In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an action of a group on an algebraic variety (or scheme) and provides techniques for forming the 'quotient' of by as a scheme with reasonable properties. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopoles. Background Invariant theory is concerned with a group action of a group on an algebraic variety (or a scheme) . Classical invariant theory addresses the situation when is a vector space and is either a finite group, or one of the classical Lie groups that acts linearly on . This action induces a linear action of on the space of polynomial functions on by the formula The polynomial invariants of the -action on are those polynomial functions on which are fixed under the 'change of variables' due to the action of the group, so that for all in . They form a commutative algebra , and this algebra is interpreted as the algebra of functions on the 'invariant theory quotient' because any one of these functions gives the same value for all points that are equivalent (that is, for all ). In the language of modern algebraic geometry, Several difficulties emerge from this description. The first one, successfully tackled by Hilbert in the case of a general linear group, is to prove that the algebra is finitely generated. This is necessary if one wanted the quotient to be an affine algebraic variety. Whether a similar fact holds for arbitrary groups was the subject of Hilbert's fourteenth problem, and Nagata demonstrated that the answer was negative in general. On the other hand, in the course of development of representation theory in the first half of the twentieth century, a large class of groups for which the answer is positive was identified; these are called reductive groups and include all finite groups and all classical groups. The finite generation of the algebra is but the first step towards the complete description of