we'll be talking about recent results on the problem of splitting brauer classes by torsors for genus one curves. in its geometric form the question to be asked is: does every severi--brauer variety contain a smooth and projective genus one curve? algebraically, this question is related to the existence of certain finite galois modules inside the linear algebraic automorphism group of the severi--brauer variety. our goal will be to motivate why this is an intuitive and interesting question, giving some new results along the way.