L2

A polar space $\Pi$ is a geometry whose elements are the totally isotropic subspaces of a vector space $V$ with respect to either an alternating, Hermitian, or quadratic form. We may form a new geometry $\Gamma$ by removing all elements contained in either a hyperplane $F$ of $\Pi$, or a hyperplane $H$ of the dual $\Pi^*$. This is a \emph{biaffine polar space}.

We will discuss two specific examples, one with automorphism group $q^6:SU_3(q)$ and the other $G_2(q)$. By considering the stabilisers of a maximal flag, we get an amalgam, or "glueing", of groups for each example. However, the two examples have "similar" amalgams - this leads to a group recognition result for their automorphism groups.

I shall discuss recent work in which bounds are obtained, generalising/specialising earlier work for general linear groups

Quaternionic quantum Hamiltonians describing nonrelativistic spin particles

require the ambient physical space to have five dimensions. The quantum

dynamics of a spin-1/2 particle system characterised by a generic such

Hamiltonian is described. There exists, within the structure of quaternionic

quantum mechanics, a canonical reduction to three spatial dimensions upon

which standard quantum theory is retrieved. In this dimensional reduction,

three of the five dynamical variables oscillate around a cylinder, thus

behaving in a quasi one-dimensional manner at large distances. An analogous

mechanism exists in the case of octavic Hamiltonians, where the ambient

physical space has nine dimensions. Possible experimental tests in search

for the signature of extra dimensions at low energies are briefly discussed.

(Talk based on joint work with Eva-Maria Graefe, Imperial.)

This meeting will mark the 80th birthday of Sir Roger Penrose. Twistor theory is one of his most remarkable discoveries and continues to have applications across pure mathematics and mathematical physics. This meeting will focus on some recent developments with speakers both on geometry and physics.

Speakers:

• Nima Arkani-Hamed (IAS, Princeton): Scattering without space-time
• Mike Eastwood (ANU): CR geometry and conformal foliations
• Nigel Hitchin (Oxford): Twistors and Octonions
• Andrew Hodges (Oxford): Polytopes and amplitudes
• Claude LeBrun (SUNY Stony Brook): On Hermitian, Einstein 4-Manifolds
• David Skinner (Perimeter Institute): Scattering amplitudes from holomorphic linking in twistor space
• Paul Tod (Oxford): Conformal cyclic cosmology

Registration will start at 1.30pm on the 21st with the first lecture at 2.15pm. The meeting will finish by 4.30pm on the 22nd. See the programme for more details.

There will be a reception at 6.30pm on the 21st July (Wadham College) followed by dinner at 7.15 in Wadham College.

Yves Couder and co-workers have recently reported the results of a startling series of experiments in which droplets bouncing on a fluid surface exhibit several dynamical features previously thought to be peculiar to the microscopic realm. In an attempt to 

develop a connection between the fluid and quantum systems, we explore the Madelung transformation, whereby Schrodinger's equation is recast in a hydrodynamic form. New experiments are presented, and indicate the potential value of this hydrodynamic approach to both visualizing and understanding quantum mechanics.