Algebraic and Geometric Methods in Data Analysis
数据分析中的代数和几何方法
基本信息
- 批准号:1702395
- 负责人:
- 金额:$ 12.25万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2017
- 资助国家:美国
- 起止时间:2017-09-01 至 2020-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
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.
生物形态的多样性及其变化的性质使它们能够通过拓扑方法进行几何数据分析。这些方法通过记录参数值(如高度、厚度、时间、距离、温度或曲率)来量化形状,这些参数值会改变几何物体的拓扑结构:孔洞出现或坍塌;连接的部件连接或分离;蛀牙形成或填充。目前这类几何方法可以处理一个变化参数,但数据通常需要多个参数。该项目开发了一个代数框架,在这种多参数拓扑数据分析的背景下进行编码和计算。所提出的方法是通用的,适用于任何科学探究的数据集,但在这里,它是为了解决进化生物学中的一个基本问题而开发的:是什么机制驱动拓扑变异的产生,数量足以使选择发挥作用?这项研究的模式生物是果蝇,即黑腹果蝇,特别是它翅膀上的静脉图案。本项目发展了多参数持久同调的数学基础。这项研究是为进化生物学中的一个特定问题服务的,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,并提供理论以及应用和算法目的的见解。参数允许在分层空间上连续变化,而不是通常在简单复合体上具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对交换代数的思考方式:通过考虑实向量或任意偏置集而不是整数向量进行多重渐变,以及通过拼接自由和内射分辨率的增强映射来获得模块的“边缘表示”。自由分辨率和内射分辨率技术的结合也改变了我们对拓扑和几何的看法,因为它使多参数持久同调的新概念在拓扑上变得透明——特征是根据出生和死亡(生成器和协生成器)来描述的,而不是通过出生和出生之间的关系(生成器和关系)来描述的——同时大大扩展了有效计算的潜力。
项目成果
期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
When is a Polynomial Ideal Binomial After an Ambient Automorphism?
环境自同构之后多项式什么时候是理想二项式?
- DOI:10.1007/s10208-018-9405-0
- 发表时间:2018
- 期刊:
- 影响因子:3
- 作者:Katthän, Lukas;Michałek, Mateusz;Miller, Ezra
- 通讯作者:Miller, Ezra
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Ezra Miller其他文献
Statistics for Data with Geometric Structure
具有几何结构的数据统计
- DOI:
10.4171/owr/2018/3 - 发表时间:
2019 - 期刊:
- 影响因子:0
- 作者:
Aasa Feragen;T. Hotz;S. Huckemann;Ezra Miller - 通讯作者:
Ezra Miller
Cohen-Macaulay quotients of normal semigroup rings via irreducible resolutions
通过不可约解析的正规半群环的 Cohen-Macaulay 商
- DOI:
10.4310/mrl.2002.v9.n1.a9 - 发表时间:
2001 - 期刊:
- 影响因子:1
- 作者:
Ezra Miller - 通讯作者:
Ezra Miller
Tableau complexes
画面综合体
- DOI:
10.1007/s11856-008-0014-5 - 发表时间:
2005 - 期刊:
- 影响因子:1
- 作者:
A. Knutson;Ezra Miller;Alexander Yong - 通讯作者:
Alexander Yong
Sticky central limit theorems at isolated hyperbolic planar singularities
孤立双曲平面奇点处的粘性中心极限定理
- DOI:
10.1214/ejp.v20-3887 - 发表时间:
2014 - 期刊:
- 影响因子:0
- 作者:
S. Huckemann;Jonathan C. Mattingly;Ezra Miller;J. Nolen - 通讯作者:
J. Nolen
GRADED GREENLEES-MAY DUALITY AND THE ČECH HULL
分级 Greenlees-May 对偶性和 ČECH 船体
- DOI:
- 发表时间:
2001 - 期刊:
- 影响因子:0
- 作者:
Ezra Miller - 通讯作者:
Ezra Miller
Ezra Miller的其他文献
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{{ truncateString('Ezra Miller', 18)}}的其他基金
CONFERENCE PROPOSAL: MEETING ON COMBINATORIAL COMMUTATIVE ALGEBRA (MOCCA 2014), September 1, 2014
会议提案:组合交换代数会议 (MOCCA 2014),2014 年 9 月 1 日
- 批准号:
1439356 - 财政年份:2014
- 资助金额:
$ 12.25万 - 项目类别:
Standard Grant
Combinatorics in geometry and algebra with applications to the natural sciences
几何和代数中的组合及其在自然科学中的应用
- 批准号:
1001437 - 财政年份:2010
- 资助金额:
$ 12.25万 - 项目类别:
Continuing Grant
CAREER: Discrete Structures in Continuous Contexts
职业:连续环境中的离散结构
- 批准号:
1014112 - 财政年份:2009
- 资助金额:
$ 12.25万 - 项目类别:
Standard Grant
CAREER: Discrete Structures in Continuous Contexts
职业:连续环境中的离散结构
- 批准号:
0449102 - 财政年份:2005
- 资助金额:
$ 12.25万 - 项目类别:
Standard Grant
Combinatorics in Cohomology and Computation
上同调和计算中的组合学
- 批准号:
0304789 - 财政年份:2003
- 资助金额:
$ 12.25万 - 项目类别:
Continuing Grant
Combinatorial Commutative Algebra and Algebraic Geometry
组合交换代数和代数几何
- 批准号:
0071549 - 财政年份:2000
- 资助金额:
$ 12.25万 - 项目类别:
Fellowship Award
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