Fibrations by Cayley submanifolds

I just submitted my third paper to the arXive, you can find it here. This completes my DPhil project by proving that Cayley fibrations are stable under small perturbations of the Spin(7)-structure. This is done by looking at how nearly singular Cayleys behave under perturbation. It turns out that at least up to C^1 they behave as expected, with the gluing described in my second paper giving a good approximation. We do this by solving the linear Cayley equation on the glued manifold and showing that the solutions (which describe the first derivative of the Cayley fibration so to speak) can also be obtained by gluing well-understood pieces. We use this to complete the longstanding programme of Kovalev to construct coassociative fibration on twisted connected sum G_2 manifolds.

Oxford go tournament 2024

Last weekend, the Oxford go tournament 2024 was held again for the first time after the pandemic. In total 54 players were present, some of which coming from as far away as Nottingham! In attendance were some of the best players in the UK, as well as almost complete beginners. Harry put in a lot of work to organise a side 9×9 tournament that was particularly popular with the youngsters. One of them played 39 (!) games in addition to their 3 tournament (19×19) games.

The playing hall was bustling with games at all ranks

Thanks to generous sponsorship we were able to hand out quite a few prizes such as vouchers for online courses or Go books by a small German publishing company (Brett&Stein, you should check them out). We were able to bring out the customary Oxford Go club scroll as decoration, and they were much appreciated.

While we had lovely weather on the day, mother nature still prompted us to stay inside, for outside the rain was not quite gone yet.

This newly formed lake is where Botley park tends to be

Organising the tournament was quite a bit of work, but I hope that the work that Harry and I did this year can translate into a more streamlined process next year. That should leave more time to make sure that the clocks are all set correctly (whoops) and that the printer is working properly when it is needed (likely no amount of preparation is going to fix this, did you know that HP need to be connected to the internet EVEN if you just want to use them with a USB cable?). At least we left some room for improvement! See you all next year.

in Go | 287 Words

Deformation theory of Cayley submanifolds

I just published my first paper on the arxive here. It’s about the deformation theory of conically singular and asymptotically conical Cayley manifolds, based on the work of McLean and Moore. I prove the usual theorems about the existence of Kuranishi charts and discuss some relations to lower-dimensional geometries, such as special Lagrangian, complex and coassociative geometries. For example, special Lagrangians in Calabi-Yau fourfolds will always be obstructed as Cayleys. It is the first in a series of three papers working towards constructing fibrations of compact Spin(7) manifolds by Cayley submanifolds. The next paper which is about the desingularisation of Cayley submanifolds will follow shortly.