Shapes of Julia sets, Thurston Sets, and Neural Networks

从流行病学到金融学再到气候学，许多学科中的复杂现象都是由动力系统模拟的。动力系统是从一个空间到它自己的函数。空间中的点代表现象的可能状态，该函数描述状态如何随时间演变。即使相对简单的动力系统也会表现出非常复杂的长期行为。动力系统的性质通常反映在相关数学集合的形状中。主要研究人员将研究在三种不同情况下产生的动态定义集合的形状：(1)区间的自映射；(2)全纯动力学；(3)神经网络。除了动力系统和几何理论的进步外，这些结果还可能导致计算机图形学或机器学习方面的新技术。这个项目的三个主要目标是：(1)建立不同动力系统族的瑟斯顿集的拓扑和几何性质；(2)刻画一个和多个复变量的Julia多项式集的形状；(3)描述网络结构如何约束神经网络的决策区域。区间上的超吸引么模自映射族的瑟斯顿集是所有这类映射的拓扑熵指数的所有伽罗华共轭的闭包。这一组的情节表明，它具有丰富而神秘的几何结构。在以前的工作中，主要研究人员刻画了复平面的哪些子集是(在强意义下)可由Julia多项式集逼近的。对于多个复变量中的多项式的无穷大吸引盆的可能的全局形状，目前还没有类似的表征，首席研究员将致力于将她的技术扩展到这种设置。同样，首席研究员将调查固定网络结构的神经网络决策区域的拓扑和几何障碍。这一裁决反映了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。

Complex phenomena in a wide range of disciplines ranging from epidemiology to finance to climatology are modeled by dynamical systems. A dynamical system is a function from a space to itself. Points in the space represent possible states of the phenomenon and the function describes how the states evolve over time. Even relatively simple dynamical systems can exhibit very complicated long-term behaviors. Properties of the dynamical systems are often reflected in the shapes of associated mathematical sets. The principal investigator will investigate the shapes of dynamically defined sets that arise in three different contexts: (1) self-maps of intervals; (2) holomorphic dynamics, and (3) neural networks. In addition to advances in the theory of dynamical systems and geometry, results could lead to new techniques in computer graphics or machine learning. The three main goals of this project are to: (1) establish topological and geometrical properties of Thurston sets for various families of dynamical systems; (2) characterize the shapes of Julia sets of polynomials in one and several complex variables; (3) describe how network architecture constrains the decision regions of neural networks. The Thurston set for the family of superattracting unimodular self-maps of an interval is the closure of the set of all Galois conjugates of the exponentials of the topological entropies of all such maps. Plots of this set reveal that it has a rich and mysterious geometric structure. In previous work, the principal investigator characterized which subsets of the complex plane are approximable (in a strong sense) by Julia sets of polynomials. No analogous characterization of the possible global shapes of basins of attraction of infinity for polynomials in several complex variables is known, and the principal investigator will work to extend her techniques to this setting. Similarly, the principal investigator will investigate topological and geometrical obstructions to approximability by decision regions of neural networks of fixed network architectures.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.