Computer Algebra Domains and Interfaces

目前计算机代数的大部分工作都是研究代数几何、线性代数和微分代数中的算法问题。现有的算法擅长计算特定的多项式、矩阵或微分方程式，但对符号对象的计算关注较少。这里我们指的是未知次数的多项式，例如x^(n^2+n)-81y^(2m)，未知维度的矩阵，未指定有限域的元素等对象。我们以前的工作已经找到了几种针对这些符号对象的算法，包括多项式GCD，因式分解和函数分解。 我们的第二组问题围绕着数学界面。第一个重点是数学手写识别和基于笔的表情操作领域。我们开发了一种适合于这一领域的符号分类方法，使用截断的勒让德-索博列夫级数来表示墨迹。它在书写墨迹笔划时进行计算，从而提供实时识别。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整的表达式之后对其进行分析，或者在编写时使用表达式模板。 另一个焦点是计算机中介的数学协作领域。当前的符号数学工具遵循数学的原型，作为一项单独的活动-一个用户驱动一项计算。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合作(有限简单群体分类的规模)和小规模合作(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根本的不同一样，计算机增强的数学协作应该与使用Maple是不同的。

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.