CAREER: Entropy in dynamics: connections with geometry, algebraic numbers, and bioscience

职业：动力学中的熵：与几何、代数数和生物科学的联系

• 批准号: 1454864
• 负责人: Daniel Thompson
• 金额: $ 44.48万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1454864&HistoricalAwards=false
• 关键词: CAREER; Entropy; dynamics; connections; geometry

Dynamics is the branch of mathematics that studies systems that evolve under time. From its origins in the late 19th century to today, the mathematical theory of dynamical systems has been inspired by problems in adjacent areas of mathematics (e.g. number theory, geometry, probability), and other natural sciences (e.g. celestial mechanics, statistical mechanics, population biology). This project belongs to that rich tradition, and focuses on deriving fundamental results for a variety of dynamical systems arising from geometry, number theory and applied settings. The common thread that unifies these topics is an 'entropic' approach to all the problems under consideration. The entropy of a dynamical system is a fundamental invariant which measures the complexity of its orbit structure. The three main research directions of the project are: (1) to establish uniqueness of measures of maximal entropy and equilibrium states in a variety of higher dimensional settings of interest to the dynamics and wider mathematical communities; (2) to give a number-theoretic description of all the possible entropies that can be achieved within certain natural classes of dynamical systems; (3) to use information-theoretic interpretations of entropy and related quantities to give insight to problems in mathematical bioscience. The project contains a substantial program of synergistic educational activities, including the development of an after-school math enrichment program for the Ohio State University Young Scholars Program. The program will develop fundamental mathematical skills for a talented population of students from under-represented groups taken from all the major urban areas of Ohio.Part (1) of the project develops a novel approach to the theory of equilibrium states, focusing on implementation to non-uniformly and partially hyperbolic systems. The project develops novel techniques to overcome some of the traditional obstructions to developing an effective theory in these higher dimensional settings. The end goal is to use these results to derive numerous global statistical properties (central limit theorems, large deviations, etc.) for the systems within our framework. Motivating examples include geodesic flows in non-positive curvature, and the Teichmueller flow on the moduli space of quadratic differentials. Part (2) concerns the algebraic description of numbers arising as entropies of post-critically finite interval maps. A surprising relationship between the degree of the map and the algebraic properties of these numbers was identified experimentally in Thurston's final paper. A rigorous explanation of this phenomenon remains an open problem, which the PI has recast in terms of identifying zeros of certain complex functions. Part (3) investigates finitary versions of entropy, and related quantities, in the mathematical biosciences. The PI will use these quantities, interpreted as measures of complexity, as tools for detecting structure in large data sets, particularly those arising in genomic analysis and in the dynamics of biologically motivated networks.

动力学是数学的一个分支，研究随时间演化的系统。从世纪末的起源到今天，动力系统的数学理论一直受到数学（例如数论，几何，概率）和其他自然科学（例如天体力学，统计力学，人口生物学）相邻领域问题的启发。这个项目属于丰富的传统，并侧重于推导各种动力系统的几何，数论和应用设置所产生的基本结果。统一这些主题的共同主线是对所有正在考虑的问题的“熵”方法。动力学系统的熵是一个基本不变量，它度量了系统轨道结构的复杂性。该项目的三个主要研究方向是：（1）在动力学和更广泛的数学界感兴趣的各种高维环境中建立最大熵和平衡态的唯一性度量;（2）对某些自然类动力系统中所有可能的熵给出数论描述;（3）利用熵和相关量的信息论解释来洞察数学生物科学中的问题。该项目包含一个实质性的协同教育活动方案，包括为俄亥俄州州立大学青年学者方案制定课后数学充实方案。该计划将发展基本的数学技能的学生从代表性不足的群体采取的所有主要城市地区的俄亥俄州的天才人口。该项目的第（1）部分开发了一种新的方法来平衡态理论，重点是实现非均匀和部分双曲系统。该项目开发了新的技术，以克服在这些更高维度的设置中开发有效理论的一些传统障碍。最终的目标是使用这些结果来推导出许多全局统计性质（中心极限定理，大偏差等）。在我们的框架内。激励的例子包括非正曲率的测地线流和二次微分模空间上的Teichmueller流。第（2）部分是关于后临界有限区间映射的熵数的代数描述。在瑟斯顿的最后一篇论文中，通过实验发现了映射的度与这些数的代数性质之间的惊人关系。对这一现象的严格解释仍然是一个悬而未决的问题，PI已经在确定某些复杂函数的零点方面进行了重新设计。第（3）部分研究了数学生物科学中熵和相关量的有限形式。PI将使用这些量，解释为复杂性的度量，作为检测大型数据集中结构的工具，特别是那些在基因组分析和生物动力网络动态中出现的数据。

项目成果

Measures of maximal entropy on subsystems of topological suspension semiflows

拓扑悬浮半流子系统的最大熵测度

• DOI: 10.4064/sm201105-13-1
• 发表时间: 2021
• 期刊: Studia Mathematica
• 影响因子: 0.8
• 作者: Kucherenko, Tamara;Thompson, Daniel J.
• 通讯作者: Thompson, Daniel J.

Fluctuations of Time Averages Around Closed Geodesics in Non-Positive Curvature

非正曲率下闭合测地线周围时间平均值的涨落

• DOI: 10.1007/s00220-021-04062-6
• 发表时间: 2021
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Thompson, Daniel J.;Wang, Tianyu
• 通讯作者: Wang, Tianyu