The Tate-Shafarevich group, Sha(E), of an elliptic curve E defined over the rational numbers plays an important role in the Birch and Swinnerton-Dyer conjecture, but very little is actually known about the structure of this group in general. In this talk we will recall the definition of Sha(E) both as the cokernel of the Kummer map and via its geometric interpretation as a collection of certain isomorphism classes of torsors of E. Using this latter description, we will introduce one strategy to attempt to understand the behavior of this group (and maybe even try to prove that it is finite). Some prior exposure to elliptic curves and/or group cohomology will be helpful. Speaker(s): Brandon Carter (UM)