Special cycles and BPS jumping loci

I'll sketch an attempt to bring the theory of special cycles, a deep part of number theory, into the domain of supersymmetric string compactifications. I'll describe a construction based on jumping loci for BPS state counts – a separate phenomenon from the better-known wall-crossing! – and explain in what cases these jumping loci generalize some parts of the theory of special cycles. Finally, I'll conclude with a host of physical and mathematical conjectures raised by this line of investigation.