，地球的表面是一个球体，人们通过计算大圆的弧长来测量它的距离（地球实际上是一个扁球体，但它非常接近完美的球形）。你可以想象通过向内推或拉小的或大的区域来扭曲几何形状，从而使地球变形。这样的变形不像我们熟悉的圆形地球那样吸引人，而且有很多方法可以使这个概念非常精确，以最小化某种总能量测量。这与物理原理直接相关，物理原理说，物理系统的状态将趋向于使总能量最小化的最终配置。 这个想法可以推广到更高维的物体，称为流形，这是我们星球表面的广义版本。例如，我们生活的空间是三维的，如果包括时间，我们就在一个四维的宇宙中。为了理解这些类型的高维物体，人们试图找到最好的方法来测量它们的距离，使用最少的能量，并最大限度地提高空间的对称性。上述项目是为了在这样的空间上定义适当的能量，并寻找最小化总能量的重要的最佳几何形状。

In more technical terms, the main component of research of this award is the study of critical points of curvature functionals on Riemannian manifolds. An important problem is to understand compactness of moduli spaces and existence of critical metrics, such as anti-self-dual and extremal Kaehler metrics. With certain geometric noncollapsing assumptions, the appropriate moduli spaces can be compactified by adding metrics with orbifold-like singularities. A long-term goal is to extend the compactness theorem to include the possibility of collapsing, and to find other applications to the differential topology of four-manifolds. In dimension four, critical points of the Weyl energy are known as Bach-flat metrics, which contains the class of anti-self-dual metrics. Such metrics have very interesting properties, and can be studied using twistor theory. In higher dimensions, the PI will investigate quadratic curvature functionals, and their variational properties, such as stability and rigidity of critical points. This has applications to volume comparison theorems and gluing results. The PI is also interested in non-compact examples of critical metrics, such as asymptotically locally Euclidean critical metrics, and obtaining optimal decay rates for such spaces. This has applications to the understanding of the structure of moduli spaces, and to the removal of singularities.An important motivation for the research of the PI is to understand the relationship between the geometry and the topology of a space. The latter, topology, is the study of properties of a space which are invariant under continuous stretching or bendings of a space, while the former, geometry, involves understanding distances and is more rigid. For example, the surface of our planet is a sphere, and one measures distances on it by computing arclengths of great circles (the Earth is actually an oblate spheroid, but it is very close to being perfectly spherical). One can imagine deforming the Earth by pushing in or pulling on small or large regions to warp the geometry. Such a deformation is less appealing that the familiar round Earth, and there are many ways to make this notion very precise in terms of minimizing some sort of total energy measurement. This is directly related to physical principles which say that the state of a physical system will tend towards a final configuration which minimizes the total energy. This idea can be generalized to higher-dimensional objects called manifolds, which are generalized versions of the surface of the our planet. For example, the space that we live in is three-dimensional, and if one includes time, we are in a four-dimensional universe. In order to understand these types of higher-dimensional objects, one attempts to find the best way to measure distances on them which use the least amount of energy, and maximize the symmetries of the space. The projects described above are to define appropriate energies on such spaces, and to seek out the important optimal geometries which minimize the total energy.