  An adaptive method for parabolic partial differential equations that combines
sparse wavelet expansions in time with adaptive low-rank approximations in the
spatial variables is constructed and analyzed. The method is shown to converge
and satisfy similar complexity bounds as existing adaptive low-rank methods for
elliptic problems, establishing its suitability for parabolic problems on
high-dimensional spatial domains. The construction also yields computable
rigorous a posteriori error bounds for such problems. The results are
illustrated by numerical experiments.
