Applications of Geometric Microlocal Analysis

几何微局部分析的应用

• 批准号: 1105050
• 负责人: Rafe Mazzeo
• 金额: $ 33.3万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105050&HistoricalAwards=false
• 关键词: Applications; Geometric; Microlocal; Analysis

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.

Mazzeo的拟议研究重点介绍了许多与关于奇异和非划界空间的几何分析有关的主题。他正在通过椭圆形和抛物线方法研究各种类型的曲率方程，无论是在紧凑的分层空间还是在具有渐近规则几何形状的完整歧管上。此处的特定主题包括具有规定奇异结构的恒定曲率和爱因斯坦指标，例如带有圆锥点或边缘的指标，或者即使在任意深度的分层空间上，也出于某些问题，以及在此类空间上的RICCI流动技术的发展。他还将在奇异空间的光谱几何形状的几个部分进行研究，包括在具有边缘和平滑的歧管上的歧管上的分析扭转研究，并在圆锥形上退化，以及多边形空间上更经典的光谱问题。该项目的其他部分包括对分段平滑的一类退化抛物线问题进行分析，例如多面体，域，是由人口遗传学中的赖特 - 法派模型引起的。他还正在研究针对非线性迪里奇（Neumann）运算符的规律性理论，该理论是在研究正确嵌入的最小表面的研究中出现的。最后，他还正在分析一类半连续的Toda样椭圆形系统的单数解决方案，该解决方案直接应用于一些新介绍的弦字段理论。总的来说，Mazzeo的研究是由中心宗旨驱动的，这些中心宗旨是由某些类型的奇异空间（特别是阶层的空间）（通常是自然的），以及最常见的对象，最常见的对象，是自然的，这是最常见的对象，并且是最常见的对象，并且是最常见的，并且是最常见的，并且是自然的。认为很重要。然而，在奇异空间上的几何分析基础仍然处于相对原始的状态，而马祖奥的工作旨在开发技术，这些技术旨在广泛地适用于许多天然的几何和分析问题，包括线性和非线性，在此类空间上。这项工作的指导是对许多特定的公认重要问题的仔细研究，这是由于纯数学中常见的问题以及在数学和物理学的界面中出现的问题引起的。 期望这些自然问题将推动一般理论的提出，从而使其可访问和有用，进而，这种新的技术应该有助于回答这些既定领域的许多兴趣问题。