(Oxford University)

A central tool in the construction of moduli spaces throughout algebraic geometry and beyond, geometric invariant theory (GIT) aims to sensibly answer the question, "How can we quotient an algebraic variety by a group action?" In this talk I will explain some basics of GIT and indicate how it can be used to build moduli spaces, before exploring one of its salient features: the non-canonicity of the quotient. I will show how the dependence on an additional parameter, a choice of so-called 'linearisation', leads to a rich 'wall crossing' picture, giving different interrelated models of the quotient. Time permitting, I will also speak about recent developments in non-reductive GIT, and joint work extending this dependence to the non-reductive setting.

One of many overlaps between logic and topology is duality: Stone duality links Boolean algebras with zero-dimensional compact Hausdorff spaces, and gives a useful topological way of describing certain phenomena in first order logic; and there are generalisations that allow one to study infinitary logics also. We will look at a couple of ways in which this duality theory is useful.'

This is the 4th talk of the study group on Beilinson's approach to p-adic Hodge theory, following the notes of Szamuley and Zabradi.

I shall finish the computation of the module of differentials of the ring of integers of the algebraic closure of Q_p and describe a universal thickening of C_p.

I shall also quickly introduce the derived de Rham algebra. Kevin McGerty will give a talk on the derived de Rham algebra in W5 or W6.