Geometric and Differential Topology

The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry.

In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.

In differential geometry, the Atiyah-Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold

In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram.

In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed).

In the mathematical field of geometric topology, the Poincaré conjecture (UK: , US: , French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere (the hypersphere that bounds the 4-ball in four-dimensional space).

In topology, a discipline within mathematics, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every simple closed curve in the plane (known as a Jordan curve) divides the plane into two regions: the interior, bounded by the curve, and an unbounded exterior, containing all of the nearby and faraway exterior points.

In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies.

In topology, a surface is a two-dimensional manifold.

In mathematics, the Poincaré-Hopf theorem (also known as the Poincaré-Hopf index formula, Poincaré-Hopf index theorem, or Hopf index theorem) is an important theorem in differential topology that relates zeros of continuous vector field on compact differential manifold to Euler characteristic of that manifold.

In topology in mathematics, Freedman's classification (or Freedman's theorem) is a central result about four-dimensional topological manifolds (short 4-manifolds).

In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it.

In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a closed, oriented, smooth manifold of dimension 4 is diagonalizable.

In differential topology, sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out").

particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney

In differential topology, the Whitney immersion theorem (named after Hassler Whitney)

In geometric topology and differential topology, an (n + 1)-dimensional cobordism