Group actions on manifolds and complexes

几何学中的统一原则之一是，复杂的系统，如行星和恒星的配置，通常可以通过它们的对称性来理解。常见的对称性包括固体在空间中的旋转或反射，以及时空的洛伦兹变换。代数拓扑学研究连续运动的离散不变量和对称群，几何拓扑学研究的是微分流形或高维曲面的性质。几何学和拓扑学是一门蓬勃发展的研究学科，与数学、科学和工程的其他领域有着广泛的联系。 流形的对称性通过群论与代数和数论联系起来，通过微分形式与偏微分方程和分析联系起来。这一建议描述了我最近在三个主要领域的工作：(I)球面乘积上的有限群作用，(Ii)4维流形上的光滑连续群作用及其与规范理论的联系，以及(Ii)高维流形上的无限离散群作用。这些新的项目包括发展一种新的用于离散群作用的粗几何，比较有限可微变换群与由代数曲面上的代数方程定义的可微变换群，以及研究高维球面上无限变换群的动力学和遍历方面。每一种情况的目的都是为了提高我们对对称性在数学和物理问题中的多种作用的理解。 这一研究建议为培养未来的研究生和研究生研究人员提供了许多高水平的机会，他们未来的工作将对我们的社会产生广泛的影响。与几何有关的数学研究技能现在被广泛应用于大学以外的环境，如复杂系统的建模、建筑设计、航空航天控制理论和计算机视觉。加拿大大学的基础研究是促进下一代数学家和科学家的关键，他们是引领加拿大知识经济的关键。

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.