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.