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

该奖项将资助2015年夏天在肯特州立大学举行的NSF/CBMS会议，主题是估值。赋值是为了更好地理解几何问题而引入的某些代数量。主要讲师将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，同时也会讲授估值结构的最新发展。这将为来自各个数学领域的研究人员提供学习该学科技术的机会，以及如何将这些技术应用于分析、几何和概率论中的广泛问题。虽然在这个领域已经写了一些经典的文本，但Alesker的工作在一定程度上彻底改变了这个领域，这使得研究人员很难找到一个独立的文本，同时描述了这个主题的艺术状态。Semyon Alesker的讲座（以及随后的专著）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门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，并发现新的应用领域。赋值理论（凸紧集上的有限加性测度）是凸几何的一个经典部分，传统上与积分几何有很强的联系。它最初的发展是由Dehn对第三个希尔伯特问题的解决所激发的。连续估值的系统发展是由哈德维格在20世纪40年代和50年代发起的，并在20世纪70年代和80年代由P. McMullen（以及其他人）进一步发展。在拟议的系列讲座中，将详细讨论哈德维格和麦克马伦的经典理论，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。本系列讲座的重点之一将是完整地证明Alesker的不可约定理，该定理证明了McMullen在1980年提出的一个猜想。Alesker最近的流形估值理论也将被描述。在整个系列讲座中，将特别关注赋值理论在凸几何和积分几何中的应用（涉及内在体积理论、不变测度的表征定理和运动学公式的问题）。对这门学科的兴趣是基于它在解决与凸几何和几何概率相关的不同问题方面取得进展的重要性。

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.