Radon Transforms and Geometric Analysis and Tomography

摘要奖：DMS-9971828主要研究者：Eric L.格林伯格私家侦探将继续研究齐次和非齐次空间的分析和几何问题。他将使用technologesof组表示时，这些都是可用的，和microlocalanalysis沿着与积分变换，和theirgeneralizations不一定齐性空间。他的计划的一个主要重点将包括Radon变换的空间，特别是他们的注入性和restrictionproperties。他将使用对称空间作为指导分析空间的规律性较低。 他还将调查问题的几何不等式，也有几个复杂的变量使用的想法密切相关，但正式distinctfrom拉东变换。最后，他将考虑通过不变多项式和枚举参数的Radon变换（或Radon变换模有限群）的轨道折叠类似物。他将继续一个积极的旅行和研讨会参观者计划，该计划对费城北部的科学界产生了重大影响。成像设备在医学、工业、科学和信息技术中的使用频率越来越高。虽然工程，计算和人体工程学的考虑是至关重要的设计，生产和利用这些设备，基本的工作原理一般是数学。通常，一个数学定理说，信息可以恢复从几何定义的积分变换或偏微分方程。Radon变换在A.Cormack的CAT扫描仪的开创性工作中发挥了关键作用。今天，这种变换及其推广在不断扩大的数学分支中起着统一的作用。私家侦探 将集中在以下几个方面：1）几何不等式，它涉及的性质，如演变，宽度，横截面的阴影区; 2）函数的多个复变量，这可能被视为数据具有很大的规律性，并发挥重要作用，在数学分析和物理学; 3）齐性空间，其中存在大量的对称性是假设和开发的最终目标; 4）有限类比，其中光滑空间的连续调和分析被涉及有限和离散集的计数变元所取代

AbstractAward: DMS-9971828Principal Investigator: Eric L. GrinbergThe P.I. will continue to study problems of analysis and geometryon homogeneous and non-homogeneous spaces. He will use techniquesof group representations when these are available, and microlocalanalysis along with integral transforms, and theirgeneralizations to not-necessarily homogeneous spaces. A majorfocus of his program will include Radon transforms on symmetricspaces, especially their injectivity and restrictionproperties. He will use symmetric spaces as a guide to analysison spaces with less regularity. He will also investigateproblems in geometric inequalities and also several complexvariables using ideas closely related to, but formally distinctfrom Radon transforms. Finally, he will consider orbitfoldanalogs of Radon transforms (or Radon transforms modulo finitegroups) by means of invariant polynomials and enumerationarguments. He will continue an active program of travel andseminar visitors that has had a significant impact on thescientific community in North Philadelphia.Imaging devices are used with increasing frequency in medicine,industry, science, and information technology. While engineering,computational, and ergonomic considerations are critical in thedesign, production and utilization of these devices, thefoundational operating principle is generallymathematical. Typically, a mathematical theorem says thatinformation may be recovered from a geometrically definedintegral transformation or partial differential equation. TheRadon transform played a critical role in A.Cormack's pioneeringwork on the CAT scanner. Today, this transform and itsgeneralizations play a unifying role in an expanding set ofbranches of mathematics. The P.I. will focus on the followingareas: 1) geometric inequalities, which relate properties likevolume, width, cross-sections shadow areas; 2) functions ofseveral complex variables, which may be viewed as data with greatregularity and which play an important role in mathematicalanalysis and physics; 3) homogeneous spaces, where the existenceof a great deal of symmetry is assumed and its exploitation isthe ultimate goal; 4) finite analogs, where the continuousharmonic analysis of smooth spaces is replaced by countingarguments involving finite and discrete sets