∂if = f− sif xi − xi+1 where si acts on f by transposing xi and xi+1 and let π̃i = ∂i(xi(1− xi+1)f) Then the Grothendieck-Demazure polynomial κα, which is attributed to A. Lascoux and M. P. Schützenberger, is defined as κα = x α1 1 x α2 2 x α3 3 ... if α1 ≥ α2 ≥ α3 ≥ ..., i.e. α is non-increasing, and κα = π̃iκαsi if αi < αi+1, where si acts on α by transposing the indices. Example 2.1. Let α = (...

Fulton and MacPherson introduced the notion of bivariant theories related to Riemann-Roch-theorems, especially in the context of singular spaces. This is powerful formalism, which is a simultaneous generalization of a pair of contravariant and covariant theories. Natural transformations of bivariant theories are called Grothendieck transformations, and these generalize a pair of ordinary natura...

This paper is motivated by questions such as P vs. NP and other questions in Boolean complexity theory. We describe an approach to attacking such questions with cohomology, and we show that using Grothendieck topologies and other ideas from the Grothendieck school gives new hope for such an attack. We focus on circuit depth complexity, and consider only finite topological spaces or Grothendieck...

The spectrum of the Ces?ro operator C is determined on spaces which arises as intersections Ap ?+ (resp. unions ?-) Bergman Ap? order 1 &lt; p induced by standard radial weights (1-|z|)?, for 0 ? 1. We treat them reduced projective limits inductive limits) weighted Ap?, with respect to ?. Proving that these admit monomials a Schauder basis paves way using Grothendieck-Pietsch criterion deduce w...

How can one prove a sharp inequality? Symmetrization, Fourier analysis, and probability are often used, and we will survey some of these methods through examples. We then survey sharp constants in Grothendieck inequalities, leading to some recent work on computing the best constant for a Grothendieck-type inequality of Khot and Naor. (Joint work with Aukosh Jagannath and Assaf Naor)

We prove a formula, originally due to Feit and Fine, for the class of the commuting variety in the Grothendieck group of varieties. Our method, which uses a power structure on the Grothendieck group of stacks, allows us to prove several refinements and generalizations of the Feit-Fine formula. Our main application is to motivic DonaldsonThomas theory.

In the Grothendieck cohomology of a flag variety G/H there are two canonical additive bases, namely, the Demazure basis [D] and the Grothendieck basis [KK]. We present explicit formulae that reduce the multiplication of these basis elements to the Cartan numbers of G.

In each dimension n ≥ 3 there are many projective simplicial toric varieties whose Grothendieck groups of vector bundles are at least as big as the ground field. In particular, the conjecture that the Grothendieck groups of locally trivial sheaves and coherent sheaves on such varieties are rationally isomorphic fails badly.

We prove that a nonlinear version of the Grothendieck-Katz conjecture (essentially in the form given by Ekhedahl, Shepherd-Barron and Taylor) is equivalent to the original Grothendieck-Katz conjecture together with a certain differential algebraic geometric/model-theoretic statement: a type over C(t) with “p-curvature 0 for almost all p” is nonorthogonal to the constants.

In [Tama], a proof of the Grothendieck Conjecture (reviewed below) was given for smooth affine hyperbolic curves over finite fields (and over number fields). The purpose of this paper is to show how one can derive the Grothendieck Conjecture for arbitrary (i.e., not necessarily affine) smooth hyperbolic curves over number fields from the results of [Tama] for affine hyperbolic curves over finit...