Degenerate Microlocal Methods in Geometric Analysis

摘要：项目负责人：Rafe mazzzeo在非紧和奇异流形几何分析中出现的线性和非线性椭圆型问题的框架下，提出了几个活动。他建议继续与Vasy合作，研究在任意秩的局部对称空间上构造解的新微局部技术。这种方法已经导致了该解的亚纯延拓的新证明，它将在散射理论和渐近对称爱因斯坦空间上的非线性椭圆方程中得到应用。这部分建议的目标是创建多用途的方法，改编自n体薛定谔理论，将高秩空间的分析开放到纯粹的微局部方法。这将导致这些空间的几何散射理论的进步。本文的另一部分讨论了非紧化Ricci平坦爱因斯坦空间和具有二次奇异点的爱因斯坦空间的变形理论。即使对于曲面，PI也希望这些方法能给出关于常曲率和里奇孤子度量存在性的新结果。最后，PI建议继续担任SUMaC的主任和老师，SUMaC是一个有数学天赋的高中生暑期住宿项目，同时也继续他的其他指导和推广活动。本提案的总体研究领域涉及研究高维弯曲空间的几何形状与这些空间上某些自然方程的解的行为之间的关系。一个例子是几何散射理论，它试图描述在大时间尺度上测量的波的行为，并观察它们如何受到底层空间曲率的影响。PI已经开发了研究自然空间类问题的方法，这些问题出现在几个不同的数学分支中。其他的例子包括对爱因斯坦方程解的研究，该方程被认为描述了大尺度空间形状的可能构型。有很多这样的解决方案；组织它们的一种方法是描述它们的变形理论，即观察它们是如何被连续改变的。PI建议对这些空间中的某些空间研究这些问题。他的方法的一个关键的新颖之处在于从一个叫做微局部分析的领域发展到这些几何问题的技术。这些主题的性质将导致与不同领域的数学家以及物理学家的互动，并且希望本提案的结果可能对这些其他领域产生一些影响。私家侦探还积极参与各级培训和指导工作。他的suma项目已经进入了第11个年头，在激励有天赋的高中生继续学习数学方面取得了成功。他还积极参与研究生培训，并与年轻的博士后研究人员合作。

AbstractAward: DMS-0505709Principal Investigator: Rafe MazzzeoThe PI proposes several activities, all fitting in the frameworkof linear and nonlinear elliptic problems arising in geometricanalysis on noncompact and singular manifolds. He proposescontinuing his collaboration with Vasy on new microlocaltechniques for constructing the resolvent on locally symmetricspaces of arbitrary rank. This approach has already led to a newproof of a meromorphic continuation of this resolvent, which inturn will have applications in scattering theory and to nonlinearelliptic equations on asymptotically symmetric Einstein spaces. Agoal of this portion of the proposal is to create versatilemethods, adapted from N-body Schroedinger theory, to open theanalysis on higher rank spaces to a purely microlocal approach.This should lead to advances in geometric scattering theory onthese spaces. The other part of the proposal discussesdeformation theory of noncompact Ricci flat Einstein spaces andof Einstein spaces with conic singularities. Even for surfaces,the PI expects these methods to give new results about theexistence of constant curvature and Ricci solitonmetrics. Finally, the PI proposes to continue his role asdirector and teacher in SUMaC, the residential summer program formathematically talented high school students, and also tocontinue his other mentoring and outreach activities.The overall area of research in this proposal concerns a study ofthe relationship between the geometry of higher dimensionalcurved spaces and the behaviour of solutions of certain naturalequations on these spaces. One example is a subject calledgeometric scattering theory, which seeks to describe thebehaviour of waves, measured at large timescales, and to see howthese are influenced by the curvature of the underlyingspace. The PI has developed methods to study such problems on anatural class of spaces which appear in several differentbranches of mathematics. Other examples include the study ofsolutions to the Einstein equations, which are supposed todescribe possible configurations for the large-scale shape ofspace. There are many such solutions; one way to organize them isto describe their deformation theory, i.e. to see how they may bealtered continuously. The PI proposes to study these questionsfor certain of these spaces. A key novelty of his methods is thedevelopment of techniques from a field called microlocal analysisto these geometric problems. The nature of these topics will leadto interactions with mathematicians in diverse areas as well asphysicists, and it is hoped that the results of this proposal mayhave some impact in these other areas. The PI is also activelyinvolved in training and mentoring at various levels. His SUMaCprogram, now entering its eleventh year, is a proven success inmotivating gifted high school students to continue theirmathematical studies. He has also been active in graduatetraining and in collaborating with young postdoctoralresearchers.