Related papers: Lidskii-type formulae for Dixmier traces

We introduce a cohomology set for groups defined by algebraic difference equations and show that it classifies torsors under the group action. This allows us to compute all torsors for large classes of groups. We also develop some tools for…

For manifolds with boundary, we define an extension of Wodzicki's noncommutative residue to boundary value problems in Boutet de Monvel's calculus. We show that this residue can be recovered with the help of heat kernel expansions and…

A fundamental result that characterizes elliptic-hyperbolic equations of Tricomi type, the uniqueness of classical solutions to the open Dirichlet problem, is extended to a large class of elliptic-hyperbolic equations of Keldysh type. The…

For almost twenty years, a search for a Lorentzian version of the well-known Connes' distance formula has been undertaken. Several authors have contributed to this search, providing important milestones, and the time has now come to put…

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups and we use structure theorems for these groups to…

The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in…

In this paper we prove a version of Connes' trace theorem for noncommutative tori of any dimension~$n\geq 2$. This allows us to recover and improve earlier versions of this result in dimension $n=2$ and $n=4$ by Fathizadeh-Khalkhali. We…

The aim of this article is to obtain variations on the classical theorems of Schur and Baer on finiteness of commutator subgroups, valid in the contexts of Lie algebras and Leibniz algebras over a field. Using non-abelian tensor products…

We present a geometric framework for discrete classical field theories, where fields are modeled as "morphisms" defined on a discrete grid in the base space, and take values in a Lie groupoid. We describe the basic geometric setup and…

We obtain general trace formulae in the case of perturbation of self-adjoint operators by self-adjoint operators of class $\bS_m$, where $m$ is a positive integer. In \cite{PSS} a trace formula for operator Taylor polynomials was obtained.…

In these notes, we present versions of trace theorems for Sobolev spaces over an interval in the real line, and also a one-dimensional version of the well-known Poincare inequality.

We introduce a Lie algebra associated with a non-orientable surface, which is an analogue for the Goldman Lie algebra of an oriented surface. As an application, we deduce an explicit formula of the Dehn twist along an annulus simple closed…

First, we shall formulate and prove Theorem of Lie-Kolchin type for a cone and derive some algebro-geometric consequences. Next, inspired by a recent result of Dinh and Sibony we pose a conjecture of Tits type for a group of automorphisms…

A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative. Some of them generalize formulas and theorems known in the…

We obtain a trace formula for algebraic differential operators which the corresponding analytic results have been proved by M. Engeli and G. Felder

Connes' distance formula is applied to endow linear metric to three 1D lattices of different topology, with a generalization of lattice Dirac operator written down by Dimakis et al to contain a non-unitary link-variable. Geometric…

The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define relative Lefschetz numbers and Reidemeister…

The Wodzicki residue is the unique trace on the algebra of classical pseudodifferential operators on a closed manifold, and Connes in 1988 proved that it coincides with the Dixmier trace. A Carnot manifold is a manifold $M$ whose tangent…

Differential-geometric structures on the space of orbits of a finite Coxeter group, determined by Groth\'endieck residues, are calculated. This gives a construction of a 2D topological field theory for an arbitrary Coxeter group.

We construct discrete analogues of the Dixmier operators, that is, commuting difference operators corresponding to a spectral curve of genus 1 whose coefficients are polynomials of the discrete variable.