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​​顿系统构成了一大类动力系统，其中能量耗散可以忽略不计，例如，天体力学中的行星运动、不可压缩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​​顿动力系统。拓扑熵是动力系统的一个重要不变量，它起源于物理学和信息论，可以测量其复杂性。PI 将开发新的方法和工具来研究动力系统的拓扑熵。哈密​​顿动力系统利用离线拓扑数据分析的思想，有可能为拓扑数据分析和图像和模式识别等应用问题做出贡献。这项工作涉及研究、教育和培训年轻科学家。在高等教育领域和知识传播领域以及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。在更技术的层面上，该项目的主题是弗洛尔理论与一方面是辛拓扑，另一方面是哈密顿动力学，特别是拓扑熵，PI 将从 Floer 理论的角度研究紧支持的哈密顿微分同胚和某些 Reeb 流。该项目建立在 PI 最近的工作基础上。重点关注 PI 引入的条形码熵，它是拓扑熵的 Floer 理论对应物，并且与其密切相关。 PI 对拓扑熵的 Floer 理论方面的方法的显着特征是，条形码熵既不是基于 Floer 同源性的指数增长（在哈密顿设置中没有增长），也不是基于图的拓扑特性，例如自由的增长周期轨道的同伦类还将研究哈密顿量或接触环境中迭代下的伽马范数的行为。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。