Symbolic Computation Group at FSU Symbolic Computation Group at FSU The widespread availability and power of computers has greatly changed the way we solve scientific and engineering problems, and is beginning to change the way we solve purely theoretical problems. This trend is expected to continue and to accelerate in the future. Problems that could not be solved before can now be solved computationally. Special cases and examples of complex phenomena can now be symbolically and computationally investigated, using the computer as a vital adjunct to mathematical intuition. A revolution in the theory and practice of mathematics is under way.
There is a growing emphasis at the national level on interdisciplinary activities. For example, the NSF has a new initiative "designed to promote comprehensive improvement in the integration of the mathematical sciences into other disciplines and in mathematics instruction through the incorporation of perspectives from other disciplines." Symbolic and Numeric Computation makes mathematics more useful and easier to use, and thus will play a major role is these developments.
The world at large is now undergoing rapid changes. In order to stay competitive, our national needs require major universities to teach and train students in the latest and most advanced methods and technology. Computational Mathematics is certain to become a major component of the mathematics of the 21st century.
Symbolic computation aims at the automation of the steps of mathematical problem solving that precede evaluating numerical models and that, to a large extent, are still the domain of human problem solvers. Symbolic computation is an emerging research tool in many areas of pure and applied mathematics, and in scientific applications. It is becoming a well-defined area for pure mathematics research. Numeric computation is the development and implementation of computational algorithms. Algorithmic development includes the formulation of a computational method for solving a computational problem, and the mathematical analysis of the algorithm to ascertain whether the computed results will be a good approximation to the exact solution. Used together, symbolic and numeric computation enhance each other.
The Symbolic Computation component of this proposal was initiated within the Pure Mathematics side of the department, and has broad support among the pure mathematicians. Its possible impact on pure mathematics in this department is best described by example. Consider the role that it has begun to play in commutative algebra and algebraic geometry. National leaders such as David Eisenbud, Craig Huneke, and Wolmer Vasconcelos make extensive use of the computer algebra system Macaulay, "the fundamental computational tool in commutative algebra." Macaulay is intended to provide a computational research tool for working mathematicians. It is designed primarily to aid in generating non-trivial examples. Such examples help mathematicians gain intuition, and formulate or gather evidence for conjectures. In certain cases, such examples have provided or motivated steps in the proof of theorems, or have provided counter-examples to conjectures. (See attached 1990 "snapshot" of projects using Macaulay.) Eisenbud has written a number of "scripts" for Macaulay, to implement new algorithms. Recent research by Eisenbud, Huneke, and Vasconcelos on the radical of an ideal and on primary decomposition has been motivated by developing new algorithms.
In geometry and topology, W. Thurston and J. Cannon have explored the structure of hyperbolic 3-manifolds by computer, and the implications of geometric rigidity in geometric group theory. A. Casson has recently used the computer to systematically enumerate and prove Thurston's geometrization conjecture of a large class of 3-manifolds. M. Thistlethwaite has used the computer to enumerate knots and links of 13-16 crossings. J. Weeks has developed the program SNAPPEA which decides whether or not the 3-manifold described by Dehn surgery on a framed link is hyperbolic. J. Sullivan has developed beautiful graphical visualization techniques on Silicon Graphics machines for triangulations, tilings and other spatially periodic structures in 3 and higher dimensions. In analysis, work is being done verifying and discovering q-series identities. These identities have both analytic and combinatorial aspects. Some of the identities in Ramanujan's notebooks fall into this category.
A good deal of the above described work is being done with the aid of computer algebra systems. Maple and Axiom are the favorites now. Leaders in this field are Doran Zeilberger (see his interesting Maple packages) and George Andrews. FSU has recently acquired a site license for Maple which has thus become the main engine for symbolic computation at FSU.
In education, mathematicians have written CD-ROM computer-interactive textbooks (Differential Geometry, T. Banchoff) which use symbolic and numeric computation.
At a fundamental level, pure mathematicians prove theorems about the internal mathematical structure of various models. Most (if not all) of these models arose from attempts to understand the "real" world. Most (if not all) pure mathematics research is believed to be "inevitably useful" in building understanding of real phenomena. Symbolic and numeric computation promises a method for "technology transfer", taking pure mathematical ideas, techniques and theorems, turning them into algorithms and computation tools for use in building increasingly sophisticated models of real phenomena.
Some of the scientific problems, such as those of weather prediction, turbulent combustion, and the mapping of the human genome, have been given national priority in the 1992 federal interagency High Performance Computing Program (HPCC), where they are identified as the grand challenges to be solved with future generations of computers.
Computational methods have gained wide acceptance in industry. For example, the Boeing 767 was designed fully on computers before wind tunnel models of it were built. The 1992 NRC Report "Research Directions in Computational Mechanics" documents the importance of computational modeling and numerical methods in mechanics (SIAM News, 9/92, pg. 18). This report outlines 14 key research areas, which have a substantial mathematical content, including CFD (Computational Fluid Dynamics) and parallel computation (ibid). Other recent reports, including the 1992 HPCC initiative and the NRC report Computing the future, emphasize the role of computations for increasing competitiveness globally and for ensuring national security. Computational methods will , in time, replace many traditional analytical methods in research and design.
Computational methods (both symbolic and numeric) are widely used in the life sciences. In computational chemistry, energy levels and molecular orbital structure can be computed, with results comparable to experimental measurement (when direct measurement is possible). Symbolic computation is useful in dealing with implicitly defined quantities (properties of eigenvectors of very large matrices, understanding how these eigenvectors change as a function of position on the potential hypersurface). Structural biology is another venue for large-scale computation, such as a computational approach to the protein folding problem. Both symbolic and numeric computation techniques are used in the human genome project, where similarities of the nucleotide code of long strings of DNA must be computed. Monte Carlo simulations of individual molecular structure (of large and small molecules) and of large systems of molecules require both improved algorithms and numerical implementation of these algorithms.
Symbolic and hybrid symbolic-numeric systems enable one to apply mathematics in larger and more complex real life problems than what has previously been possible. This makes mathematics more useful and opens new opportunities for mathematicians who are able to take full benefit of these methods.
According to our experience, mathematicians and mathematics students, who can use both symbolic and numeric tools fluently, get easily jobs outside the academic world. This is not only good for the field of symbolic computation, since often the best persons are attracted away from research to good positions in industry or elsewhere. On the other hand, this fact makes the field of symbolic and numeric computing attractive.
Last updated: April 18, 1996 by Mika Seppälä