Stability and tilts on triangulated categories

In representation theory, a theorem of Chindris' states that stability for modules is preserved under a tilting equivalence if weights are chosen in a suitable way. In algebraic geometry, a fundamental result on elliptic curves states that the Fourier-Mukai transform from a Poincaré line bundle preserves stability of coherent sheaves. In this talk, I will explain how these two results are connected, and how understanding this connection helps us prove new results on Bridgeland stability and polynomial stability on elliptic surfaces.