Parabolic quantitative rectifiability

The purpose of this thesis is to develop a parabolic analog of uniform rectifiability. First, we provide a very general result concerning corona decompositions and the big pieces functor. This result is stated in a very general metric space setting, but is extremely useful in the parabolic theory because various quantitative properties are "stable" under the big pieces functor. We then show that parabolic uniformly rectifiable sets admit corona decompositions in terms of parabolic uniformly rectifiable parabolic Lipschitz graphs. This result, in conjunction with the first discussed, shows that parabolic uniformly rectifiable sets have "big pieces squared" of parabolic uniformly rectifiable parabolic Lipschitz graphs. We move on to show that in domains with parabolic uniformly rectifiable boundaries solutions to the heat equation which are bounded and continuous up the boundary necessarily have good gradient estimates at the boundary. Finally, we show that parabolic Ahlfors-regular sets satisfying either the parabolic uniform rectifiability condition, or an appropriate time-directed interior and exterior thickness condition, necessarily have "big pieces of Lipschitz" graphs. In the case that the underlying set is parabolic uniformly rectifiable we can show that the graphs are also parabolic uniformly rectifiable. This thesis is essentially a compilation of four papers in which the results above are proved.