The reading seminar will kick off this term with a series of informal talks in which people will introduce the background of the various areas they are working on. The first talk will be give by me (Joel) at 11am on Tuesday 13th September, in the Salle de Profs, 9th floor of building NO. The abstract is below. If you would like to give a talk in this seminar then please email me, I’m always looking for volunteers! (Also note that in a while, the reading seminar will switch to focus on a single topic for several talks.)

Given an oriented Riemannian 4-manifold M, its twistor space is a certain 2-sphere bundle Z, over M. Z caries two natural almost complex structures and features of the Riemannian geometry of M can be translated into the almost complex geometry of Z. Traditionally, the focus is on the case when one of these almost complex structures makes Z a genuine complex manifold. This happens when the starting metric is anti-self-dual (part of the curvature tensor vanishes). Then one can use techniques of complex geometry to answer questions about M. More recently symplectic techniques have found applications. There is a natural closed 2-form on Z which sometimes tames one of the two almost complex structures. This corresponds to a curvature inequality for M. I will try and describe both the complex and symplectic approaches to twistor spaces, giving a few of the details along the way.