Differential Equations and the Geometry of Manifolds

用更专业的术语来说，这个奖项的主要研究内容是黎曼流形上曲率泛函的临界点的研究。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。在一定的几何非坍缩假设下，适当的模空间可以通过添加具有类轨道奇点的度量来紧化。一个长期的目标是扩展紧性定理，使其包含坍缩的可能性，并找到四流形微分拓扑的其他应用。在四维中，Weyl能量的临界点被称为Bach-flat度量，它包含了一类反自对偶度量。这样的度量具有非常有趣的性质，并且可以使用扭转理论进行研究。在更高的维度上，PI将研究二次曲率泛函及其变分性质，如临界点的稳定性和刚性。这适用于体积比较定理和粘合结果。PI也对临界指标的非紧化例子感兴趣，例如渐近局部欧几里得临界指标，并获得这种空间的最优衰减率。这对模空间的结构的理解和奇点的去除都有应用。研究PI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，是对空间在连续拉伸或弯曲下不变的性质的研究，而前者，几何学，涉及到对距离的理解，并且更加刚性。例如，我们星球的表面是一个球体，人们通过计算大圆的弧来测量它上的距离（地球实际上是一个扁球体，但它非常接近完美的球体）。你可以想象通过推或拉小或大的区域来扭曲地球的几何形状。这样的变形不像我们熟悉的地球那样吸引人，有很多方法可以使这个概念非常精确，就最小化某种总能量测量而言。这与物理原理直接相关，即物理系统的状态将趋向于使总能量最小的最终构型。这个想法可以推广到被称为流形的高维物体上，流形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。为了理解这些高维物体，人们试图找到最好的方法来测量它们上的距离，使用最少的能量，并最大化空间的对称性。上述项目是在这样的空间上定义适当的能量，并寻找重要的最佳几何形状，使总能量最小化。

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.