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Karol Borsuk was a mathematician whose work helped develop topology, particularly the study of retracts and the relationship between geometric and topological structure. His collaboration with Stanisław Ulam produced results about continuous mappings and topological invariants.^[ref] The Borsuk-Ulam theorem states that every continuous map from the sphere $S^n$ to ${\mathbb{R}}^n$ takes the same value at some pair of antipodal points. The equal-value pair is forced by global topology, even though the map may have no explicit symmetry. This provides a tool for existence proofs rather than a recipe for locating the pair.

Stanisław Ulam helped develop the modern computational use of Monte Carlo methods. The firsthand account by Nicholas Metropolis describes Ulam's proposal to revive statistical sampling with electronic computers and its subsequent development with John von Neumann.^[ref]

In particle-transport calculations, one samples successive events—scattering, absorption and branching—to produce many possible histories. Averaging their outcomes estimates a macroscopic quantity. The link between a physical random process, probability distributions and an executable algorithm made difficult calculations accessible without enumerating every possible trajectory.^[ref]