Elliptic curves are important sources of questions (and answers!) for mathematicians in number theory and algebraic geometry. The points of an elliptic curve have an "addition" operation, and the points of finite order with respect to this addition operation are called torsion points. This structure opens the door for even more questions, such as "what are the torsion points of an elliptic curve?" and "what do the components of these points look like?"

This introductory talk will cover the definition of an elliptic curve with examples, the definition of the group law, and two main results regarding the torsion points of elliptic curves: the Nagell-Lutz Theorem and Mazur's Theorem. Proofs will be discussed as time allows. No prior knowledge of elliptic curves is required.