The Laplacian of a graph is a matrix that describes that graph, turning a graphical abstraction into a set of linear equations that describes the graph's connectivity in exactly the same way. It's useful in many areas of, naturally, graph theory, which in turn contributes to various fields from computer science to electrical engineering to mathematics.
Apparently it was only in 2004 that "researchers first proposed an algorithm that solved graph Laplacians in 'nearly linear time,' meaning that the algorithm's running time didn't increase exponentially with the size of the problem." The proof was also hundreds of pages long.
The breakthrough here? According to the researchers: "We were able to replace it with something that would fit on a blackboard."