AF: Medium: Collaborative Research:Numerical Algebraic Differential Equations

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Many basic physical principles, like conservation of mass or momentum for a fluid, are captured mathematically as systems algebraic differential equations. Simplifying and solving these systems (which means reducing the number or complexity of the equations, and finding inputs that satisfy all equations) are fundamental to applications in many areas, including cellular biology, approximation for chemical reaction systems, combinatorics, and analysis. The theoretical and algorithmic study of such systems spans more than a century, using three methods: purely symbolic, numerical, and hybrid symbolic-numeric. Symbolic methods (the quadratic formula being the simplest example) give the strongest guarantees of reliability, at a high (even exorbitant) cost in computational time and memory, since the same algorithm solves both mathematically hard and easy instances. Numerical methods (the basis for computational simulation) allow small errors or approximations for speed; small intermediate errors produce corrupted outputs on singular and ill-conditioned (that is, nearly singular) input instances. In this project, a hybrid symbolic-numeric approach will be developed. Hybrid algorithms are more adaptive and have lower complexity than symbolic algorithms, and can avoid the errors of numerical algorithms.In more technical detail, the three investigators apply existing and develop new methods of symbolic-numeric computation and differential algebra, producing algorithms that run on all inputs. They bring together existing methods of numerical algebraic geometry and software packages, such as Bertini, with recent theoretical results in differential algebra that provide upper bounds needed for guaranteed results. New near-optimal root isolation techniques are developed, implemented, and applied to solve systems of differential equations with finitely many solutions. The work spans from theory to producing practical tools.As part of this project the three investigators mentor and train students in symbolic and numeric computation at CUNY (noted for serving minority and low-income students) and NYU, and more broadly in New York City and Long Island, by activities ranging from developing a Symbolic-Numeric Computing course for graduate students at the Computer Science program of the CUNY Graduate Center and NYU, to advising high school students in projects.