Collaborative Research: Symbolic Computations in Algebra and Topology

合作研究：代数和拓扑中的符号计算

• 批准号: 0311996
• 负责人: Henry Schenck
• 金额: $ 8.54万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0311996&HistoricalAwards=false
• 关键词: Collaborative; Research; Symbolic; Computations; Algebra

Schenck This is a collaborative project between Henry Schenck andAlexandru Suciu. The investigators study the interplay betweenthe topology of a manifold X and certain algebraic structuresrelated to X. From a theoretical standpoint, such an undertakinginvolves a mainstream question of algebra, geometry, andtopology: how geometric, topological, or combinatorial aspects ofa manifold manifest in algebraic properties of objects such asthe cohomology ring, fundamental group, and resonance varieties.The focus is on the case where X is the complement of anarrangement of lines or rational curves in the projective plane,or a configuration space. The investigators develop a softwarepackage of algorithms to study the aforementioned algebraicinvariants of X. The software is used to generate tables ofarrangements (similar to the tables used in knot theory),providing an extensive list of examples and invariants. Theinvestigators use these tables to search for counterexamples toopen conjectures, and to spot patterns leading to theorems. Thetables and code are a community resource, available online, andgenerate considerable synergy between disparate groups(algebraists, topologists, combinatorialists) involved in MSRI'sspecial semester on hyperplane arrangements (Fall 2004). There is also a practical benefit: hyperplane arrangementsand configuration spaces are ubiquitous in pure and appliedmathematics, arising in numerous areas including braid groups,knot theory, robotics, approximation theory, and mathematicalmodelling. For example, in approximation theory one canapproximate a function of several variables, say k of them in ak-dimensional region, by dividing the region into pieces and oneach piece approximating the function by polynomials; theresulting piecewise polynomials are called splines. Technically,the region is divided into simplices using hyperplanes; the setof splines on the resulting simplicial complex is an algebraicobject that depends strongly on the geometry of the chosenhyperplanes. In robotics, arrangements arise in motion planning(finding a collision-free motion between two placements of agiven robot among a set of objects). Configuration spaces show upin multidimensional billiards (describing the periodictrajectories of a mass-point in a domain in Euclidean space).Information about the structure of the cohomology ring translatesinto bounds on the complexity of the motion planning problem, orbounds on the number of periodic trajectories. Thus, finding fastalgorithms to compute algebraic invariants associated toarrangements and configuration spaces could have real worldapplications. The problems the investigators study are also wellsuited to introducing graduate (and undergraduate!) students toresearch and the use of computational tools. Students conductcomputational experiments, discover patterns and the structure ofthe problem, and thus have motivation to learn new theoreticaltools.

这是Henry Schenck和alexandru Suciu的合作项目。研究人员研究流形X的拓扑结构与X相关的某些代数结构之间的相互作用。从理论的角度来看，这样的工作涉及到代数、几何和拓扑的主流问题：流形的几何、拓扑或组合方面如何体现在对象的代数性质中，如上同调环、基本群和共振变异体。重点是X是投影平面或位形空间中直线或有理曲线排列的补的情况。研究人员开发了一个算法软件包来研究前面提到的x的代数不变量。该软件用于生成排列表（类似于结理论中使用的表），提供了一个广泛的示例和不变量列表。研究人员使用这些表格来寻找反例来打开猜想，并找出导致定理的模式。表格和代码是一个社区资源，可以在线获得，并在MSRI的超平面排列特别学期（2004年秋季）中涉及的不同小组（代数学家、拓扑学家、组合学家）之间产生相当大的协同作用。这也有一个实际的好处：超平面排列和构型空间在纯数学和应用数学中无处不在，出现在许多领域，包括辫群、结理论、机器人、近似理论和数学建模。例如，在近似理论中，我们可以近似一个有多个变量的函数，比如k维区域中的k个变量，方法是将区域分成若干块，每一块用多项式近似该函数；由此产生的分段多项式称为样条。从技术上讲，使用超平面将区域划分为简单体；所得到的简单复合体上的样条集合是一个代数对象，它强烈依赖于所选超平面的几何形状。在机器人技术中，运动规划（在一组物体中找到给定机器人的两个位置之间的无碰撞运动）中出现了安排。构型空间显示在多维台球中（描述欧几里得空间域中质量点的周期轨迹）。关于上同环结构的信息转化为运动规划问题的复杂度的边界，周期轨迹的数量的边界。因此，寻找快速算法来计算与排列和配置空间相关的代数不变量可能具有现实世界的应用。研究人员研究的问题也非常适合介绍研究生（和本科生！）学生进行研究和使用计算工具。学生进行计算实验，发现问题的模式和结构，从而有动力学习新的理论工具。