The notation "S^n" represents the n-dimensional sphere in mathematics. It is defined as the set of points in (n+1)-dimensional Euclidean space that are at a constant distance (radius) from a central point, typically the origin. For example, S^1 is a circle, S^2 is a standard sphere, and S^3 represents a 3-dimensional sphere, which is more abstract and difficult to visualize. S^n is significant in various fields, including topology and geometry. It helps mathematicians study properties of spaces that remain unchanged under continuous transformations. The study of S^n also connects to concepts like homotopy, homology, and manifolds, which are essential in advanced mathematics.