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.

Schenck这是Henry Schenck Andalexandru Suciu之间的合作项目。 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共振品种。重点是X是线条或构图平面中有理曲线或配置空间中的曲线的补充的情况。 研究人员开发了算法的软件处理，以研究X的上述代数变量。该软件用于生成arrangements的表（类似于结理论中使用的表），提供了广泛的示例列表和不变的列表。 评估者使用这些表来搜索反例构想的猜想，并发现导致定理的模式。 Thetables和代码是一种社区资源，可以在线获得，并在不同的群体（代数，拓扑家，组合主义者，组合主义者）之间产生了相当大的协同作用，参与了MSRI的超平面安排（2004年秋季）。 也有一个实际的好处：超平面布置和配置空间无处不在，在纯和应用的隔离学中，在包括编织组，结理论，机器人技术，近似理论和数学模型等许多领域产生。 例如，在近似理论中，可以通过将该区域分成碎片并在each片段上通过多项式近似函数，从而在AK维区域中占据了几个变量的函数；各个分段多项式称为花键。 从技术上讲，该区域使用超平面分为简单。所得的简单复合物上的setof花键是一个代数对象，它在很大程度上取决于选择性层析平面的几何形状。 在机器人技术中，安排在运动计划中出现（在一组对象之间找到两个放置机器人的放置之间的无冲动运动）。 配置空间显示了多维台球（描述了欧几里得空间中一个域中的质量点的周期性标记）。关于同胞环的结构的信息转化有关运动计划问题的复杂性的界限，对定期轨迹的数量发出的数量。 因此，寻找封装的封装不变式相关的代数不变性和配置空间可能具有现实世界的应用。 研究人员的研究还可以很好地介绍研究生（和本科生！）学生ToreSearch和计算工具的使用。 学生进行计算实验，发现模式和问题的结构，因此有学习新理论工具的动机。