Linear partial differential equations on singular spaces

在这个项目中，PI将研究与具有奇异黎曼度量的流形相关的线性偏微分方程组理论的几个不同方面。其中包括具有圆锥奇点和边缘奇点的流形上波动方程的奇点的传播和绕射，以及具有散射度规的流形上含时薛定谔方程基本解的结构。我们将根据需要研究和改进傅里叶-布罗斯-伊戈尔尼泽变换等工具。描述波传播的线性偏微分方程组(如经典波动方程和薛定谔方程)在物理学和几何学中具有基本的重要性。波在弯曲空间中的传播常常以一种较小的方式涉及下面空间的几何形状。当空间是“奇异的”时，无论是在广度上无限的意义上，还是在有限区域内具有退化性质的意义上，几何与偏微分方程解的性质之间的关系尤其令人着迷，但人们对此知之甚少。

In this project, the PI will investigate several different aspects of thetheory of linear partial differential equations associated to manifoldswith singular Riemannian metrics. These include the propagation anddiffraction of singularities for the wave equation on manifolds with conicand edge singularities, and the structure of the fundamental solution forthe time-dependent Schrodinger equation on manifolds with scatteringmetrics. Tools such as the Fourier-Bros-Iagolnitzer transform will bestudied and refined as needed.Linear partial differential equations (PDEs) that describe wave propagation(such as the classical wave equation and the Schrodinger equation) are offundamental importance in physics and geometry. The propagation of waveson curved spaces often involves the geometry of the underlying space in asubtle manner. When the space is `singular,' either in the sense of beinginfinite in extent or of having degenerate properties in a finite region,the relationship between geometry and the properties of solutions to PDEsis especially fascinating and poorly understood.