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Geometric invariant theory
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, geometric invariant theory (or GIT) is a method for constructing quotients by <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group actions</a> in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, used to construct <a href="/facts/Moduli_space/9sqrdpwR">moduli spaces</a>. It was developed by <a href="/facts/David_Mumford/WJoJ8K2u">David Mumford</a> in 1965, using ideas from the paper (Hilbert 1893) in classical <a href="/facts/Invariant_theory/5fIUh9h3">invariant theory</a>. </p><p>Geometric invariant theory studies an <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">action of a group</a> G on an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a> (or <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a>) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was to construct <a href="/facts/Moduli_space/9sqrdpwR">moduli spaces</a> in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a> as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with <a href="/facts/Symplectic_geometry/runXFCea">symplectic geometry</a> and <a href="/facts/Equivariant_topology/01HreDnt">equivariant topology</a>, and was used to construct moduli spaces of objects in <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, such as <a href="/facts/Instanton/IuXhVcRF">instantons</a> and <a href="/facts/Monopole_(mathematics)/EGoRN9pI">monopoles</a>. </p>