Computer Algebra Domains and Interfaces

目前计算机代数的大部分工作都在研究代数几何、线性代数和微分代数中的算法问题。目前的算法在计算特定的多项式、矩阵或微分方程方面表现出色，但很少关注符号对象的计算。我们的意思是对象，如多项式的未知次数，例如x^（n^2 +n）-81 y ^（2 m），矩阵的未知维数，元素的一个未指定的有限域，等等。我们早期的工作已经发现了几个算法，这些种类的符号对象，包括多项式GCD，因式分解和功能分解。 我们的第二组问题围绕着数学接口。第一个重点是数学手写识别和基于笔的表达式操作领域。 我们已经开发出一种方法来符号分类非常适合这个领域，使用截断Legendre-Sobolev系列表示墨水痕迹。 这在书写墨水笔划时进行计算，提供实时识别。 我们现在计划研究实时布局分析。 这一领域的早期工作主要是在编写完表达式后对完整表达式进行分析，或者在编写时使用表达式模板。 另一个重点是计算机介导的数学合作领域。 当前的符号数学工具遵循数学的原型，即一个用户驱动一个计算。 然而，现代数学在很大程度上依赖于大型合作。 从根本上说，需要新的想法来支持大规模的合作（有限简单群的分类规模）和小规模的合作（讨论解决一个问题的方法的规模）。 就像使用Facebook与使用文字处理器有着本质的不同一样，计算机增强的数学协作也应该与使用Maple有着本质的不同。

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.