Topology

Topology studies properties of spaces that are invariant under deformations. A special role is played by manifolds, whose properties closely resemble those of the physical universe. Stanford faculty study a wide variety of structures on topological spaces, including surfaces and 3-dimensional manifolds. The notion of moduli space was invented by Riemann in the 19th century to encode how Riemann surfaces vary in families; today, the study of geometric and homotopy-theoretic aspects of moduli spaces is an important subject with strong ties to algebraic and symplectic geometry. It also leads to interesting dynamical systems and group theory. More algebraic aspects of topology study homotopy theory and algebraic K-theory, and their applications to geometry and number theory.