I will describe my work, all joint with D. Gaiotto and some also joint with E. Witten, to understand the homotopy type of the space of (1+1)d N=(0,1) SQFTs --- what a condensed matter theorist would call "phases" of SQFTs. Our motivating hypothesis (due in large part to Stolz and Teichner) is that this space models the spectrum called "topological modular forms". Our work includes many nontrivial checks of this hypothesis. First, the hypothesis implies constraints on the possible values of elliptic genera, and suggests (but does not imply) the existence of holomorphic SCFTs saturating these constraints; we have succeeded in constructing such SCFTs in low central charge. Second, the hypothesis implies the existence of torsion-valued "secondary invariants" beyond the elliptic genus that protect SQFTs from admitting deformations that spontaneously break supersymmetry. I will explain such an invariant in terms of holomorphic anomalies and mock modularity.