In this seminar, we will build on the foundations studied in the Algebra III lectures on moduli problems and geometric invariant theory. We will study links with toric geometry and symplectic geometry, and cover other questions in GIT such as how the quotient depends on the choice of linearisation and how one can construct quotients for non-reductive actions. The main goals of this seminar are as follows.

1. To explore the relationship between toric geometry and GIT. More precisely, we will explain how any projective normal toric variety can be constructed as a GIT quotient of an affine space, where the action is linearised by a character (see [C] for example). We would also like to explain a result which describes the automorphism groups of toric varieties.

2. To study how the GIT quotient depends on the choice of linearisation of the action, known as variation of GIT. We will describe a wall and chamber structure on the space of linearisations of a reductive group action such that the quotient only changes as one crosses a wall. Furthermore, we will explain the birational transformations between the GIT quotients produced by these wall-crossings. [DH,T]

3. To give an introduction to symplectic geometry and symplectic quotients [CdS]. Then we will prove the Kempf-Ness Theorem which gives a homeomorphism between a projective GIT quotient over the complex numbers and a symplectic quotient.

4. To understand how to construct quotients of non-reductive groups in algebraic geometry. We will explain some of the differences between reductive and non-reductive group actions and how one can construct quotients of non-reductive group actions. We will also explain some motivation for studying non-reductive group actions coming from moduli problems. [BDHK]