(1.1). This paper concerns three aspects of the action of a compact group K on a space X. The first is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality.… (More)
Cellular structures or tessellations are ubiquitous in nature. Metals and ceramics commonly consist of space-filling arrays of single-crystal grains separated by a network of grain boundaries, and foams (froths) are networks of gas-filled bubbles separated by liquid walls. Cellular structures also occur in biological tissue, and in magnetic, ferroelectric… (More)
We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero-and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result… (More)
A combinatorial formula for the Pontrjagin classes of a triangulated manifold is given. The main ingredients are oriented matroid theory and a modified formulation of Chern-Weil theory.
We define an 'enriched' notion of Chow groups for algebraic varieties, agreeing with the conventional notion for complete varieties, but enjoying a func-torial push-forward for arbitrary maps. This tool allows us to glue intersection-theoretic information across elements of a stratification of a variety; we illustrate this operation by giving a direct… (More)
Assuming a certain " purity " conjecture, we derive a formula for the (complex) coho-mology groups of the affine Springer fiber corresponding to any unramified regular semisimple element. We use this calculation to present a complex analog of the fundamental lemma for function fields. We show that the " kappa " orbital integral that arises in the… (More)
We propose a measure of shape which is appropriate for the study of a complicated geometric structure, defined using the topology of neighborhoods of the structure. One aspect of this measure gives a new notion of fractal dimension. We demonstrate the utility and computability of this measure by applying it to branched polymers, Brownian trees, and… (More)