Logic, Topology and Genomics

逻辑、拓扑和基因组学

• 批准号: 1620271
• 负责人: Saugata Basu
• 金额: $ 20万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620271&HistoricalAwards=false
• 关键词: Logic; Topology; Genomics

The goal of the project is to apply methods of logic and topology to several important problems in genomics and medicine. The first application is mining large scale clinical databases for information that would be usable for clinicians and/or biological researchers. The PI plans to design and implement a method applying ideas from persistent homology theory in a logic-based framework. The starting point of this approach is the notion of "redescriptions" introduced by Parida (senior consultant for the project) and Ramakrishnan in the context of knowledge discovery. A mathematical reformulation leads to certain filtered complexes arising from set systems, which are then amenable to analysis using tools from topology. A second application will be in the area of phylogenetics. Studying population admixtures is a very active area of research in population genomics. The PI will use topological methods to not only detect but also discriminate ancient from recent admixture, and validate the approach by testing it on large simulated populations where the admixtures are known in advance. Generating such populations pose unique challenges that have been tackled by Parida and her group recently.A common practical difficulty encountered in many applications of topological data analysis is computing persistent homology groups of filtrations of very large simplicial complexes. The sizes of these complexes makes the computations of their persistent homology bar-codes using current generation publicly available software impossible. The second part of the project will address this shortcoming. The PI will investigate a new approach towards improving efficiency of computing persistent homology groups over existing algorithms. This approach will be useful in a wide variety of applications where topological data analysis is currently being used. The PI plans to implement this algorithm and develop it into a general purpose software-package for computing approximations of persistent homology invariants of filtrations of large complexes. The project will bring together tools from two different areas of mathematics -- logic and topology -- in a novel way, as a method towards analyzing large data-sets. In addition, the PI will also study the underlying mathematical problems that come up -- on the interface of logic and topology which are fundamentally interesting in their own rights, and should have other applications as well. The PI also intends to work with a graduate student and involve them in all aspects of the proposed research.

该项目的目标是将逻辑和拓扑方法应用于基因组学和医学中的几个重要问题。第一个应用是挖掘大型临床数据库以获取可供临床医生和/或生物研究人员使用的信息。 PI 计划在基于逻辑的框架中设计和实现一种应用持久同源理论思想的方法。 这种方法的出发点是 Parida（该项目的高级顾问）和 Ramakrishnan 在知识发现背景下引入的“重新描述”概念。数学重构导致集合系统产生某些经过过滤的复合体，然后可以使用拓扑工具进行分析。 第二个应用将在系统发育学领域。研究群体混合物是群体基因组学中一个非常活跃的研究领域。 PI 将使用拓扑方法不仅检测而且区分古代和现代的混合物，并通过在预先已知混合物的大型模拟群体上进行测试来验证该方法。生成这样的群体带来了独特的挑战，Parida 和她的团队最近已经解决了这些挑战。在拓扑数据分析的许多应用中遇到的一个常见的实际困难是计算非常大的单纯复形过滤的持久同源群。这些复合物的大小使得使用当前一代公开软件不可能计算它们的持久同源条形码。该项目的第二部分将解决这个缺点。 PI 将研究一种新方法，以提高现有算法计算持久同源组的效率。这种方法将在当前使用拓扑数据分析的各种应用中有用。 PI 计划实现该算法并将其开发为通用软件包，用于计算大型复合物过滤的持久同源不变量的近似值。该项目将以一种新颖的方式汇集来自两个不同数学领域（逻辑和拓扑）的工具，作为分析大型数据集的方法。此外，PI 还将研究逻辑和拓扑接口上出现的潜在数学问题，这些问题本身就很有趣，并且也应该有其他应用。 PI 还打算与一名研究生合作，让他们参与拟议研究的各个方面。

项目成果

Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare

可定义超曲面上多项式的零点：病理现象存在，但很少见

• DOI: 10.1093/qmath/haz022
• 发表时间: 2019-10
• 期刊: The Quarterly Journal of Mathematics
• 作者: Basu, Saugata;Lerario, Antonio;Natarajan, Abhiram
• 通讯作者: Natarajan, Abhiram