Collaborative Research: Floer Theory and Topological Entropy

合作研究：弗洛尔理论和拓扑熵

• 批准号: 2304207
• 负责人: Basak Gurel
• 金额: $ 24.76万
• 依托单位: The University of Central Florida Board of Trustees
• 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=2304207&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​​顿系统构成了一系列的动态系统，可以忽略能量耗散。例如，天体力学中的行星运动，不可压缩的理想流体的流动以及电磁场中带电粒子的运动通常被视为哈密顿动态系统。拓扑熵是动态系统的重要不变性，它测量了其复杂性和起源于物理和信息理论。 PI将开发新的方法和工具来研究哈密顿动态系统的拓扑熵，并利用拓扑数据分析的思想。相反，这项研究有可能为拓扑数据分析的领域做出贡献，并应用了包括图像和模式识别的问题。这项工作涉及研究，教育和培训年轻科学家的整合。它将在高等教育和知识传播，领域和广泛的科学界的领域产生影响，并且将增加来自人为不足的数学群体的个人的参与。在技术层面上，该项目的主要主题是浮点理论与一方面和汉密尔顿动力学以及尤其是拓扑的拓扑之间的相互作用。 PI将研究紧凑型的汉密尔顿二型差异和某些REEB流的拓扑熵，从浮动理论的角度来看。该项目建立在PI的最新工作的基础上，并重点介绍了PIS引入的条形码熵，PIS是拓扑熵的浮动理论对应物，并且与之紧密相关。 PIS对拓扑熵的漂浮理论方面方法的关键新特征是，条形码熵基于浮点同源性的指数增长 - 在哈密顿式环境中没有增长 - 也没有地图的拓扑特性，例如周期性轨道的自由同苯二型类别的生长。 PI还将研究在哈密顿量或接触设置中迭代下伽马 - 标准的行为。大多数项目都需要开发适用于其他问题的新技术，并与与对称几何和动态之外的领域进行互动。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响评估审查标准，认为通过评估被认为是宝贵的支持。