Algorithms.

Matrix multiplication underlies everything from computer graphics to machine learning. The naive algorithm takes O(n³) operations for n×n matrices, but faster algorithms exist — the matrix multiplication exponent ω tracks the best asymptotic speed anyone has proven achievable.

The traveling salesman problem — finding the shortest route visiting every city once — is NP-hard to solve exactly, so research focuses on how close a fast algorithm can guarantee to get to the true optimum for realistic (metric) instances.