Abstract: I will explain how to reduce a version of the Bayer-Macri-Toda conjecture for a Calabi-Yau threefold, which ensures the existence of Bridgeland stability conditions on it, to verifying a Brill-Noether-type inequality for curves. This enables us to prove the existence of Bridgeland stability conditions for a broad class of Calabi-Yau threefolds. Based on the joint work with Soheyla Feyzbakhsh, Naoki Koseki, and Nick Rekuski.