Rational Dynamics on Complex Surfaces

复杂曲面上的有理动力学

• 批准号: 2246893
• 负责人: Jeffrey Diller
• 金额: $ 40.31万
• 依托单位: University of Notre Dame
• 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: 2023-05-01 至 2026-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246893&HistoricalAwards=false
• 关键词: Rational; Dynamics; Complex; Surfaces

The project involves research on dynamical systems of several complex variables. Dynamical systems are processes that can be modeled mathematically by a set of states that evolve over time according to fixed rules. Dynamical systems are ubiquitous in science and engineering, arising in the context of weather prediction, macroeconomics, mechanics, and neural networks, to name just a small selection of application domains. Topics of interest in this area include making predictions about the future state of a system based on present conditions, understanding how sensitively future states depend on initial states, and characterizing the full array of eventual states. Dynamical systems given by polynomial or rational formulas constitute an important and interesting special class and are a focus of study in this project. Such systems are natural from a mathematical standpoint, and also arise in various scientific and mathematical applications including interior point methods in linear programming, thermodynamics, and root-finding algorithms. The study of rational dynamics in several complex variables draws on tools and techniques from a range of mathematical areas, and advances in multivariable complex dynamics in turn yield new insights in those areas. In addition to the expected research advances, the project contributes to the development of human resources via the support of doctoral students and STEM outreach activities at the Riverbend Community Math Center, a local non-profit organization devoted to working with K-12 mathematics students and teachers.The project involves the dynamics of rational mappings in the setting of several complex variables. One major thrust of the proposed work concerns the difficult problem of constructing measures of maximal entropy for rational maps on complex projective surfaces. Of particular interest are rational maps on complex surfaces that do not admit algebraically stable models. It is anticipated that any such map must preserve additional geometric structure, such as a fibration onto a curve or an invariant meromorphic canonical form. Progress towards this conjecture is an important goal of the project. In addition, the project aims to develop analytic tools tailored to such structures, with an eye towards producing invariant currents and measures, as were previously constructed in the setting of algebraically stable maps.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.

该项目涉及多复变量动力系统的研究。动态系统是可以通过一组状态在数学上建模的过程，这些状态根据固定的规则随时间演变。动力系统在科学和工程中无处不在，出现在天气预报，宏观经济学，力学和神经网络的背景下，仅举一小部分应用领域的例子。在这一领域感兴趣的主题包括基于当前条件对系统的未来状态进行预测，了解未来状态对初始状态的敏感程度，以及表征最终状态的全部阵列。由多项式或有理公式给出的动力系统是一个重要而有趣的特殊类，也是本项目的研究重点。这样的系统从数学的观点来看是自然的，并且也出现在各种科学和数学应用中，包括线性规划中的内点方法、热力学和求根算法。多复变量的理性动力学研究借鉴了一系列数学领域的工具和技术，而多变量复杂动力学的进展反过来又在这些领域产生了新的见解。除了预期的研究进展外，该项目还通过支持博士生和Riverbend社区数学中心的STEM外展活动来促进人力资源的开发，Riverbend社区数学中心是一家致力于与K-12数学学生和教师合作的当地非营利组织。该项目涉及在多个复杂变量的设置中理性映射的动态。所提出的工作的一个主要推力涉及的困难的问题，构造措施的最大熵的合理映射复杂的射影曲面。 特别有趣的是复杂表面上的有理映射，这些表面不允许代数稳定的模型。可以预见的是，任何这样的映射必须保持额外的几何结构，如纤维化到曲线或不变的亚纯标准型。这一猜想的进展是该项目的一个重要目标。此外，该项目旨在开发针对此类结构量身定制的分析工具，着眼于产生不变的电流和措施，就像以前在代数稳定地图的设置中构建的那样。该奖项反映了NSF的法定使命，并被认为值得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来支持。