In
mathematics, noncommutative residue, defined independently by M. and , is a certain trace on the algebra of
pseudodifferential operators on a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
that is expressed via a local density. In the case of the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
, the noncommutative residue had been studied earlier by M. and Y. in the context of one-dimensional
integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
s.
See also
*
Dixmier trace
References
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*{{Citation , last1=Wodzicki , first1=Mariusz , title=K-theory, arithmetic and geometry (Moscow, 1984--1986) , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , series=Lecture Notes in Math. , doi= 10.1007/BFb0078372 , mr=923140 , year=1987 , volume=1289 , chapter=Noncommutative residue. I. Fundamentals , pages=320–399
Noncommutative geometry