Affine Invariants in Geometric Analysis

几何分析中的仿射不变量

• 批准号: 327635-2012
• 负责人: Stancu, Alina
• 金额: $ 2.19万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=695245
• 关键词: Affine; Invariants; Geometric; Analysis

Given the explosion of applications of affine geometry to image processing, information technology and stochastic geometry, new research is needed to investigate novel affine invariants for convex bodies. Additionally, there is a need to develop new techniques to tackle long-standing problems which resisted conventional methods. Our project addresses precisely these issues on the subject of affine invariants understood here at large as to include centro-affine and equi-affine invariants. While part of our study will employ new curvature flows to study affine invariants for convex bodies with sufficiently regular boundary, we will start an original study of new affine invariants for convex bodies which are lacking smooth positive Gauss curvature. The long-term goal is fitting the two classes of problems into a global coherent theory which includes all convex bodies disregarding a particular boundary structure. The direct consequences of the proposed project will be novel research results in the area of affine and isoperimetric inequalities, and solutions to generalized Minkowski problems arising in the Brunn-Minkowski-Firey theory of convex bodies. A particular attention will be given to open problems in the class of polytopes and connections to the theory of valuations. Adding to the significance of the project, our methods will range from convex, discrete and differential geometry to partial differential equations and curvature flows, while following the common thread of invariance under certain groups of transformations.

鉴于仿射几何形状在图像处理，信息技术和随机几何形状上的应用爆炸，需要进行新的研究来研究凸体的新型仿射不变。此外，有必要开发新技术来解决抗拒常规方法的长期存在的问题。 我们的项目准确地解决了这些问题，这些问题在这里广泛了解的仿期不变剂的主题包括质心和等等型不变性。尽管我们的一部分研究将采用新的曲率流来研究具有足够规则边界的凸体的仿射不变剂，但我们将开始对缺乏平滑阳性高斯曲率的凸面的新仿射不变的原始研究。长期目标是将两个类别的问题纳入一个全球连贯的理论中，其中包括所有无视特定边界结构的凸体。 拟议项目的直接后果将是仿射和等距不平等领域的新研究结果，以及在Brunn-Minkowski-Firey凸体理论中引起的通用Minkowski问题的解决方案。将特别关注多面体类别的开放问题以及与估值理论的联系。 除了项目的重要性外，我们的方法范围从凸，离散和微分几何形状到部分微分方程和曲率流，同时遵循某些转换组的共同线程。