Small Cosmological Constants in String Theory

We construct vacua of string theory in which all moduli are stabilized and the magnitude of the cosmological constant is exponentially small.  The vacua are supersymmetric AdS_4 solutions in flux compactifications of type IIB string theory on orientifolds of Calabi-Yau hypersurfaces.  I will explain the advances in computing topological data in Calabi-Yau compactifications that led to these solutions, then speculate about implications for the cosmological constant problem.  The vacuum energy is small because we ensure the exact cancellation of all perturbative contributions, through an explicit choice of integer parameters determined by the topology and quantized fluxes.  The nonperturbative contributions that remain are exponential in these integers.  Finding cosmological constants of small magnitude in this landscape is exponentially easier than in Bousso-Polchinski landscapes.  Extending this approach to positive cosmological constants in realistic universes is a difficult open problem.