We apply noncommutative deformation theory to general moduli problems.

It is well known that ordinary deformation theory of modules applies to the theory of moduli, and that it solves problems in very special algebraic situations.

In most algebraic situations, e.g. geometric invariant theory, the ordinary deformation theory is not sufficient. Olav Arnfinn Laudal generalized the deformation functor

Def_{M} : ℓ → Sets,

which goes from the category of local artinian (pointed) k-algebras to the category of sets and where M is an A-module, to a noncommutative deformation functor

Def_{M} : a_{r} → Sets,

which goes from the category of r-pointed, not necessarily commutative, artinian k-algebras to the category of sets, and where M = {M1, . . . , Mr} is a family of r (right) A-modules. The study of this generalization is interesting in its own rights, and it turns out that it more or less solves the problems in geometric invariant theory (e.g. when an action of a group is not free).

Lecture 1 introduces noncommutative deformation theory, and the class of resulting algebras. Lecture 2 gives the details of computation of the resulting algebras, and ends with a result on local representability. In Lecture 3 we observe that the representations of the (noncommutative) algebras resulting from the noncommutative deformation theory can be glued together to a noncommutative scheme-theory, thereby unifying representation theory and algebraic geometry. In lecture 4, we introduce Noncommutative Geometric Invariant Theory (NGIT) and prove that this solves some particular problems when a group acts non-freely. Also, we give an example on the moduli of endomorphisms.

The lectures are included in a text which will be available on the conference cite or on ArXiv. The four lectures is the background for the dynamical applications of noncommutative algebraic geometry, which is the second part of the text.