Radon Transforms and Geometric Analysis and Tomography

项目编号：dms -9971828首席研究员：Eric L. grinberg P.I.将继续研究齐次和非齐次空间的分析和几何问题。他将使用群表示技术，以及微局部分析和积分变换，并将其推广到不一定是齐次的空间。他的计划的主要焦点将包括对称空间上的Radon变换，特别是它们的注入性和限制性。他将使用对称空间作为分析不规则空间的指南。他还将研究几何不等式和一些复杂变量的问题，使用与Radon变换密切相关但形式上不同的思想。最后，他将通过不变多项式和枚举参数考虑Radon变换（或Radon变换模有限群）的轨道折叠类比。他将继续积极的旅行和研讨会访客计划，这对北费城的科学界产生了重大影响。成像设备在医学、工业、科学和信息技术领域的应用越来越频繁。虽然工程、计算和人体工程学的考虑在这些设备的设计、生产和利用中是至关重要的，但基本的操作原理通常是数学的。通常，一个数学定理说，信息可以从几何定义的积分变换或偏微分方程中恢复。在A.Cormack关于CAT扫描仪的开创性工作中，氡变换发挥了关键作用。今天，这个变换和它的推广在不断扩展的数学分支中起着统一的作用。P.I.将专注于以下领域：1)几何不等式，它与进化、宽度、横截面阴影区域等属性相关；2)多个复杂变量的函数，这些函数可以看作是具有很强规律性的数据，在数学分析和物理中起着重要作用；3)同质空间，假设存在大量对称，并将其利用为最终目标；4)有限类比，其中光滑空间的连续调和分析被涉及有限集和离散集的计数参数所取代

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