Shapes of Julia sets, Thurston Sets, and Neural Networks

Julia 集、瑟斯顿集和神经网络的形状

• 批准号: 1901247
• 负责人: Kathryn Lindsey
• 金额: $ 14.98万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901247&HistoricalAwards=false
• 关键词: Shapes; Julia; sets; Thurston; Sets

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.

从流行病学到金融学再到气候学的广泛学科中的复杂现象都是由动力系统建模的。 动力系统是从空间到其自身的函数。 空间中的点表示现象的可能状态，函数描述状态如何随时间演变。 即使是相对简单的动力系统也可能表现出非常复杂的长期行为。 动力系统的性质往往反映在相关数学集合的形状中。 主要研究者将研究在三种不同背景下出现的动态定义集的形状：（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，研究结果还可能导致计算机图形学或机器学习的新技术。 该项目的三个主要目标是：（1）建立各种动力系统族的Thurston集的拓扑和几何性质;（2）描述一个和多个复变量多项式的Julia集的形状;（3）描述网络结构如何约束神经网络的决策区域。 区间上的超吸引幺模自映射族的瑟斯顿集是所有此类映射的拓扑熵的指数的所有伽罗瓦共轭的集合的闭包。 这一套的情节揭示了它有一个丰富而神秘的几何结构。 在之前的工作中，首席研究员描述了复平面的哪些子集可以（在很强的意义上）由朱莉娅多项式集逼近。 没有类似的表征可能的全球形状的吸引力的无穷多个复杂的变量的多项式是已知的，和主要研究人员将努力扩大她的技术，以这种设置。 同样，首席研究员将调查拓扑和几何障碍的近似性的决策区域的神经网络的固定网络architecture.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

项目成果

The shape of Thurston's Master Teapot

• DOI: 10.1016/j.aim.2020.107481
• 发表时间: 2019-02
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Harrison Bray;D. Davis;Kathryn A. Lindsey;Chenxi Wu

A characterization of Thurston’s Master Teapot

• DOI: 10.1017/etds.2022.73
• 发表时间: 2019-09
• 期刊: Ergodic Theory and Dynamical Systems
• 影响因子: 0.9
• 作者: Kathryn A. Lindsey;Chenxi Wu

Bicritical Rational Maps With a Common Iterate

具有公共迭代的双临界有理图

• DOI: 10.1093/imrn/rnad041
• 发表时间: 2023
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Koch, Sarah;Lindsey, Kathryn;Sharland, Thomas
• 通讯作者: Sharland, Thomas

On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks

• DOI: 10.1137/20m1368902
• 发表时间: 2020-08
• 期刊: SIAM J. Appl. Algebra Geom.
• 作者: J. E. Grigsby;Kathryn A. Lindsey