Applications of Geometric Microlocal Analysis

Mazzeo提出的研究集中在与奇异和非紧空间上的几何分析有关的一些主题上。他正在研究各种类型的曲率方程，无论是在紧致分层空间上，还是在具有渐近正则几何的完备流形上，都是通过椭圆和抛物线方法。这里的特定主题包括具有规定的奇异结构的常曲率和爱因斯坦度量，例如具有圆锥点或边的常曲率和爱因斯坦度量，或者即使在任意深度的分层空间上的某些问题的常曲率度量和爱因斯坦度量，以及此类空间上Ricci流技术的发展。他还将在奇异空间的谱几何的几个部分进行研究，包括研究带边流形上的解析扭转和圆锥退化的光滑流形，以及多边形空间上更经典的谱问题。这个项目的其他部分包括分析由种群遗传学中的Wright-Fisher模型产生的一类分段光滑区域上的退化抛物问题，例如多面体区域。他还研究了一种非线性Dirichlet-to-Neumann算子的正则性理论，该算子产生于双曲空间中适当嵌入的极小曲面的研究中。最后，他还分析了一类半线性类Toda椭圆组的奇异解，它直接应用于一些新引入的弦场理论。总的来说，Mazzeo的研究受到这样一个中心原则的驱动，即某些类型的奇异空间--特别是那些被称为分层空间的空间--与几何中最常见的光滑流形一样自然地产生，并且这两类空间应该被认为是同等重要的。然而，奇异空间的几何分析基础仍然处于相对原始的状态，而Mazzeo的工作旨在开发出广泛适用于此类空间上的许多自然几何和分析问题的技术，包括线性和非线性。这项工作是在对许多公认重要的特殊问题的仔细审查的指导下进行的，这些问题既源于纯数学中普遍研究的问题，也源于数学和物理交界处出现的问题。人们的期望是，这些自然问题将推动一般理论的形成，从而使其易于理解和有用，反过来，这套新的技术应该有助于回答这些既定领域中的许多感兴趣的问题。

Mazzeo's proposed research focuses on a number of themes related to geometric analysis on singular and noncompact spaces. He is studying various types of curvature equations, both on compact stratified spaces and on complete manifolds with asymptotically regular geometries, via both elliptic and parabolic methods. Particular topics here include constant curvature and Einstein metrics with prescribed singular structure, for example with conic points or edges, or for certain problems even on stratified spaces of arbitrary depth, and also the development of Ricci flow techniques on such spaces. He will also conduct research in several parts of spectral geometry on singular spaces, including the study of analytic torsion on manifolds with edges and on smooth manifolds degenerating conically, and on more classical spectral problems on the space of polygons. Other parts of this project include the analysis of a class of degenerate parabolic problems on piecewise smooth, e.g. polyhedral, domains, arising from the Wright-Fisher model in population genetics. He is also investigating the regularity theory for a nonlinear Dirichlet-to-Neumann operator which arises in the study of properly embedded minimal surfaces in hyperbolic space. Finally, he is also analyzing the singular solutions of a class of semilinear Toda-like elliptic systems, which has direct application to some newly introduced string field theories.In general terms, Mazzeo's research is driven by the central tenet that certain types of singular spaces -- specifically the ones known as stratified spaces -- arise just as naturally as smooth manifolds, which are the most common objects of study in geometry, and both classes of spaces should be considered as comparably important. However, the foundations of geometric analysis on singular spaces are still in a relatively primitive state, and Mazzeo's work is aimed at developing techniques which are meant to be broadly applicable to many natural geometric and analytic problems, both linear and nonlinear, on such spaces. This work is guided by a close examination of many particular problems of recognized importance, arising from both commonly studied questions in pure mathematics and from problems emerging at the interface of mathematics and physics. The expectation is that these natural problems will drive the formulation of the general theory so as to make it accessible and useful, and in turn, this new set of techniques should help answer many problems of interest in these established fields.