Abstract: Calabi-Yau manifolds have played a central role in both string theory and mathematics for decades, but in spite of this no Ricci-flat metric on a compact non-toroidal Calabi-Yau manifold is known. I will discuss a new physically motivated approach toward the determination of such metrics for K3 surfaces. The key remaining step is the determination of a BPS spectrum of a heterotic little string theory on T^2. I will use string dualities to provide a number of mathematical reformulations of this problem, ranging from open string reduced Gromov-Witten theory for the mirror K3 surface (in accordance with the SYZ conjecture) to Donaldson-Thomas theory for auxiliary Calabi-Yau threefolds. Finally, I will discuss new approximations to K3 metrics near the semi-flat limit that require only a minimal knowledge of this BPS spectrum.