Rational Dynamics on Complex Surfaces

该项目涉及对多个复变量的动力系统的研究。动态系统是可以由一组状态以数学方式建模的过程，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，仅举几个应用领域的例子。这一领域感兴趣的主题包括根据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度，以及描述最终状态的完整阵列。由多项式或有理公式给出的动力系统构成了一类重要而有趣的特殊类，也是本课题研究的重点。从数学的观点来看，这样的系统是自然的，也出现在各种科学和数学应用中，包括线性规划中的内点法、热力学和求根算法。对多个复变量的有理动力学的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步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，该项目还通过支持博士生和河湾社区数学中心的STEM外联活动来促进人力资源的发展，该中心是当地的一个非营利性组织，致力于与K-12数学学生和教师合作。该项目涉及几个复杂变量设置下的有理映射的动态。所提出的工作的一个主要主题是关于构造复杂射影曲面上有理映射的最大熵的困难问题。特别感兴趣的是复杂曲面上的有理映射，这些映射不允许使用代数稳定的模型。可以预料，任何这样的映射都必须保持附加的几何结构，例如曲线上的纤维或不变的亚纯标准形。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这种结构量身定做的分析工具，着眼于产生不变的电流和测量，就像以前在代数稳定地图的背景下构建的那样。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.