This paper is a continuation of [1], where we have studied a general class of (pseudo)differential operators with nonlocal coefficients, referred to as operators with shifts, and obtained a local index formula (i.e., a formula expressing the index as the integral of a differential form explicitly determined by the principal symbol of the operator) for matrix elliptic operators of this kind. In ...

Multigrid methods are typically used to solve partial differential equations, i.e., they approximate the inverse of the corresponding partial differential operators. At least for elliptic PDEs, this inverse can be expressed in the form of an integral operator by Green's theorem. This implies that multigrid methods approximate certain integral operators , so it is straightforward to look for var...

We prove various pointwise estimates for solutions of linear elliptic difference inequalities with random coefficients. These estimates include discrete versions of the maximum principle of Aleksandrov and Harnack inequalities and Holder estimates of Krylov and Safonov for elliptic differential operators with bounded coefficients.

This paper concerns a rather concrete phenomenon in abstract operator algebras. The main examples of the algebras we study are algebras of singular integral operators (pseudo-differential operators of order zero). As everyone knows, the Fredhohn index of a pseudo-differential operator depends only on its symbol and the Atiyah-Singer Index Theorem gives an explicit formula for computing the depe...

This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential operators on a compact manifold, namely those operators P with the property that dim ker P, dim coker P < ∞. In general these integers are very difficult to compute without some very precise information abou...