Group actions on manifolds and complexes

流形和复形上的群作用

• 批准号: RGPIN-2016-05111
• 负责人: Hambleton, Ian
• 金额: $ 2.91万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=652891
• 关键词: Group; actions; manifolds; complexes

One of the unifying principles in geometry is that complex systems, such as configurations of planets and stars can often be understood by means of their symmetries. Familiar symmetries include the rotations or reflections of solids in space and the Lorentz transformations of space-time. Discrete invariants and groups of symmetry of continuous motions are studied in algebraic topology, while geometric topology is concerned with the properties of differential manifolds, or higher-dimensional surfaces. Geometry and topology is a flourishing subject for research, with broad connections to other areas of mathematics, science and engineering.***Symmetries of manifolds are related to algebra and number theory through group theory, and to partial differential equations and analysis through differential forms. This proposal describes my recent work in three main areas (i) finite group actions on products of spheres, (ii) smooth and continuous group actions on 4-dimensional manifolds and their connections to gauge theory, and (ii) infinite discrete group actions on high-dimensional manifolds. The new projects include the development of a new coarse geometry for discrete group actions, a comparison of finite groups of differentiable transformations with those defined by algebraic equations on algebraic surfaces, and the study of dynamical and ergodic aspects of infinite groups of transformations on high-dimensional spheres. The goal in each case is to improve our understanding of the many roles of symmetry in mathematical and physical problems.***This research proposal provides many high-level opportunities for training of prospective graduate students and postgraduate researchers, whose future work will have a broad impact on our society. Mathematical research skills related to geometry are now widely used outside of the university setting, such as in the modelling of complex systems, in architectural design, in control theory for aerospace, and in computer vision. Fundamental research at Canadian universities is the key to promoting the next generation of mathematicians and scientists, who are vitally needed to lead the Canadian knowledge-based economy.******************

几何学中的统一原理之一是，通常可以通过它们的对称性来理解复杂的系统，例如行星和恒星的配置。熟悉的对称性包括空间中固体的旋转或反射以及时空的洛伦兹变换。在代数拓扑中研究了离散的不变性和连续运动对称性组，而几何拓扑与差分流形或较高维表面的特性有关。几何和拓扑是研究的繁荣主题，与数学，科学和工程学的其他领域具有广泛的联系。***流形的对称性通过组理论与代数和数字理论有关，以及通过不同形式的部分微分方程和分析。 该提案描述了我最近在三个主要领域的工作（i）对球体产品的有限群体行动，（ii）对4维流形的平稳而连续的群体行动及其与规格理论的联系，以及（ii）无限的离散小组对高维流形的行动。新项目包括开发用于离散组动作的新的粗几何形状，对代数表面上的代数方程定义的有限转换组的有限群体的比较，以及对高维球体上无限转换的动态和怪异方面的研究。在每种情况下，目的是提高我们对对称性在数学和身体问题中的许多作用的理解。与几何相关的数学研究技能现在已在大学环境之外广泛使用，例如在复杂系统的建模，建筑设计中，航空航天控制理论和计算机视觉中的控制理论。加拿大大学的基本研究是促进下一代数学家和科学家的关键，他们在领导基于加拿大知识的经济中至关重要。******************