Noncommutative residue
Encyclopedia
In mathematics
, noncommutative residue, defined independently by M. and , is a certain trace on the algebra of pseudodifferential operators on a compact
differentiable manifold
that is expressed via a local density. In the case of the circle
, the noncommutative residue had been studied earlier by M. and Y. in the context of one-dimensional integrable system
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, noncommutative residue, defined independently by M. and , is a certain trace on the algebra of pseudodifferential operators on a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
that is expressed via a local density. In the case of the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
, the noncommutative residue had been studied earlier by M. and Y. in the context of one-dimensional integrable system
Integrable system
In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems.In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical...
s.