Consider the classical heat equation in dimension d, and formally replace the external forcing term by white noise. The resulting "stochastic PDE" (SPDE, for short) is the so-called stochastic heat equation. It has been known for some time that the stochastic heat equation suffers from a "curse of dimensionality": It has function solutions if and only if the ambient dimension is one. First we present a rigorous formulation of this SPDE, and explain why it has function solutions in only one dimension. Then, we discuss some newly-found connections between systems of solutions and classical notions from geometric measure theory [joint work with Robert Dalang and Eulalia Nualart]. Time permitting, we also address the mentioned curse of dimensionality, in greater length, by presenting an unexpected connection to classical probabilistic potential theory and the theory of local times [joint work with Mohammud Foondun and Eulalia Nualart].