Geometric Cut-and-Paste Problems
A common approach to problem-solving is to split a problem into smaller sub-problems, solve each of the smaller problems, and assemble the answers into a solution to the original problem. This last step is often very difficult, as there are multiple ways of gluing the pieces of the solution together. The mathematical area of K-theory studies the different ways of putting such solutions back together, as well as the relations behind differently-assembled pieces.