noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Hirzebruch signature theorem?
For $X$ a smooth manifold, $E \to X$ a vector bundle and $D : \Gamma(E) \to \Gamma(E)$ a differential operator on sections of $E$, its symbol is the bundle morphism
given at any point $x \in X$ on a cotangent vector of the form $(\mathbf{d}f)_x \in \Gamma(T^* X)_x$ by
where in the commutator on the right we regard multiplication by $f$ as an endomorphism of $\Gamma(E)$.
The symbol may naturally be thought of as an element in the K-theory of $X$ (Freed).
For instance chapter 2.5 of
Nigel Higson, John Roe, Lectures on operator K-theory and the Atiyah-Singer Index Theorem (pdf)