Collaborative Research: Symbolic Computations in Algebra and Topology

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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.