Mathematical Sciences: Topology, Arithmetic Groups and Toric Varieties

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9704535 McConnell The project has three parts. First, let G = SL(n,R), let Gamma be an arithmetic subgroup of G, and let X be the symmetric space for G. The spaces X/Gamma are one setting where automorphic forms can be defined; they provide a topological approach to parts of the Langlands conjectures. For each Hecke operator T on the cohomology of X/Gamma, MacPherson and McConnell have defined a cell complex W(T) which allows one to find the operator by cellular techniques, i.e., using only a finite amount of combinatorial data. They will investigate W(T) and extend the definition to more G. In the second part, Ash and McConnell are computing the cohomology of X/Gamma in degree five for certain Gamma for SL(4). The goal is to determine the cuspidal classes and the Hecke action on them. The third part concerns Sheafhom, a suite of computer programs McConnell has developed. Sheafhom provides models of chain complexes, spectral sequences, sheaves, and other objects. The largest application to date is its algorithm for finding the intersection homology (IH) of toric varieties in any perversity. One goal is to compute the intersection product on IH of toric varieties, with applications to convex polytopes. Sheafhom has also been used in the Ash-McConnell work. A computer algebra system is a program for calculation with algebra, as opposed to numbers. Any program can find 2 + 2, but a computer algebra system combines whole formulas: the input (2x + 7) + (4x - 3) becomes 6x + 4 automatically. 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 everything that every mathematician needs. In recent years, several disciplines have been given their own special-purpose systems-- Cayley/Magma for algebraists, Pari for number theorists, and a dozen others. McConnell has written Sheafhom, a computer algebra system for algebraic topology. The program, some 10,000 lines long, is written in Lisp, one of the liveliest (and most efficient) programming languages. The goal is to apply Sheafhom to study convex polytopes. Convex polyhedra are solid bodies with flat faces, like cubes, pyramids, or hundred-faced diamonds. A convex polytope is the same kind of body in the fourth or higher dimension. Since 1980, algebraic geometry has become a major tool for studying polytopes. This is surprising, because 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; these will advance our understanding of convex polytopes. McConnell's project actually has two other parts that are more loosely related to Sheafhom. These are connected with the Langlands conjectures, a very deep set of ideas that relates number theory to other, more geometric parts of mathematics. ***