Collaborative Research: Symbolic Computations in Algebra and Topology

这是亨利·申克和亚历山大·苏乔的合作项目。研究人员研究了流形X的拓扑和与X有关的某些代数结构之间的相互作用。从理论的观点来看，这样的研究涉及一个代数、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、基本群和共振域。重点放在X是射影平面或配置空间中直线或有理曲线的排列的补集的情况。研究人员开发了一个算法软件包来研究X的上述代数不变量。该软件用于生成排列表(类似于纽结理论中使用的表)，提供了广泛的示例和不变量列表。研究人员使用这些表格来搜索反例，以打开猜想，并发现导致定理的模式。这些表格和代码是一个社区资源，可在网上获得，并在MSRI关于超平面安排的特殊学期(2004年秋季)中参与的不同群体(代数学家、拓扑学家、组合学家)之间产生相当大的协同作用。还有一个实际的好处：超平面排列和配置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、逼近理论和数学建模。例如，在逼近理论中，一个人可以通过将区域分成多段并在每一段上用多项式逼近函数来逼近多个变量的函数，例如k个变量在Ak维区域中；这些分段多项式被称为样条线。从技术上讲，使用超平面将区域划分为单纯形；得到的单纯复形上的样条集是一个代数对象，它强烈依赖于所选择的超平面的几何。在机器人学中，安排出现在运动规划中(在一组物体中找到一个机器人的两个放置位置之间的无碰撞运动)。位形空间表现在多维台球中(描述欧氏空间中区域中质点的周期轨迹)。关于上同调环结构的信息转化为关于运动规划问题的复杂性的界，或关于周期轨迹的数目的界。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会有实际的应用。研究人员研究的问题也非常适合介绍研究生(和本科生！)学生学习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。

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.