Basic Twistor Geometry; Physical Motivations
Twistor theory is motivated by the idea that the union
between space-time structure and quantum-mechanical principles may well
involve non-standard quantization procedures. Two guiding principles underlying
the twistor approach are holomorphicity (complex analyticity) and non-locality,
these seeming to be features that an appropriate "quantized space-time"
ought to have. In this lecture, I shall concentrate on the basic twistor
geometry, in which space-time ideas are translated into a very
different-looking geometrical picture, this having close connections to some well-known classical geometry.
Twistor Cohomology and linear Physical Fields
Abstract: Massless physical field equations
(such as the wave equation, Maxwell's equations, the linearized Einstein
gravitational field, and the massless neutrino equation) find a remarkable
translation into twistor terms; they are represented as elements of holomorphic
sheaf cohomology. Such cohomology groups have for many years had important
roles to play in complex-manifold
theory; now we see that they play an important role, also, in the description of basic physical fields.
Non-Linear Fields and Googlies; a New Input from String Theory?
Abstract: Some of these constructions have non-linear versions, and these lead to twistor solutions to the problem of finding anti-self-dual solutions of the Einstein vacuum equations (Ricci-flatness) and the Yang-Mills equations, relevant to strong and weak interactions. But we also need to see how to solve the self-dual versions of these equations, which has been termed the "googly problem", and this has proved remarkably resistant to any coherent mathematical treatment. Some recent new input has come from an unexpected source, starting with a 100-page paper by Edward Witten, in December 2003. I shall try to address the role of these new ideas.
Refreshments will be served at 3:30p.m. preceding each lecture.