Combinatorics, Topology and Genomics
Laxmi Parida (May 21, 2018)
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Many problems in genomics are combinatorial in nature, i.e., the interrelationship amongst the entities is at par, if not more important, than the value of the entity themselves.
Graphs are the most commonly used mathematical object to model such relationships. However, often it is important to capture not just pairwise, but more general $k$-wise relationships. TDA provides a natural basis to model such interactions and, additionally, it provides mechanisms for extracting signal patterns from noisy data.
I will discuss two applications of this model, one to population genomics and the other to meta-genomics. In the former, we model the problem of detecting admixture in populations and in the latter we deal with the problem of detecting false positives in microbiome data.
The unifying theme is the use of topological methods -- in particular, persistent homology. I will explain the underlying mathematical models and describe the results obtained in both cases.