BIGDATA: Collaborative Research:F: Manifold Hypotheses and Foundations of Data Science with Applications

It is often said that progress in science is characterized by successive steps of measurement, arithmetization, algorithmization, and algebraization – each step representing in a succinct manner the intuitions collected in the earlier step. In sciences, various breakthrough in technology, e.g., sequencing, high-throughput measurement of DNA/RNA abundance, electron and scanning tunneling microscopy, astronomical studies, space telescopes, collection of social-media data, on-line observations of human interactions, etc., have made it possible to obtain a quantitative arithmetic picture of the “states” of a complex structure (cell, organism, population, social groups, universe) at a certain instant and under certain conditions. As the complexity of the systems studied have scaled, the “bigness” of the data has grown spectacularly; many scalable exact and approximate algorithms have been proposed; a unifying foundational study of the emerging “data science” has become prominent, and yet, it shies away from the final step of the algebraization of data sciences. Such an approach could center around the so-called “Manifold Hypothesis,” that seeks a differential algebraic structure in the state-space – to be inferred from the sampled data point clouds. We wish to build on “Manifold Hypothesis” to create an algebraic (geometric/topological) investigation of existing and emerging Big Data approaches in computer science, statistics, computational science, and mathematics, along with innovative applications in domain sciences, namely, cancer biology, linguistics and the physical sciences that lead towards the further development of the field of data science. Thus, the main emphasis of our study is “Foundational” (F): focusing on fundamental theories, techniques, methodologies, technologies of broad applicability to Big Data problems.