In algebraic geometry it is often useful to be able t o construct quotients of algebraic varieties by l inear algebraic group actions\; in particular modu li spaces (or stacks) can be constructed in this way. When the linear algebraic group is reductive\ , and we have a suitable linearisation for its act ion on a projective variety\, we can use Mumford&# 39\;s geometric invariant theory (GIT) to construc t and study such quotient varieties. The aim of th is talk is to describe how Mumford'\;s GIT can be extended effectively to actions of linear algeb raic groups which are not necessarily reductive\, with the extra data of a graded linearisation for the action. Any linearisation in the traditional s ense for a reductive group action can be regarded as a graded linearisation in a natural way.

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The classical examples of moduli spaces w hich can be constructed using Mumford'\;s GIT a re the moduli spaces of stable curves and of (semi )stable bundles over a fixed curve. This more gene ral construction can be used to construct moduli s paces of unstable objects\, such as unstable curve s (with suitable fixed discrete invariants) or uns table bundles (with fixed Harder-Narasimhan type). LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR