Computer Algebra Domains and Interfaces

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Much of the current work in computer algebra examines algorithmic questions in algebraic geometry, linear algebra and differential algebra. Current algorithms are excellent at computing with particular polynomials, matrices or differential equations, but little attention has been paid to computing with symbolic objects. By this we mean objects such as polynomials of unknown degree, e.g. x^(n^2 +n) - 81y^(2m), matrices of unknown dimension, elements of an unspecified finite field, etc. Our earlier work has found several algorithms for these sorts of symbolic objects, including polynomial GCD, factorization and functional decomposition. Our second set of questions revolves around mathematical interfaces. The first focus is in the area of mathematical handwriting recognition and pen-based expression manipulation. We have developed an approach to symbol classification well-suited to this domain, using truncated Legendre-Sobolev series to represent ink traces. This computes as ink strokes are written, providing real-time recognition. We now plan to investigate real-time layout analysis. Earlier work in this area has been predominantly on the analysis of complete expressions, after they are written, or use expression templates at the time of writing. Another focus is in the area of computer-mediated mathematical collaboration. Current symbolic mathematics tools follow archetype of mathematics as a solitary activity --- one user drives a computation. However, modern mathematics relies heavily on large collaborations. Fundamentally new ideas are needed to support both large-scale collaboration (of the scale of the classification of finite simple groups) and small-scale collaboration (of the scale of discussing approaches to solve one problem). Just as using Facebook is fundamentally different from using a word processor, computer-enhanced mathematical collaboration should be as different from using Maple.