Logic, Topology and Genomics

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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.