Algebraic and Geometric Methods in Data Analysis

生物形态的多样性及其变异的性质使其适合于通过拓扑方法进行几何数据分析。 这些方法通过记录几何对象拓扑变化的参数值（如高度、厚度、时间、距离、温度或曲率）来量化形状：孔出现或塌陷;连接的组件连接或分叉;空腔形成或填充。目前这种几何方法可以处理一个变化的参数，但数据往往需要一个以上。 本计画发展一个代数架构来编码及计算多参数拓扑资料分析。 所提出的方法是通用的，适用于任何科学调查的数据集，但它是在进化生物学中的一个基本问题：什么机制驱动生成足够数量的拓扑变体选择采取行动？ 本研究的模式生物是果蝇，果蝇，特别是其翅膀上的静脉图案。本项目为多参数持续同源性奠定了数学基础。 这项调查是在进化生物学中的一个具体问题，使用果蝇翅膀作为模型系统，但所提出的方法是通用的，应用同调代数，真实的代数几何，组合学和计算技术编码，控制，并提供理论上的洞察力，以及应用和算法的目的。 参数被允许连续变化的分层空间，而不是通常的离散设置与一个参数上的单纯复杂。 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：通过考虑真实的向量或任意偏序集而不是整数向量的多重分级，以及通过将自由和内射分辨率的增强映射拼接在一起以获得模块的“边缘表示”。 自由解析法和内射解析法的结合也改变了我们对拓扑和几何的看法，因为它使得多参数持续同源的新概念在拓扑上透明--特征是根据出生和死亡（生成元和共生成元）而不是出生和出生之间的关系（生成元和关系）来描述的--同时大大扩展了有效计算的潜力。

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.