We study the topological heterotic ring in (0,2) Landau-Ginzburg models without a (2,2) locus. The ring elements correspond to elements of the Koszul cohomology groups associated to a zero-dimensional ideal in a polynomial ring, and the computation of half-twisted genus zero correlators reduces to a map from the first nontrivial Koszul cohomology group to complex numbers. This map is a generali...

The purpose of this proposal is to study algebraic symplectic varieties, which arise naturally in algebraic geometry (Hilbert schemes), representation theory (quiver varieties, Springer theory), combinatorics and polyhedral geometry (hypertoric varieties), and string theory (moduli spaces of gauge theories and of Higgs bundles). Our primary interest will be a certain category of sheaves on thes...

In this paper we study the relationships between operations in K-theory and ordinary mod p cohomology. In particular, conditions are given under which the mod p associated graded ring of a filtered λ-ring is an unstable algebra over the Steenrod algebra. This result partially extends to the algebraic setting a topological result of Atiyah about operations on K-theory and mod p cohomology for to...

Let I denote an ideal of a local Gorenstein ring (R, m). Then we show that the local cohomology module H I (R), c = height I, is indecomposable if and only if V (Id) is connected in codimension one. Here Id denotes the intersection of the highest dimensional primary components of I. This is a partial extension of a result shown by Hochster and Huneke in the case I the maximal ideal. Moreover th...

We prove a conjecture of Knutson asserting that the Schubert structure constants of the cohomology ring of a two-step flag variety are equal to the number of puzzles with specified border labels that can be created using a list of eight puzzle pieces. As a consequence, we obtain a puzzle formula for the Gromov-Witten invariants defining the small quantum cohomology ring of a Grassmann variety o...

It is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies. We look at twisted tensor products, and investigate to which extent this is still true. We give an explicit description of the Ext-algebra of the tensor product of two modules, and under certain additional conditions, describe an essential part of the Hochschild cohomology ring of a t...