Hilbert's Fifth Problem

Hilbert's Fifth Problem asks whether every mathematical structure called a “topological Lie group”—a space with both group and geometric properties—can also be described smoothly, like a curve or surface. In simple terms, it questions if groups that are only assumed to be continuous can actually have a well-defined, differentiable structure. The problem was solved affirmatively in the 1950s, proving that such groups are indeed “Lie groups,” meaning their structure can be described smoothly. This discovery unified the concepts of symmetry, geometry, and algebra, deepening our understanding of continuous symmetry in mathematics and physics.