Parametric Computation in Axiom Toward Indefinite Symbolic Computing
NSF Grant 0430722 (July 30, 2004)
Gilbert Baumslag, Timothy Daly, William Sit, Sean Cleary, Douglas Troeger
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.