I will review our current mathematical understanding of waves on black hole backgrounds, starting with the classical boundedness theorem of Kay and Wald on Schwarzschild space-time and ending with recent boundedness and decay theorems on a wider class of black hole space-times.
There are many theorems concerning cycles in graphs for which it is natural to seek analogous results for directed graphs. I will survey
recent progress on certain questions of this type. New results include
(i) a solution to a question of Thomassen on an analogue of Dirac’s theorem
for oriented graphs,
(ii) a theorem on packing cyclic triangles in tournaments that “almost” answers a question of Cuckler and Yuster, and
(iii) a bound for the smallest feedback arc set in a digraph with no short directed cycles, which is optimal up to a constant factor and extends a result of Chudnovsky, Seymour and Sullivan.
These are joint work respectively with (i) Kuhn and Osthus, (ii) Sudakov, and (iii) Fox and Sudakov.
A graph homomorphism is a mapping of vertices which takes edges to edges. The endomorphisms of a graph (homomorphisms to itself) form a submonoid of he full transformation monoid on the vertex set. In the other direction, there is a construction of a graph from a transformation monoid, which will be described in the talk. Composing these maps gives closure operators on graphs and on monoids which have some interesting properties. There are also connections with finite automata and permutation groups.