  We present a family of integral equation-based solvers for the linear or
semilinear heat equation in complicated moving (or stationary) geometries. This
approach has significant advantages over more standard finite element or finite
difference methods in terms of accuracy, stability and space-time adaptivity.
In order to be practical, however, a number of technical capabilites are
required: fast algorithms for the evaluation of heat potentials, high-order
accurate quadratures for singular and weakly integrals over space-time domains,
and robust automatic mesh refinement and coarsening capabilities. We describe
all of these components and illustrate the performance of the approach with
numerical examples in two space dimensions.
