We present an introductory overview of research in
hypercomputation theory, especially as it relates to the relationship
between physics and computation. On the quantum side, we argue that a
computational model of quantum motion exists, showing that if
super-Turing computation is possible it must (somewhat paradoxically)
derive from uncomputable properties of classical action. On the
cosmological side, we review the work of Hogarth, Andréka, Németi
al, showing that realistic models of spacetime permit the solution of
formally undecidable problems. Conversely, very simple computational
considerations imply that closed timelike curves (CTCs), if they
exist, can act as very low capacity information storage systems.
The material, which is based on a recent talk at the BMC 2012, will be
presented in easily accessible form, and questions from the audience
(especially those we cannot answer) are very welcome.