Geometry and Topology of Manifolds

我的研究计划的总体目标是提高关于流形的几何和拓扑的知识。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、医学、科学和工程的其他领域有着活跃的联系。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。这项建议描述了我最近在四个主要领域的长期目标所做的工作：(I)3和4维的光滑和连续的群作用，以及它们与规范理论的联系；(Ii)具有直角Artin基本群的4-流形的分类；(Iii)球面乘积上的有限群作用；以及(Iv)光滑和拓扑Kervaire流形的对称性。在接下来的五年里，我计划开辟几个新的方向，包括流形的同伦自等价空间的研究，曲面上曲面丛中的群作用的研究，以及无限离散群的外科理论中局部-整体方法的发展。这些调查路线涉及基本问题，如果实现这些目标，很有可能产生重大影响。这一提议为未来的研究生和研究生提供了高水平的跨学科机会，他们未来的工作将塑造我们的社会。几何拓扑学在生物学中有应用，通过复制DNA链的打结和连接。代数拓扑学通过“持久同调”为分析大数据集提供了有效的方法。几何观点是必不可少的：大多数现实情况下的数学模型是多维的，涉及空间和时间限制(例如，最近用于森林火灾预测和控制的“保持拓扑”的神经网络模型)。加拿大大学的基础研究对于培养下一代数学家和科学家至关重要。我们的培训活动基于公平和包容的原则，将有助于消除成功的障碍，并在以知识为基础的经济中促进更多元化的劳动人口。

The overall objective of my research program is the advancement of knowledge about the geometry and topology of manifolds. 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 active connections to other areas of mathematics, medicine, 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 recent work towards my long-term goals in four main areas (i) smooth and continuous group actions in dimensions 3 and 4, and their connections to gauge theory, (ii) classification of 4-manifolds with right angled Artin fundamental groups, (iii) finite group actions on products of spheres, and (iv) symmetries of smooth and topological Kervaire manifolds. Over the next five years I plan to open up several new directions, including the study of the space of homotopy self-equivalences of manifolds, the study of group actions in surface bundles over surfaces, and the development of local-global methods in surgery theory for infinite discrete groups. These lines of inquiry address basic problems, with a high potential for significant impact if the goals are achieved. This proposal offers high-level interdisciplinary opportunities for prospective graduate students and postgraduates, whose future work will shape our society. Geometric topology has applications in biology, through the knotting and linking of replicating DNA strands. Algebraic topology provides effective methods for analyzing large data sets, through "persistent homology". A geometrical perspective is essential: mathematical models in most realistic situations are multi-dimensional and involve spatial as well as temporal constraints (e.g. the recent "topology-preserving" neural net models for forest fire prediction and control). Fundamental research at Canadian universities is critical to providing the next generation of mathematicians and scientists. Our training activities, based on principles of equity and inclusion, will help to remove barriers to success and promote a more diverse workforce in a knowledge-based economy.