  Classical approach of solvability problem has shed much light on what we can
solve and what we cannot solve mathematically. Starting with quadratic
equation, we know that we can solve it by the quadratic formula which uses
square root. Polynomial is a generalization of quadratic equation. If we define
solvability by using only square roots, cube roots etc, then polynomials are
not solvable by radicals (square root, cube root etc). We can classify
polynomials into simple (solvable by radicals) and complex (not solvable by
radicals). We will use the same metaphor to separate what is solvable (simple
part) and what is not solvable (complex part).
  This paper is a result of our presentation at a University of Houston
seminar. In this paper, we will study seismic complexity through the eyes of
solvability. We will investigate model complexity, data complexity and operator
complexity. Model complexity is demonstrated by multiple scattering in a
complex model like Cantor layers. Data complexity is studied through Betti
numbers (topology/cohomology). Data can be decomposed as simple part and
complex part. The simple part is solvable as an inverse problem. The complex
part could be studied qualitatively by topological method like Betti numbers.
Operator complexity is viewed through semigroup theory, specifically through
idempotents (opposite of group theory). Operators that form a group are
invertible (solvable) while semigroup of operators is not invertible (not
solvable) in general.
