Thursday, July 28, 2005

Stable Algebraic Varieties, Nearly Kahler Manifolds and Twistors

Today I have been on an excursion, for it was our day of rest. The organisers of the symposium arranged for us to leave Grey College at 9.15am and travel by coach to Bamburgh castle, which was the seat of the kings of Northumbria and also the setting for a peculiar worm legend, after that we were whisked off to Holy Island (at which point there followed a series of very poor "Holy Island X Batman!" jokes, where X was some local attraction on the island) and finally we made a brief aborted stop at Alnwick castle. Some outdoor scenes of Hogwart's from Harry Potter was filmed at Alnwick, but due to our sharp exit we didn't even see the outside of the castle. It rained for most of the day, still it was fun to get out of the lecture theatre for a bit.

Yesterday morning we listened to Richard Thomas talk to us about "Special metrics and stability of algebraic varieties". He told us about Geometric Invariant Theory (GIT) and how it may be used to understand a stability condition on both algebraic varieties and simultaneously on related Kahler and Kahler-Einstein metrics. The essential idea was that the stability of a vector bundle can be made into a well-defined notion even over non-compact groups by using a GIT :), and then extending the analogue of this idea to algebraic varieties, and Kahler-Einstein metrics. The detail about the different kinds of stability conditions that can exist can be read about in his paper with Ross entitled "A study of the Hilbert-Mumford criterion for the stability of projective varieties".

In the afternoon yesterday we heard talks from Misha Verbitsky on "An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry" which was based on material from his paper; and Gabriele Travaglini who talked to us about the progress in understanding the calculation of loop diagrams coming from the twistor string programme. Travaglini's talk was entitled "From Twistors to Amplitudes" and was a review of the progress in understanding why some very complex loop Feynman diagrams turn out to have very simple contributions (i.e. zero) to amplitude calculations, as seen from the twistor viewpoint by making use of maximal helicity violating (MHV) diagrams to simplify otherwise horrendous calculations.

In the evening yesterday we had a pub quiz, where after some very suspicious recounting, victory was snatched from the grasp of two of the other (less suspicious) teams by means of a tie-break. The tie-break question was to guess the number of Durham's in the USA, it turns out there are 21, and after a dubiously large team (of the suspicious counting) claimed victory, Richard Thomas, whose team lost out in the 3-way tie-break, magnanamously conceded defeat by pointing out that the winning team had 19 participants, while most of the other teams had only 4 members. Good fun was had by all.