Mini-Workshop
CONFORMAL STRUCTURES AND ODEs
ABSTRACTS
Boris Doubrov (Minsk), Conformal geometries associated with
3-dimensional vector distributions
Abstract: I'll mainly speak about the case of 3-dimensional vector
distributions D on 6-dimensional manifolds with an integrable square root (a
subdistribution D'\subset D, such that [D',D']\subset D). This is a bit more
general language to describe a pair of two second-order PDE's on 1 function
of two variables. It also makes transparent various links between different
SL(4,R)=SO(3,3) geometries.
Maciej Dunajski (Cambridge and IMPAN), G2 structures,
rational curves, and ODEs
Abstract: Consider the space M of parabolas y=ax2+bx+c,
with (a,b,c) as coordinates on M.
Two parabolas generically intersect at two (possibly complex) points, and we
can define a conformal structure on M by declaring two points to be null
separated iff the corresponding parabolas are tangent. A simple calculation
of discriminant shows that this conformal structure is flat. In this talk I
shall show how replacing parabolas by rational plane curves of higher degree
allows constructing curved conformal structures in any odd dimension. In
dimension seven one can use this "twistor" construction to find G2
structures in a conformal class.
Bronislaw Jakubczyk (IMPAN, Warsaw), Characteristic cones of
non-integrable distributions
Abstract: We will survey recent results concerning invariants of
non-integrable distributions in terms of their characteristic directions. We
will analyse in detail the case of distributions of corank 2, where the cone
of characteristic directions is a Veronese curve in a projectivised
characteristic subdistribution, or is given by a finite family of (conformal)
vector fields.
Jacek Jezierski (Warsaw University), On conformal Yano-Killing tensors
Abstract: Properties of (skew-symmetric) conformal Yano-Killing
tensors will be reviewed.
Possible applications in General Relativity will be presented. The examples
of CYK tensors in Minkowski,
Kerr, Taub-NUT, de Sitter and anti-de Sitter spacetimes will be discussed.
Wojciech Krynski (IMPAN and Vienna), Affine contact equivalence of
ODEs and Veronese webs
Abstract: I will discuss invariants of ordinary differential equations
up to contact transformations preserving the canonical affine distribution on
the jets space. I will also consider a problem of classification of Veronese
webs which were introduced by Gelfand and Zakharevich in connection with
bi-hamiltonian systems. I will show that the two topics can be treated in an
unified way. The talk is based on a joint work with B. Jakubczyk.
George Sparling (Pittsburgh), Causal Geometries
Paul Tod (Oxford), Thursday lecture: Three-dimensional Einstein-Weyl
Geometry
Abstract: Einstein-Weyl geometry is a conformally-invariant
generalisation of the theory of Einstein spaces. In three dimensions, there
are many connections with integrable systems and super-symmetry. I shall
review some aspects of this theory.
Paul Tod, Friday lecture: Penrose's Conformally Cyclic Cosmology and
Weyl Curvature Hypothesis
Abstract: Conformally Cyclic Cosmology and the Weyl Curvature
Hypothesis are two sets of ideas
due to Penrose and separated by about 30 years. According to the Weyl
Curvature Hypothesis,
and as a matter of observation, the conformal structure of the Big Bang
is severely constrained.
If the observations of a positive cosmological constant are correct,
the conformal structure
of the remote future has very similar features. I shall review the two sets
of ideas
from the standpoint of mathematical General Relativity and see how they relate.