In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, noncommutative residue, defined independently by M. and , is a certain trace on the algebra of
pseudodifferential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in m ...
s 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 ...
s.
See also
*
Dixmier trace
In mathematics, the Dixmier trace, introduced by , is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces are examples of singular traces.
Some applications of Dixmie ...
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 in ...
, 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