Degenerate Microlocal Methods in Geometric Analysis

AbstractAward：DMS-0505709首席研究员：Rafe Mazzze PI提出了几项活动，所有活动都适合非紧和奇异流形几何分析中出现的线性和非线性椭圆问题的框架。他proposescontinue他的合作与Vasy的新microlocaltechniques建设的预解式上的地方discriminicspaces任意秩。这种方法已经导致了一个新的证明亚纯延续这个预解式，这反过来将有应用在散射理论和非线性椭圆方程的渐近对称爱因斯坦空间。这一部分的目标是创建一个可扩展的方法，从N体薛定谔理论，打开对更高秩空间的分析，以一个纯粹的微局部的方法。这将导致在这些空间的几何散射理论的进展。另一部分讨论了非紧Ricci平坦Einstein空间和具有锥奇点的Einstein空间的形变理论。即使对于曲面，PI也希望这些方法能给出关于常曲率和Ricci孤子度量存在性的新结果。最后，PI建议继续担任SUMaC的主任和教师，SUMaC是一个为数学天才高中生开设的暑期住宿项目，同时也继续他的其他指导和推广活动。这项建议的总体研究领域涉及高维弯曲空间几何与这些空间上某些自然方程解的行为之间的关系。其中一个例子是几何散射理论，它试图描述波的行为，在大的时间尺度上测量，并了解这些是如何受到底层空间曲率的影响。PI已经开发了方法来研究这些问题的自然类的空间出现在几个不同的分支数学。其他的例子包括研究爱因斯坦方程的解，这些方程被认为是描述大尺度空间形状的可能配置。有许多这样的解;组织它们的一种方法是描述它们的变形理论，也就是说，看看它们如何不断地改变。PI建议为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奇是技术的发展，从一个领域称为微局部分析这些几何问题。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，希望这项建议的结果可能会在这些其他领域产生一些影响。PI还积极参与各级培训和指导。他的SUMaC计划，现在进入第十一个年头，是一个成功的激励天才高中生继续他们的数学学习。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.