Convex Geometry and Geometric Invariant Theory

这个项目的总主题是探索凸几何和组合格论之间的深层联系，并在每个方向上追求应用。从几何的观点来看，这个项目将特别关注凸体、星形集、混合体和对偶混合体的结构，目的是刻画在各种群作用下不变的赋值和集函数，并将这些结果应用于几何概率、凸几何、几何层析和Grassmannians分析等问题。这个项目还将继续研究凸几何和代数组合理论之间许多以前未被开发的深层次联系，特别是有限晶格上不变赋值和运动学公式的组合理论的发展(其中不变性是关于自同构群的作用)。这项研究的一个中心目标是发展和应用Hadwiger的不变赋值刻画定理和偏序集背景下的经典运动学公式的组合类似物，特别是凸几何中出现的组合结构。凸几何和赋值理论处理如何测量(或在有限特征的情况下，列举)并最终表征几何对象的内在特征的基本问题。例如，从有限的数据重建关于几何对象的信息，例如关于投影和阴影的信息(体视学)或切片和横截面的信息(断层摄影)。这些技术反过来又导致了许多应用，如生物技术(如分子生物学)、经济和金融(分析有限资源在人口中的有效和公平分配)以及计算机图形学(信息的可视化显示)。

The overall theme of this project is to explore the deep connections between convex geometry and combinatorial lattice theory, and to pursue applications in each direction. From the geometric viewpoint, this project will focus particular attention on the structure of convex bodies, star-shaped sets, mixed volumes, and dual mixed volumes, with a goal of characterizing valuations and set functions that are invariant under various group actions and applying these results to problems in geometric probability, convex geometry, geometric tomography, and analysis on Grassmannians. This project will also pursue an investigation of the many deep and previously unexploited connections between convex geometry and algebraic combinatorial theory, with the particular end of the development of a combinatorial theory of invariant valuations and kinematic formulas on finite lattices, (where the invariance is with respect to the action of an automorphism group). A central goal of this investigation is the development and application of combinatorial analogues to Hadwiger's characterization theorem for invariant valuations and to classical kinematic formulas in the context of partially ordered sets, with a special focus on the combinatorial structures that arise in convex geometry. Convex geometry and the theory of valuations treat the fundamental question of how to measure (or in the case of finite features, to enumerate) and ultimately to characterize intrinsic features of geometric objects. Examples include the reconstruction of information about a geometric object from limited data, such as information about projections and shadows (stereology) or slices and cross-sections (tomography). These techniques lead in turn to many applications, such as those in biotechnology (such as molecular biology), economics and finance (analysis of efficient and equitable distributions of limited resources over a population), and computer graphics (the visual display of information).