The Geometry and Dynamics of Data-Driven Modeling

The fundamental problem in science is that of using measurements and observations to draw conclusions that at rst appear to be hidden from the observer. The task of turning raw measurements into useful conclusions is exactly the task underlying data driven modeling. The full task involves the ability to turn various assumptions into useful constraints which allow the data to be converted into conclusions. I plan to use geometric methods and dynamical systems concepts to build pieces of a general data driven modeling program and use these to solve scientiic problems of current interest. Note: throughout this document, `geometric methods' is used in a wide sense to cover the methods of diierential geometry, diierential topology, topology (algebraic and point set), etc.]