Logic, Topology and Genomics

该项目的目标是将逻辑和拓扑学的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规模临床数据库，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应用持久同调理论的思想来设计和实现一种方法。这一方法的起点是Parida(项目高级顾问)和Ramakrishnan在知识发现方面引入的“重新描述”概念。数学上的重新表述导致了从集合系统中产生的某些经过过滤的复形，然后可以使用拓扑学中的工具进行分析。第二个应用将在系统发育领域。研究种群混合体是种群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并区分古代和最近的混合物，并通过在预先知道混合物的大型模拟种群上进行测试来验证该方法。生成这样的种群带来了独特的挑战，这是Parida和她的团队最近解决的问题。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形复形的滤子的持久同调群。这些复合体的大小使得使用当前一代公开可用的软件来计算它们的持久同源条形码是不可能的。该项目的第二部分将解决这一缺陷。PI将研究一种新的方法，以提高计算持久同调群的效率，而不是现有的算法。这种方法将在目前使用拓扑数据分析的各种应用中有用。PI计划实现这一算法，并将其开发成一个通用软件包，用于计算大型复合体过滤的持久同调不变量的近似。该项目将汇集两个不同数学领域的工具--逻辑和拓扑学--以一种新颖的方式，作为分析大型数据集的一种方法。此外，PI还将研究出现的基本数学问题--在逻辑和拓扑学的界面上，这些问题本身就很有趣，也应该有其他应用。国际和平研究所还打算与一名研究生合作，让他们参与拟议研究的所有方面。

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.