What is the Concept of Spectral Geometry?

[mhill]

What is Spectral Geometry ??

in many cases of Connes' work he introduced the concept (??) of spectral geometry, replacing the derivatives by commutators so

df \rightarrow (f,A) what does 'A' here mean ?? , it is similar to the Heisenberg

equation of motion ?? \dot f = (f,H)

Also instead of integrals he used expressions like

\int T = Res_{s=0} Tr( f|D|^{-s})

also he defined an 'infinitesimal operator' (??) dx or integral of infinitesimal operator as the value of the log(e) inside Tr_{e} or something similar.

the .pdf bear the name ' NONCOMMUTATIVE GEOMETRY AND PHYSICS' by the Physicist Alain Connes, i have tried googling but the papers that appeared had a heavy content on algebra and Galois theory.

 

[Greg Bernhardt]

Spectral geometry is a branch of mathematics that studies the geometric properties of a spectrum. A spectrum is a vector space of functions on a given domain; spectral geometry is the study of the geometry of such vector spaces. A spectral geometry is defined by a choice of a spectral family, which is a locally finite collection of spectra of functions on a given domain, such that for all such functions, the limit of the scalar product of the function with any vector in the collection is nonzero.