Parametric Computation in Axiom Towards Indefinite Symbolic Computing

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Current research in computation focuses on how to program a computer to perform algebraic calculations that involve concrete mathematical objects. In a concrete situation, explicit numerical data is available to effect and control the calculations at run time. No such information is available when the objects are unspecified (indefinite), and yet mathematicians routinely carry out manual calculations with them as preliminary steps to obtain insightful theorems. For the most part, computations involving indefinites have not been studied in depth. This research explores both the scope and the methods by which algebraic manipulations of indefinite objects can be automated and lays the foundation for an entirely new level of abstraction in symbolic computation.Computer implementation of indefinite computation considerably expands the scope of computer use in algebra and is an intellectual challenge of the highest order. Every tiny inroad into understanding and developing methods to do such computations will have very wide applications. The planned research consists of several phases: (1) Investigation and analysis of examples. (2) Experimental prototypes in specific cases. (3) Defining a practical and categorical framework for indefinite computation. (4) Eventually full implementation in the open source system Axiom. (5) Applications to open problems. The investigators will apply state of the art algorithms in algebra and in computer science to define the framework. Cutting edge methods for solving parametric equations, symbolic summation, recurrence equations, Grobner basis, cylindrical algebraic decomposition, dynamical evaluation, and lazy evaluation will be integrated as necessary. Specific open problems of great importance in group theory and differential algebra will be studied in order to better understand some of the basic difficulties. Students will participate through courses, seminars, and hands-on implementation.