### A brief introduction to topological data analysis and biological applications

12 June 2015

Raul Rabadan

Department of Biomedical Informatics

Columbia University College of Physicians and Surgeons

#### Abstract

We present a brief introduction to topological data analysis techniques with applications to study some biological problems. The last few years have witnessed a revolution in the acquisition of biological data. High throughput biological data is usually presented as collections of points in a very high dimensional space. Topology is a branch of mathematics that aims to characterize properties of spaces up to continuous deformations. Topological data analysis is a recently developed framework that adopts some of the tools of topology to study finite metric spaces, i.e. points and distances between them. We will introduce some basic concepts in topology and topological data analysis that are needed to understand the applications to the biological problems developed later. Biological applications include the use of topology to study evolution of viruses (influenza and HIV), recombination in humans, and how condensed topological representations can provide insights into the study of large cross-sectional data in cancer.

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