Topological Data Analysis for Genomics and Evolution: Topology in Biology

Biology has entered the age of Big Data. A technical revolution has transformed the field, and extracting meaningful information from large biological data sets is now a central methodological challenge. Algebraic topology is a well-established branch of pure mathematics that studies qualitative descriptors of the shape of geometric objects. It aims to reduce comparisons of shape to a comparison of algebraic invariants, such as numbers, which are typically easier to work with. Topological data analysis is a rapidly developing subfield that leverages the tools of algebraic topology to provide robust multiscale analysis of data sets. This book introduces the central ideas and techniques of topological data analysis and its specific applications to biology, including the evolution of viruses, bacteria and humans, genomics of cancer, and single cell characterization of developmental processes. Bridging two disciplines, the book is for researchers and graduate students in genomics and evolutionary biology as well as mathematicians interested in applied topology.

Table of Contents:
Introduction; Part I. Topological Data Analysis: 1. Basic notions of algebraic topology; 2. Topological data analysis; 3. Statistics and topological inference; 4. Manifold learning and metric geometry; Part II. Biological Applications: 5. Evolution, trees, and beyond; 6. Cancer genomics; 7. Single cell expression data; 8. Three dimensional structure of DNA; 9. Topological data analysis beyond genomics; 10. Conclusions.

About the Author :
Raul Rabadan is a Professor at Columbia University, New York. He is Director of the Program for Mathematical Genomics at Columbia University, New York, and the NCI Physics and Oncology Center for Topology of Cancer Evolution and Heterogeneity. Dr Rabadan received his Ph.D. in Theoretical Physics in 2001 and went on to conduct research at the European Laboratory for Particle Physics (CERN) in Switzerland, and at the Institute for Advanced Study (IAS), Princeton, New Jersey. At Columbia University, he leads a highly interdisciplinary laboratory with researchers from the fields of mathematics, physics, computer science, engineering, and medicine, with the common goal of solving biomedical problems through quantitative computational models. Andrew J. Blumberg is a Professor in the Department of Mathematics at the University of Texas, Austin. He completed his Ph.D. at the University of Chicago under the supervision of Peter May and Michael Mandell, and was later a National Science Foundation postdoctoral fellow at Stanford University, California. He also spent a year as a member at the Institute for Advanced Study (IAS), Princeton, New Jersey. His pure mathematics research focuses primarily on homotopy theory and algebraic topology and his applied research focuses on the development of topological and geometric techniques for studying genomic data.

Review :
'The time is right to bring new approaches to the analysis of biological data. Topological data analysis reveals the structure of data. This book shows how algebraic topology opens new doors, presenting ideas and directions that make testable predictions and explore life processes.' Arnold J. Levine, Institute for Advanced Study, New Jersey 'This fascinating book describes how advances in mathematics, especially in fields such as topology, are transforming our understanding of biology. Rabadan, one of the founders of the field, shows us how the evolution of cancer, and of viruses and bacteria, can be deeply understood through these novel mathematical techniques. Rabadan's capacity to create a synthesis of many threads, and lay out future challenges, makes this an intriguing and compelling read.' Siddhartha Mukherjee, Columbia University Medical Center, New York and author of The Gene: An Intimate History and The Emperor of All Maladies: A Biography of Cancer 'This is a very important work that shows the way to applications of topological data analysis in genomics. It should be studied carefully by anyone working on the biomedical applications of topological data analysis.' Gunnar Carlsson, Stanford University, California 'This is an important book. Modern experimental biology produces large amounts of data and of many disparate types, requiring new methods of analysis. In explaining biology to mathematicians and data scientists, and subtle new statistical analyses based on the flexible form of geometry called topology to biologists, carefully and clearly, without sacrificing accuracy, the authors have written a unique book that is cutting edge, truly interdisciplinary, and a resource for both communities. I found it fascinating, and will insist that my students read it.' Shmuel Weinberger, Andrew MacLeish Distinguished Service Professor of Mathematics, University of Chicago '… has been a useful strategy, it is ill-equipped to deal with the very high dimensionality of genomic data. The book will be of interest to biologists and mathematicians ranging from advanced undergraduates to experienced researchers seeking to add new analytic strategies in their work, or to establish collaborations across disciplines.' D. P. Genereux, Choice 'The book works well for mathematicians who want to understand how some basic ideas from topology can be applied to the study of data from problems in biology … It is convenient to have all of the necessary background in a single publication … the volume succeeds in being a stepping stone between a textbook and a survey article in an exciting new field' Terence Gaffney, The Quarterly Review of Biology