CBMS Conference: Introduction to the theory of valuations on convex sets

CBMS 会议：凸集估值理论简介

• 批准号: 1444411
• 负责人: Benjamin Jaye
• 金额: $ 3.5万
• 依托单位: Kent State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-11-15 至 2015-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1444411&HistoricalAwards=false
• 关键词: CBMS; Conference; Introduction; theory; valuations

This award will fund an NSF/CBMS conference at Kent State University in the summer of 2015 on the topic of valuations. Valuations are certain algebraic quantities introduced in order to better understand geometric problems. The main lecturer will be Semyon Alesker from Tel Aviv University. In his lectures, Alesker will give self-contained introduction to the subject, while also lecturing on the most recent developments on the structure of valuations. This will provide researchers from a variety of areas of mathematics with the opportunity to learn the techniques of the subject, and how they can be applied to a wide range of problems in analysis, geometry, and probability. While several classical texts have been written in this area, the extent to which Alesker's work has revolutionized the field makes it difficult for researchers to find a text which is self-contained, and at the same time describes the state of the art in the subject. The lectures (and subsequent monograph) of Semyon Alesker are going to fill this gap, benefitting many researchers and students. Interaction among participants at the conference from different parts of the mathematical sciences community is likely to bring fresh perspectives to an evolving field, stimulating the development of new avenues of research and uncovering new areas of application.The theory of valuations (finitely-additive measures on convex compact sets) is a classical part of convex geometry with traditionally strong relations to integral geometry. Its initial development was motivated by Dehn's solution of the 3rd Hilbert problem. The systematic development of continuous valuations was initiated by Hadwiger in the 40's and 50's, and further developed by P. McMullen (amongst others) in the 1970s and 1980s. In the proposed lecture series, the classical theory of Hadwiger and McMullen will be discussed in detail, before moving onto to the more recent developments which have arisen since the work of Klain and Schneider in the mid-1990s. One of the highlights of the lecture series will be an essentially complete proof of Alesker's irreducibility theorem, which, amongst its many consequences, proved a conjecture of McMullen dating from 1980. Alesker's more recent theory of valuations on manifolds will also be described. Throughout the lecture series, special attention will be drawn to applications of the theory of valuations to convex and integral geometry (questions involving the theory of intrinsic volumes, characterization theorems of invariant measures, and kinematic formulas). The interest in the subject is based upon its importance in order to achieve progress in solving different problems related to convex geometry and geometric probability.

该奖项将资助2015年夏天在肯特州立大学举行的关于估值主题的NSF/CBMS会议。 赋值是为了更好地理解几何问题而引入的某些代数量。 主讲人是来自特拉维夫大学的Semyon Alesker。 在他的讲座中，Alesker将对这一主题进行独立的介绍，同时还将讲述估值结构的最新发展。 这将为来自不同数学领域的研究人员提供学习该主题技术的机会，以及如何将其应用于分析，几何和概率中的各种问题。虽然几个经典的文本已经写在这方面，在何种程度上阿列斯克的工作已经彻底改变了该领域使得研究人员很难找到一个文本是独立的，并在同一时间描述了国家的艺术在这个问题上。 Semyon Alesker的讲座（和随后的专著）将填补这一空白，使许多研究人员和学生受益。 来自数学科学界不同部分的与会者之间的互动可能会为一个不断发展的领域带来新的视角，刺激新的研究途径的发展，并揭示新的应用领域。估值理论（凸紧集上的有限可加测度）是凸几何的经典部分，传统上与积分几何有着密切的关系。 它最初的发展是由Dehn的解决方案的第三希尔伯特问题。 连续估值的系统发展是由Hadwiger在40年代和50年代发起的，并由P. McMullen（以及其他人）在20世纪70年代和80年代进一步发展。 在拟议的系列讲座中，将详细讨论Hadwiger和McMullen的经典理论，然后再讨论自20世纪90年代中期Klain和Schneider的工作以来出现的最新发展。 讲座系列的亮点之一将是一个基本上完整的证明阿列斯克的不可约定理，其中，在其许多后果，证明了一个猜想的麦克马伦追溯到1980年。 阿列斯克的最近理论的估值流形也将被描述。 在整个系列讲座中，特别注意将提请应用理论的估值凸和积分几何（问题涉及理论的内在卷，特征定理不变的措施，和运动公式）。对这一主题的兴趣是基于它的重要性，以便在解决与凸几何和几何概率有关的不同问题方面取得进展。