  Mock modular forms have found applications in numerous branches of
mathematical sciences since they were first introduced by Ramanujan nearly a
century ago. In this proceeding we highlight a new area where mock modular
forms start to play an important role, namely the study of three-manifold
invariants. For a certain class of Seifert three-manifolds, we describe a
conjecture on the mock modular properties of a recently proposed quantum
invariant. As an illustration, we include concrete computations for a specific
three-manifold, the Brieskorn sphere $\Sigma(2,3,7)$. This note is partially
based on the talk by the first author in the conference "Srinivasa Ramanujan:
in celebration of the centenary of his election as FRS" held at the Royal
Society in 2018.
