L3

Static vacuum data and their conformal classes play an important role in the discussion of the smoothness of gravitational fields at null infinity. We study the question under which conditions such data admit non-trivial conformal rescalings which lead again to such data. Some of the restrictions implied by this requirement are discussed and it is shown that there exists a 3-parameter family of static vacuum data which are not conformally flat and which admit non-trivial rescalings.

I will discuss ongoing work to provide a proof for the following

conjecture: if the development of a time symmetric, conformally flat

initial data set admits a smooth null infinity, then the initial data

is Schwarzschildean in a neighbourhood of infinity. The strategy

to construct a proof consists in a detailed analysis of a

certain type of expansions that can be obtained using H. Friedrich's

"cylinder at infinity" formalism. I will also discuss a toy model for

the analysis of the Maxwell field near the

spatial infinity of the Schwarzschild spacetime

By classical results of Tarski and Artin-Schreier, the elementary theory of the field of real numbers can be axiomatized in purely Galois-theoretic terms by describing the absolute Galois group of the field. Using work of Ax-Kochen/Ershov and a p-adic analogue of the Artin-Schreier theory the same can be proved for the field $\mathbb{Q}_p$ of p-adic numbers and for very few other fields.

Replacing, however, the absolute Galois group of a field K by that of the rational function field $K(t)$ over $K$, one obtains a Galois-theoretic axiomatiozation of almost arbitrary perfect fields. This gives rise to a new approach to longstanding decidability questions for fields like

$F_p((t))$ or $C(t)$.

Hypergraph Transversals have been studied in Mathematics for a long time (e.g. by Berge) . Generating minimal transversals of a hypergraph is an important problem which has many applications in Computer Science, especially in database Theory, Logic, and AI. We give a survey of various applications and review some recent results on the complexity of computing all minimal transversals of a given hypergraph.