Mathematical Sciences: Cohomology of Sheaves, and Applications

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9401460 McConnell This project involves at the outset the upgrading of a new computer algebra system, Sheafhom, for research in the derived category of sheaves on a topological space. In Sheafhom the basic items are finite-dimensional vector spaces, cochain complexes, regular cell complexes, sheaves, and morphisms among these objects. There are functions like kernel, image, cohomology, and tensor and wedge products. The system works with important complexes of sheaves, such as the intersection cohomology sheaves and other perverse sheaves. Second, the project involves bringing Sheafhom to bear on convex polytopes and the intersection homology of toric varieties. McMullen's conjecture characterized the f-vectors of simplicial convex polytopes. Stanley's proof of (one direction of) McMullen's conjecture used the cohomology of the toric variety associated to a simplicial polytope. Key facts were that the cohomology, modulo the Lefschetz class, is a graded ring generated in degree two. For an arbitrary rational convex polytope, one can ask whether the intersection cohomology of its toric variety, modulo the Lefschetz class, is a ring, and whether it is generated in degree two. The project will build on preliminary results on these questions, and Sheafhom will play a part in approaching them. A computer algebra system is a program for algebraic calculation. Any program can add numbers, but a computer algebra system combines whole formulas: the input (2x + 7) + (4x - 3) automatically becomes 6x + 4. This capacity is more abstract, hence more flexible. Excellent general-purpose systems, like Maple or Mathematica, are available, but of course they don't have every function that every mathematician needs. In recent years several disciplines have acquired their own special-purpose systems--Cayley/Magma for algebraists, Pari for number theorists, and a dozen others. The investigator intends to develop Sheafhom, a computer algebra system for algebra ic topology. Second, he intends to apply Sheafhom to study convex polytopes. Convex polyhedra are familiar solid bodies like cubes, pyramids, or hundred-faced diamonds. A convex polytope is the same kind of body in the fourth dimension or higher. Since 1980, algebraic geometry has become a major tool for studying polytopes. This is surprising, since algebraic geometry includes some of the most abstract mathematics known, while polytopes, like crystals, are very concrete objects. Sheafhom will make possible some difficult computations in algebraic topology and geometry that will advance our understanding of convex polytopes. ***