The little Grothendieck problem (a special case of Boolean quadratic optimization) consists of maximizing ∑ ij Cijxixj over binary variables xi ∈ {±1}, where C is a positive semidefinite matrix. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C ∈ Rdn×dn a positive semidefinite matrix, the objective is to maximi...

We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate K-theory of Grassmannians to a bialgebra of stable Grothendieck polynomials, which is a K-theory parallel of the ring of symmetric functions.

0. Introduction. This paper continues the study of the noncommutative infinitesimal cohomology we introduced in [3]. This is the cohomology of sheaves on a noncommutative version of the commutative infinitesimal site of Grothendieck ([8]). Grothendieck showed that, for a smooth scheme X of characteristic zero, the cohomology of the structure sheaf on the infinitesimal site gives de Rham cohomol...

We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco–Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.

We construct an algebraic commutative ring T -spectrum BO which is stably fibrant and (8, 4)-periodic and such that on SmOp/S the cohomology theory (X, U) 7→ BO(X+/U+) and Schlichting’s hermitian K-theory functor (X, U) 7→ KO [q] 2q−p(X, U) are canonically isomorphic. We use the motivic weak equivalence Z×HGr ∼ −→ KSp relating the infinite quaternionic Grassmannian to symplectic K-theory to equ...

The unity of opposites in the title is essentially that between logic and geometry, and there are compelling reasons for maintaining that geometry is the leading aspect. At the same lime, in the present joint work with Myles Tierney there are important influences in the other direction: a Grothendieck " topology " appears most naturally as a modal operator, of the nature " it is locally the cas...

If $f : S' \to S$ is a finite locally free morphism of schemes, we construct symmetric monoidal "norm" functor $f_\otimes \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ the pointed unstable motivic homotopy category over $S$. $f$ étale, show that it stabilizes to \mathcal{S}\mathcal{H}(S') \mathcal{S}\mathcal{H}(S)$, $\mathcal{S}\mathcal{H}(S)$ $\mathbb{P...