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: 0311142
• 负责人: Alexandru Suciu
• 金额: $ 14.17万
• 依托单位: Northeastern University
• 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=0311142&HistoricalAwards=false
• 关键词: Collaborative; Research; Symbolic; Computations; Algebra

Suciu 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.

Suciu 这是亨利申克和亚历山大苏丘之间的合作项目。 研究者们研究流形X的拓扑和与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，这样的一个映射涉及到代数、几何和拓扑学的主流问题：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、基本群和共振簇。重点是X是射影平面或构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（类似于纽结理论中使用的表），提供了大量的例子和不变量。 研究人员使用这些表格来寻找反例，以打开定理，并发现导致定理的模式。 表格和代码是一个社区资源，可在线获得，并在MSRI的超平面安排特别学期（2004年秋季）中涉及的不同群体（代数学家，拓扑学家，组合学家）之间产生相当大的协同作用。 它还有一个实际的好处：超平面展开和构形空间在纯数学和应用数学中无处不在，出现在许多领域，包括辫子群、纽结理论、机器人学、近似理论和几何建模。 例如，在逼近理论可以近似一个函数的几个变量，说k他们在k维区域，通过划分该地区成块，并在每一块近似的功能多项式; thegreenerated分段多项式被称为样条。 从技术上讲，该区域被划分为单形使用超平面;所产生的单纯复形上的样条集是一个代数对象，强烈依赖于所选择的超平面的几何形状。 在机器人技术中，安排出现在运动规划中（在一组对象中找到给定机器人的两个位置之间的无碰撞运动）。 配置空间表现在多维台球（描述一个质点在欧氏空间中的运动轨迹）上，关于上同调环结构的信息包含在运动规划问题复杂性的范围内，或者周期轨迹数量的范围内。 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真实的应用。 调查人员研究的问题也非常适合介绍研究生（和本科生！）让学生学会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。