Collaborative Research: Floer Theory and Topological Entropy

合作研究：弗洛尔理论和拓扑熵

• 批准号: 2304206
• 负责人: Viktor Ginzburg
• 金额: $ 34.04万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304206&HistoricalAwards=false
• 关键词: Collaborative; Research; Floer; Theory; Topological

Hamiltonian systems constitute a broad class of dynamical systems where energy dissipation can be neglected. For example, the planetary motion in celestial mechanics, the flow of an incompressible ideal fluid and the motion of a charged particle in an electro-magnetic field are usually treated as Hamiltonian dynamical systems. Topological entropy is an important invariant of a dynamical system, measuring its complexity and originating in physics and information theory. The PIs will develop new methods and tools to study topological entropy of Hamiltonian dynamical systems, utilizing ideas from topological data analysis. Conversely, this research has a potential to contribute to the field of topological data analysis and applied questions including image and pattern recognition. The work involves integration of research, education and training young scientists. It will have impact in the areas of higher education and dissemination of knowledge, within the field and to a wider scientific community, and it will increase participation of individuals from underrepresented groups in mathematics.On a more technical level, the main theme of the project is the interaction between Floer theory and symplectic topology on one side and Hamiltonian dynamics and, in particular, topological entropy on the other. The PIs will study topological entropy of compactly supported Hamiltonian diffeomorphisms and certain Reeb flows from the perspective of Floer theory. The project builds on the PIs’ recent work and focuses on barcode entropy introduced by the PIs, which is a Floer theoretic counterpart of topological entropy and is closely related to it. The key new and distinguishing feature of the PIs’ approach to Floer theoretic aspects of topological entropy is that barcode entropy is based on neither exponential growth of Floer homology – there is no growth in the Hamiltonian setting – nor on topological properties of the map such as the growth of free homotopy classes of periodic orbits. The PIs will also study the behavior of the gamma-norm under iterations in the Hamiltonian or contact setting. Most of the projects will require developing new techniques applicable to other questions, and interactions with areas outside symplectic geometry and dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

哈密顿系统是一大类可以忽略能量耗散的动力系统。例如，天体力学中的行星运动、不可压缩理想流体的流动和带电粒子在电磁场中的运动通常被视为哈密顿动力学系统。拓扑熵是动态系统的一个重要不变量，用来衡量系统的复杂性，起源于物理学和信息论。pi将利用拓扑数据分析的思想，开发新的方法和工具来研究哈密顿动力系统的拓扑熵。相反，这项研究有可能有助于拓扑数据分析领域和应用问题，包括图像和模式识别。这项工作包括把研究、教育和培训年轻科学家结合起来。它将在高等教育和知识传播领域、在该领域内以及对更广泛的科学界产生影响，并将增加来自代表性不足群体的个人对数学的参与。在更技术性的层面上，该项目的主要主题是Floer理论和辛拓扑之间的相互作用，一方面是汉密尔顿动力学，特别是拓扑熵。pi将从花理论的角度研究紧支持哈密顿微分同态和某些Reeb流的拓扑熵。该项目建立在pi最近工作的基础上，重点研究pi引入的条形码熵，这是拓扑熵的花理论对应物，并且与拓扑熵密切相关。pi方法在拓扑熵的花理论方面的关键新特征是条形码熵既不是基于花同调的指数增长——在哈密顿集合中没有增长——也不是基于映射的拓扑性质，如周期轨道的自由同伦类的增长。pi还将研究在哈密顿量或接触设置下迭代的伽马范数的行为。大多数项目将需要开发适用于其他问题的新技术，以及与辛几何和动力学以外的领域的相互作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。