Group actions on manifolds and complexes

几何学中的一个统一原则是，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。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。离散不变量和连续运动的对称群在代数拓扑中进行研究，而几何拓扑则与微分流形或高维曲面的性质有关。几何和拓扑学是一个蓬勃发展的研究课题，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相关，通过微分形式与偏微分方程和分析相关。这个建议描述了我最近在三个主要领域的工作：（i）有限群作用于球面的乘积，（ii）四维流形上的光滑和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。新的项目包括开发一个新的粗几何离散组行动，比较有限群的微分变换与代数方程定义的代数曲面，以及研究动态和遍历方面的无限群的变换高维领域。在每一种情况下的目标是提高我们对对称性在数学和物理问题中的许多作用的理解。 这项研究计划为未来的研究生和研究生研究人员的培训提供了许多高层次的机会，他们未来的工作将对我们的社会产生广泛的影响。与几何相关的数学研究技能现在广泛应用于大学环境之外，例如复杂系统的建模，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，他们是领导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.