noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
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
from the tensor product of vector bundles with the cotangent bundle which is 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 $X = \mathbb{R}^n$ a Cartesian space and $D$ the Dirac operator of the flat connection, the symbol of $D$ reproduces the symbol map between differential forms and Clifford algebra elements.
For $E$ a real trivial line bundle then the principal symbol is equivalently just a real-valued smooth function on the cotangent bundle. Since any cotangent bundle is canonically a symplectic manifold, in this case the symbol may be regarded as a Hamiltonian funtion. The corresponding Hamiltonian flow is called the bicharacteristic flow of the given differential operator.
For instance chapter 2.5 of
Nigel Higson, John Roe, Lectures on operator K-theory and the Atiyah-Singer Index Theorem (pdf)