Applications of Geometric Microlocal Analysis

Mazzeo提出的研究集中在一些主题有关的几何分析奇异和非紧空间。他正在研究各种类型的曲率方程，无论是在紧凑的分层空间和完整的流形与渐近规则的几何形状，通过椭圆和抛物的方法。这里的特定主题包括常曲率和爱因斯坦度量与规定的奇异结构，例如与圆锥点或边缘，或某些问题，甚至在分层空间的任意深度，也发展的里奇流技术等空间。他还将进行研究的几个部分谱几何奇异空间，包括研究分析扭转流形的边缘和光滑流形退化圆锥，并在更经典的频谱问题的空间多边形。该项目的其他部分包括分析一类退化抛物问题的分段光滑，如多面体，域，所产生的赖特-费舍尔模型在人口遗传学。他还调查了正则性理论的非线性狄利克雷到诺依曼算子，这是在研究中出现的适当嵌入极小曲面的双曲空间。最后，他还分析了一类半线性Toda-like椭圆系统的奇异解，这类系统直接应用于一些新引入的弦场论。一般来说，Mazzeo的研究是由中心原则驱动的，即某些类型的奇异空间-特别是被称为分层空间的空间-就像光滑流形一样自然出现，这是几何学中最常见的研究对象，两类空间都应该被认为是同等重要的。然而，奇异空间上的几何分析的基础仍然处于相对原始的状态，Mazzeo的工作旨在开发技术，这意味着广泛适用于许多自然的几何和分析问题，包括线性和非线性，在这样的空间。这项工作的指导下，仔细检查许多特定的问题，公认的重要性，所产生的共同研究的问题，在纯数学和新出现的问题的接口数学和物理。 人们期望这些自然问题将推动一般理论的形成，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.