Introduction to geometric invariant theory

"Geometric invariant theory" (GIT) is a way of taking quotients (by group actions) in algebraic geometry. The algebraic side of this story is about finding the subalgebra of invariants (for an algebra with a group action) and hence the name "invariant theory". This might look dry and technical, but it is beautifully geometric and is also related (rather unexpectedly) to symplectic geometry and convex geometry. Two Davids had major contributions to the development of GIT: David Hilbert and David Mumford. In this talk (aimed at graduate students in algebra and geometry), I will give an overview of basic concepts and results in geometric invariant theory. I will assume only familiarity with basics of algebraic geometry (such as affine and projective varieties and coordinate rings and homogeneous coordinate rings etc.).