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。动力系统的数学理论从19世纪末到今天，一直受到邻近数学领域(如数论、几何学、概率论)和其他自然科学(如天体力学、统计力学、人口生物学)问题的启发。这个项目属于这一丰富的传统，专注于从几何、数论和应用环境中得出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有正在考虑的问题采取“熵”方法。动力学系统的熵是衡量其轨道结构复杂性的一个基本不变量。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社区感兴趣的各种高维环境中建立最大熵和平衡态的测量的唯一性；(2)给出在动力系统的某些自然类中可以达到的所有可能的熵的数论描述；(3)使用信息论对熵和相关量的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外数学充实计划。该计划将为来自俄亥俄州所有主要城市地区的未被充分代表的群体的有才华的学生培养基本的数学技能。项目的(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施。该项目开发了新的技术，以克服在这些更高维度的环境中开发有效理论的一些传统障碍。最终目标是使用这些结果来推导出大量的全局统计性质(中心极限定理、大偏差等)。对于我们框架内的系统。激励的例子包括非正曲率的测地线流，以及二次微分模空间上的Teichmueller流。第二部分是关于后临界有限区间映射的数的代数刻画。在瑟斯顿的最后一篇论文中，实验确定了映射的次数和这些数的代数性质之间令人惊讶的关系。对这种现象的严格解释仍然是一个悬而未决的问题，PI在识别某些复杂函数的零点方面已经重塑了这一问题。(3)研究数学生物科学中的有限形式的熵和相关的量。PI将使用这些被解释为复杂性度量的量作为工具来检测大型数据集的结构，特别是在基因组分析和生物动机网络的动力学中出现的那些结构。

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.