(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)
Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs. Aomoto diloga-rithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane.
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)
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)
Suppose that a d-dimensional convex polytope P ⊂ R d is rational, i.e. its vertices are all rational points. Then P gives rise to a polynomial g(P) = 1 + g 1 (P)q + g 2 (P)q 2 + · · · with non-negative coefficients as follows. Let X P be the associated toric variety (see §6 – our variety X P is d + 1-dimensional and affine). The coefficient g i is the rank… (More)
Let k be an algebraically closed field, G a connected reductive group over k, and A a maximal torus in G. We write g for the Lie algebra of G and a for X * (A) ⊗ Z R. Let F = k((ǫ)) be the field of formal Laurent series over k and let o = k[[ǫ]] be the subring of formal power series. We fix an algebraic closure ¯ F of F. We write G and g for G(F) and g(F)… (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)
If an algebraic torus T acts on a complex projective algebraic variety X then the affine scheme Spec H * T (X; C) associated to the equivariant cohomology is often an arrangement of linear subspaces of the vector space H T 2 (X; C). In many situations the ordinary cohomology ring of X can be described in terms of this arrangement.