Geometry and Topology of Manifolds

流形的几何和拓扑

• 批准号: RGPIN-2022-04539
• 负责人: Hambleton, Ian
• 金额: $ 2.26万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758984
• 关键词: Geometry; Topology; Manifolds

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.

我的研究计划的总体目的是关于多种形态和拓扑拓扑的知识的进步。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。几何和拓扑是研究繁荣的主题，并与其他数学，医学，科学和工程领域具有积极的联系。流形的对称性与代数和数字理论有关，以及通过差分形式的部分微分方程和分析。该提议描述了我在四个主要领域的长期目标（i）在第3和4方面的平稳而连续的小组行动，以及它们与量规理论的联系，（ii）对4个模型的分类，具有直角的Artin基本组，（iii）Spheres产品的有限群体动作，以及（IV）Symmetries和（IV）Smplace和Topolical Kervere Sormodic sormodic sormodic sormodic sormole sormole sormods sormods sormods sormods sormods sormods sormods sormolds sormolds sormolds sormolds sormolds。 在接下来的五年中，我计划打开多个新方向，包括研究歧管同质拷贝自我等量的研究，对表面上的表面束中的群体作用的研究以及无限离散组手术理论中局部全球方法的发展。这些探究线解决了基本问题，如果实现目标，则具有很大的重大影响潜力。该建议为准研究生和研究生提供了高级跨学科的机会，他们的未来工作将塑造我们的社会。几何拓扑通过复制DNA链的打结和联系在生物学中应用。代数拓扑提供了通过“持续同源性”来分析大数据集的有效方法。几何视角是必不可少的：在大多数现实情况下，数学模型是多维的，涉及空间和时间约束（例如，最近用于森林火灾预测和控制的“拓扑拓扑”神经网络模型）。 加拿大大学的基本研究对于提供下一代数学家和科学家至关重要。基于公平和包容性原则，我们的培训活动将有助于消除成功的障碍，并在基于知识的经济中促进更多样化的劳动力。