Applications of Geometric Microlocal Analysis

Mazzeo提出的研究重点是与奇异和非紧空间的几何分析相关的一些主题。他正在研究各种类型的曲率方程，既在紧化分层空间上，也在具有渐近正则几何的完全流形上，通过椭圆和抛物方法。这里的特定主题包括恒定曲率和爱因斯坦度量与规定的奇异结构，例如与圆锥点或边，或某些问题，甚至在任意深度的分层空间，以及里奇流技术在这些空间的发展。他还将在奇异空间上的光谱几何的几个部分进行研究，包括边流形的解析扭转研究和二次退化光滑流形的解析扭转研究，以及更经典的多边形空间上的光谱问题。本项目的其他部分包括分析一类在分段光滑上的退化抛物问题，例如多面体，区域，产生于群体遗传学中的Wright-Fisher模型。他还研究了非线性dirichlet - toneumann算子的正则性理论，该算子是在双曲空间中适当嵌入最小曲面的研究中出现的。最后，他还分析了一类半线性类toda椭圆系统的奇异解，这对一些新引入的弦场理论有直接的应用。一般来说，Mazzeo的研究是由一个核心原则驱动的，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——和光滑流形一样自然地出现，这是几何中最常见的研究对象，这两类空间都应该被认为是相当重要的。然而，奇异空间几何分析的基础仍然处于相对原始的状态，Mazzeo的工作旨在开发一种技术，这种技术意味着可以广泛适用于这些空间上的许多自然几何和解析问题，无论是线性的还是非线性的。这项工作的指导是对许多公认重要的特殊问题的仔细研究，这些问题既来自纯数学中通常研究的问题，也来自数学和物理界面出现的问题。期望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，这套新技术将有助于回答这些既定领域中感兴趣的许多问题。

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.