Consider a rational map φ on the projective line, from which we form a (discrete) dynamical system via iteration, and let K be a number field. A fundamental question in arithmetic dynamics is the uniform boundedness conjecture of Morton and Silverman, which states that there is a constant independent of φ (depending only on its degree) giving an upper bound for the number of K-rational preperiodic points of φ. This is a deep conjecture, and no specific case of it is known. I have proposed a specific version of the conjecture: that in the case of a degree-2 rational map and K = Q, the upper bound is 12.

In this talk, which assumes no previous knowledge of arithmetic dynamics, I will describe why this question is so difficult and sketch work that has been done to date, including giving justification for my refined uniform boundedness conjecture. The techniques used so far, which have clear limitations, involve constructing algebraic curves parameterizing maps $\phi$ together with points of period n for varying n (so-called dynamic modular curves).