Algebraic and Geometric Methods in Data Analysis

生物形态的多样性及其变异的本质适合用拓扑学方法进行几何数据分析。这些方法通过记录参数(如高度、厚度、时间、距离、温度或曲率)的值来量化形状，几何对象的拓扑在这些参数值上发生变化：洞出现或坍塌；连接的组件连接或发散；空洞形成或填充。目前这种类型的几何方法可以处理一个变化的参数，但数据通常需要不止一个。该项目开发了一个代数框架，用于在此多参数拓扑数据分析的背景下进行编码和计算。建议的方法是通用的，适用于任何科学研究的数据集，但它是为了服务于进化生物学中的一个基本问题而开发的：什么机制驱动了足够数量的拓扑变异的产生，以便选择发挥作用？这项研究的模式生物是果蝇，特别是它翅膀上的静脉图案。这个项目为多参数持久同源性建立了数学基础。这项研究服务于进化生物学中的一个特定问题，使用果蝇翅膀作为模型系统，但提出的方法是通用的，应用同调代数、实代数几何、组合学和计算技术来编码、控制和提供理论上以及应用和算法目的的洞察力。参数允许在分层空间上连续变化，而不是通常在单纯复数上只有一个参数的离散设置。因此，该项目同时以两种独立的方式改变了我们思考交换代数的方式：通过考虑实数向量或任意偏序集而不是用于多重分级的整数向量，以及通过将自由和内射分辨率的增广图拼接在一起来获得模块的“边缘表示”。自由分解和内射分解技术的结合也改变了我们对拓扑和几何的思考方式，因为它使多参数持久同调的新概念在拓扑上是透明的--特征是根据出生和死亡(生成器和协生成器)而不是出生和出生之间的关系(生成器和关系)来描述的--同时极大地扩展了有效计算的可能性。

The diversity of biological forms and the nature of their variation lend themselves to geometric data analysis by topological methods. These methods quantify shape by recording the values of parameters (such as height, thickness, time, distance, temperature, or curvature) across which the topology of the geometric object changes: holes emerge or collapse; connected components join or diverge; cavities form or fill. Current geometric methods of this sort can handle one varying parameter, but data often call for more than one. This project develops an algebraic framework to encode and compute in the context of this multiparameter topological data analysis. The proposed methodology is general, applicable to datasets from any scientific inquiry, but it is being developed here in service to a fundamental question in evolutionary biology: what mechanism drives the generation of topological variants in sufficient quantity for selection to act? The model organism for this investigation is the fruit fly, Drosophila melanogaster, specifically the pattern of veins in its wings.This project develops mathematical foundations for multiparameter persistent homology. This investigation is in service to a specific question in evolutionary biology, using fruit fly wings as the model system, but the proposed methodology is general, applying homological algebra, real algebraic geometry, combinatorics, and computational techniques to encode, control, and provide insight theoretically as well as for applied and algorithmic purposes. The parameters are allowed to vary continuously on stratified spaces, instead of the usual discrete setup with one parameter on a simplicial complex. As such, the project transforms the way we think about commutative algebra in two independent ways simultaneously: by considering real vectors or an arbitrary poset instead of integer vectors for multigradings, and by splicing together the augmentation maps of free and injective resolutions to get "fringe presentations" of modules. The combination of techniques from free resolutions and injective resolutions also transforms how we think about topology and geometry, since it renders the new conception of multiparameter persistent homology transparent topologically -- features are described in terms of birth and death (generators and cogenerators) rather than by birth and relations between births (generators and relations) -- while greatly expanding the potential for effective computation.