This is a collaborative project between Henry Schenck and Alexandru Suciu. The investigators study the interplay between the topology of a manifold X and certain algebraic structures related to X. From a theoretical standpoint, such an undertaking involves a mainstream question of algebra, geometry, and topology: how geometric, topological, or combinatorial aspects of a manifold manifest in algebraic properties of objects such as the cohomology ring, fundamental group, and resonance varieties. The focus is on the case where X is the complement of an arrangement of lines or rational curves in the projective plane, or a configuration space. The investigators develop a software package of algorithms to study the aforementioned algebraic invariants of X. The software is used to generate tables of arrangements (similar to the tables used in knot theory), providing an extensive list of examples and invariants. The investigators use these tables to search for counterexamples to open conjectures, and to spot patterns leading to theorems. The tables and code are a community resource, available online, and generate considerable synergy between disparate groups (algebraists, topologists, combinatorialists) involved in MSRI's special semester on hyperplane arrangements (Fall 2004). There is also a practical benefit: hyperplane arrangements and configuration spaces are ubiquitous in pure and applied mathematics, arising in numerous areas including braid groups, knot theory, robotics, approximation theory, and mathematical modelling. For example, in approximation theory one can approximate a function of several variables, say k of them in a k-dimensional region, by dividing the region into pieces and on each piece approximating the function by polynomials; the resulting piecewise polynomials are called splines. Technically, the region is divided into simplices using hyperplanes; the set of splines on the resulting simplicial complex is an algebraic object that depends strongly on the geometry of the chosen hyperplanes. In robotics, arrangements arise in motion planning (finding a collision-free motion between two placements of a given robot among a set of objects). Configuration spaces show up in multidimensional billiards (describing the periodic trajectories of a mass-point in a domain in Euclidean space). Information about the structure of the cohomology ring translates into bounds on the complexity of the motion planning problem, or bounds on the number of periodic trajectories. Thus, finding fast algorithms to compute algebraic invariants associated to arrangements and configuration spaces could have real world applications. The problems the investigators study are also well suited to introducing graduate (and undergraduate!) students to research and the use of computational tools. Students conduct computational experiments, discover patterns and the structure of the problem, and thus have motivation to learn new theoretical tools.
