Related papers: The Derived Ring of Differential Operators
A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially…
Divided difference operators are degree-reducing operators on the cohomology of flag varieties that are used to compute algebraic invariants of the ring (for instance, structure constants). We identify divided difference operators on the…
We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction, used in [GS]; the other involving quantum hamiltonian reduction of…
The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and…
We introduce the notion of discrete Baker-Akhiezer (DBA) modules, which are modules over the ring of difference operators, as a certain discretization of Baker-Akhiezer modules which are modules over the ring of differential operators. We…
The purpose of this note is to extend the results of V. Guillemin on elliptic self-adjoint pseudodifferential operators of order one, from operators defined on smooth functions on a closed manifold to operators defined on smooth sections in…
In the theory of species, differential as well as integral operators are known to arise in a natural way. In this paper, we shall prove that they precisely fit together in the algebraic framework of integro-differential rings, which are…
We consider a circle of ideas involving differential algebra, local Noetherian rings, and their generic formal fibers. Connecting these ideas gives rise to what we term a "twisted" subring $R$ of a ring $S$. Each such subring $R$ arises as…
We study linear ordinary differential equations which are analytically parametrized on Hermitian symmetric spaces and invariant under the action of symplectic groups. They are generalizations of the classical Lam\'e equation. Our main…
In this article, we examine the geometry of a group of Fourier-integral operators, which is the central extension of $Diff(S^1)$ with a group of classical pseudo-differential operators of any order. Several subgroups are considered, and the…
We describe globally nilpotent differential operators of rank 2 defined over a number field whose monodromy group is a nonarithmetic Fuchsian group. We show that these differential operators have an S-integral solution. These differential…
We construct a functor, from the category of schemes to the category of graded rings, that is an initial object for having a theory of Chern classes with an additive first Chern class. For any scheme $X$, the graded ring that our functor…
Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of…
A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a…
The Grothendieck construction of a diagram $X$ of categories can be seen as a process to construct a single category $\Gr(X)$ by gluing categories in the diagram together. Here we formulate diagrams of categories as colax functors from a…
A gradient dependent formula is derived for the spinless one-particle density-matrix operator z from the differential virial theorem. A gradient dependent formula is also derived for a spinless one-particle density-matrix operator that can…
For $k$ a field of positive characteristic and $X$ a smooth variety over $k$, we compute the Hochschild cohomology of Grothendieck's differential operators on $X$. The answer involves the derived inverse limit of the Frobenius acting on the…
In this paper we use families of finite subgroups to study Grothendieck rings associated to certain discrete groups, such as the arithmetic ones.
Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star,…
We define the concept of completely regular ordinary differential operators and give various criteria for operators to belong to this class. We give also criteria for Birkhof regularity of ordinary differential operators in terms of the…