Complex Analysis and Geometry

这个项目研究由方程给出的几何图形，很像我们三维空间中的平面或曲面上的曲线。然而，这里研究的数字是更高的，有时是无限维的，并且它们是以涉及复数的方程给出的，而不是曲线和曲面可能出现的实数。尽管这些高维和复杂的几何图形很难想象，因为它们不适合我们熟悉的空间，但事实证明，它们与我们对物理世界的描述最相关，无论是在原子和亚原子水平(量子理论)，还是在天文尺度(相对论)上。几何学及其应用中的一个核心概念是曲率。直线的曲率为零：一般来说，曲率衡量的是图形与直线的偏差。该项目将探索在不同情况下出现的图形的曲率特性，以及已知曲率为正或负时可以得出的影响。该项目包括三个部分。第一个问题涉及紧致复流形上的Kahler度量空间，紧致复流形是已知具有非正曲率的无限维空间。我们的目标是找到这个空间中连接点的最短曲线，空间的对称性，并研究对称性如何与某些被广泛研究的泛函相互作用，例如马布奇的能量。第二和第三部分涉及一般的全纯Hilbert丛或Banach丛，它们被赋予了度量。一个问题是，如果丛及其度量经历了一定的变换，度量的曲率属性将如何变化。另一种是利用曲率性质，通过外推技术估计调和分析和复分析中出现的算子。这些方法将来自复杂的分析和几何、算子理论和偏微分方程式。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project studies geometric figures that are given by equations, much like curves in the plane or surfaces in our three dimensional space. However, the figures studied here are higher, sometimes infinite dimensional, and they are given in terms of equations that involve complex numbers, rather than the real numbers that would occur with curves and surfaces. Even if these high dimensional and complex geometric figures are hard to visualize, as they do not fit in the space we are familiar with, they have proved to be most relevant to the description of our physical world, both on the atomic and subatomic levels (quantum theory), and on the astronomical scale (relativity). A central notion in geometry and its applications is that of curvature. A straight line has zero curvature: in general, curvature measures deviation of figures from straight. The project will explore curvature properties of figures that arise in various contexts, and implications that can be drawn if the curvature is known to be positive or negative.The project has three parts. The first concerns the space of Kahler metrics on a compact complex manifold, an infinite dimensional space that is known to have non-positive curvature. The goal is to find shortest curves connecting points in this space, symmetries of the space, and investigate how symmetries interact with certain much studied functionals, such as Mabuchi's energy. The second and third parts concern general holomorphic Hilbert or Banach bundles, endowed with metrics. One problem is how curvature properties of the metrics change if the bundle and its metric are subjected to certain transformations. Another is to exploit curvature properties to estimate operators that arise in harmonic and complex analysis, by the technique of extrapolation. The methods will come from complex analysis and geometry, operator theory, and partial differential equations.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.