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

动力学是研究随时间演化的系统的数学分支。从19世纪末的起源到今天，动力系统的数学理论一直受到相邻数学领域（如数论、几何、概率论）和其他自然科学（如天体力学、统计力学、种群生物学）问题的启发。该项目属于这一丰富的传统，并侧重于从几何，数论和应用设置中获得各种动力系统的基本结果。统一这些主题的共同主线是对所考虑的所有问题的“熵”方法。一个动力系统的熵是衡量其轨道结构复杂性的基本不变量。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测度的唯一性；(2)给出在动力系统的某些自然类别中可以得到的所有可能熵的数论描述；(3)利用熵和相关量的信息理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计划，包括为俄亥俄州立大学青年学者计划开发一个课后数学丰富计划。该项目将培养来自俄亥俄州所有主要城市地区代表性不足群体的优秀学生的基本数学技能。该项目的第(1)部分开发了平衡状态理论的新方法，重点是对非均匀和部分双曲系统的实现。该项目开发了一些新技术，以克服一些传统的障碍，在这些高维环境中发展有效的理论。最终目标是使用这些结果为我们框架内的系统推导出许多全局统计特性（中心极限定理、大偏差等）。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)部分关注作为后临界有限区间映射熵的数的代数描述。在瑟斯顿的最后一篇论文中，通过实验确定了映射的程度和这些数的代数性质之间的惊人关系。对这种现象的严格解释仍然是一个悬而未决的问题，PI已经根据确定某些复杂函数的零点来重新定义了这个问题。第(3)部分研究了在数学生物科学中熵和相关量的有限版本。PI将使用这些数量，解释为复杂性的度量，作为检测大型数据集结构的工具，特别是那些在基因组分析和生物动机网络动力学中出现的数据。

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.