CAREER: CISE-CCF-AF-Algebra: DMS-Algebra: Computational Differential Algebra

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Partial differential algebraic equations (PDAEs) have many important applications, such as those in cellular biology, chemistry, mathematical physics, mechanics and dynamical systems, control theory, differential geometry, and analysis. Algorithms on PDAEs have been instrumental in simplifying and solving practical problems but are increasingly taxed as the size of problems increases. Improvements in efficiency will reduce the time for scientists to find properties, and perhaps solutions, to the systems they are working on. The research consists of developing a computational theory for systems of PDAEs, including those with parameters, and to design algorithms that provide efficient descriptions of their solutions. For non-parameterized systems, the PI will first obtain complexity estimates for existing differential elimination algorithms. For systems with parameters, he will study the structure of their differential Galois groups, which are linear groups consisting of solutions to PDAEs themselves, via representation theory. Better understanding of them can then be applied to improve the efficiency of existing algorithms or create new ones. Despite over a century of numerous studies on PDAEs, as pioneered by Holder, Janet, Riquer, Ritt, Kolchin, and recently furthered by Singer, Boulier, and Hubert, among many others, there do not yet exist methods computationally efficient enough to explore and understand the differential algebraic behavior of solutions of these systems. As a result of the proposed research activities, new improved complexity upper bounds will highlight some bottlenecks of existing algorithms, allowing deeper exploration of designs for more efficient algorithms. The research activities will further develop the theory of radical differential ideals which is essential for any substantial progress in theoretical and algorithmic developments. Among the open problems this project will investigate is the Ritt Problem, which if solved, would provide unique, irredundant, and effective representations of radical differential ideals.