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标是有关流形的几何和拓扑知识的进步。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，医学，科学和工程的其他领域有着积极的联系。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。这个建议描述了我的长期目标在四个主要领域的近期工作：（一）光滑和连续的3维和4维的群作用，以及它们与规范理论的联系，（二）直角Artin基本群的4维流形的分类，（三）有限群作用于球面的乘积，以及（四）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，包括同伦空间的研究自等价的流形，研究群体行动的表面束在表面上，和发展的局部全球方法在外科理论的无限离散群体。这些调查涉及基本问题，如果目标得以实现，很有可能产生重大影响。该提案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“持久同源性”为分析大型数据集提供了有效的方法。几何角度是必不可少的：在大多数现实情况下的数学模型是多维的，涉及空间和时间的限制（例如，最近的“拓扑保持”神经网络模型的森林火灾预测和控制）。 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。我们的培训活动以公平和包容原则为基础，将有助于消除成功的障碍，并在知识型经济中促进更多样化的劳动力。