Fourier Analytic Approach to the Geometric Tomography

摘要：本文的主要目的是利用谐波分析方法研究凸几何和几何层析成像中的问题。几何层析成像是一个数学领域，人们根据这些固体的剖面和投影信息来研究固体的性质。它借鉴了理论数学许多领域的思想和方法，如积分与微分几何、统计学和傅立叶分析。但也许它最大的重叠是与凸几何。这个提议的一个主要组成部分是研究PI最近的工作中自然产生的问题，这些问题是用傅里叶解析方法来测量恒星体截面的最一般测量。这种方法在调和分析、概率论和凸几何之间提供了许多新的联系。PI打算继续他在这一领域的工作，并考虑由切片问题和同构Busemann-Petty问题引发的一些问题。PI还提出了一种新的方法来研究凸集关于高斯测度的几何性质。作为一个应用，PI考虑与高斯相关猜想有关的问题。这种方法的思想是用几何层析成像的语言重述猜想，并应用最近发展的傅立叶分析技术。几何层析成像在现实生活中有很多应用。它的目标与许多相关且通常是实用领域的目标相似。最著名的例子之一是经典计算机辅助断层扫描，其目的是通过线积分重建物体的密度。其他的例子还有晶体学、机器人、立体学和电子显微镜。这一建议的思想之一是通过几何层析成像将凸几何的理论结果与实际领域的结果联系起来。拟议研究的更广泛影响的另一个组成部分在于研究生和本科生的培训。事实上，在这个提案中提到的很多问题都是以这样一种方式陈述的，不仅对毕业生而且对本科生都是直观的。另一方面，许多问题的答案是相当违反直觉的，这激发了学生对这门学科和数学的兴趣！

ABSTRACT:The main goal of this proposal is to study problems in Convex Geometry and Geometric Tomography using methods of Harmonic Analysis. Geometric Tomography is the area of mathematics where one investigates properties of solids based on the information about sections and projections of these solids. It borrows ideas and methods from many fields of Theoretical Mathematics, such as integral and differential geometry, statistics, and Fourier analysis. But perhaps its biggest overlap is with Convex Geometry. A major component of this proposal is to study problems which arise naturally from the recent work of the PI on Fourier analytic approach to the case of the most general measure of sections of star bodies. This method provided a number of new links between harmonic analysis, probability theory and convex geometry. PI intends to continue his work in this area and to consider a number of questions motivated by Slicing Problem and Isomorphic Busemann-Petty problem. PI also proposes a new approach to the study of the geometric properties of convex sets with respect to the Gaussian measure. As an application PI consider problems connected to the Gaussian Correlation Conjecture. The idea, of this method, is to restate the conjecture in the language of Geometric Tomography and to apply recently developed Fourier Analytic techniques. Geometric Tomography has a lot of real life applications. It has goals similar to those of many related and often practical areas. One of the best known examples is Classical Computer Aided Tomography, which aims to reconstruct the density of objects by means of their line integrals. Other examples are crystallography, robotics, stereology, and electron microscopy. One of the ideas of this proposal is to connect theoretical results from Convex Geometry to those practical areas via Geometric Tomography. Another component of the broader impact of proposed research lies in the training of graduate and undergraduate students. Indeed, a lot of problems addressed in this proposal stated in such a way that they are intuitively clear not only to graduate but to undergraduate students. On the other hand, the answers for many of those problems are quite counterintuitive which stimulates an interest of students to the subject and Mathematics in general!