noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Hirzebruch signature theorem?
Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
By pseudo-differential analysis? an elliptic operator acting on sections of two vector bundles on a manifold is a Fredholm operator and hence has closed kernel and cokernel of finite dimension. The difference of these two dimensions is the analytical index of the operator.
More generally, for $(E_p, D_p)$ an elliptic complex, its analytical index is the alternating sum
This index does not the depend of the Sobolev space used to get a bounded operator (by elliptic regularity the kernel is made up of smooth sections and the same for the cokernel as it is the kernel of the adjoint). By topological K-theory one can associate to it also a topological index. The Atiyah-Singer index theorem say that this two indexes coincide.
analytical index