Abstract: This pair of talks is aimed at introducing the concept of height to graduate students who are new to number theory and discussing an open problem which may interest students. The concept of height first arose in Diophantine geometry, particularly in the study of rational points on elliptic curves. Over the years, as mathematicians realized a "dictionary" between arithmetic geometry and the arithmetic of dynamical systems, height became a key tool in studying arithmetic dynamics as well. Results like equidistribution theorems, which focus on the behavior of points of low height, have become key tools in both arithmetic geometry and in arithmetic dynamics. Many well-known problems on geometric side have inspired more recent analogues, often open, on the dynamical side; for example, the celebrated Mazur-Merel theorem on bounding the torsion of group of an elliptic curve naturally generalizes to the uniform boundedness conjecture of Morton and Silverman, which is still unproven even in very simple cases such as for the family of quadratic maps over the rationals. In the first talk, we will go over the basic definitions of the Weil height, give some examples on both the geometric and the dynamical side, and talk about some common problems with heights and their applications. In the second talk, we will explore an open problem: finding a dynamical analogue of the height gap theorem of Amoroso and Dvornicich for algebraic numbers in cyclotomic extensions. We will sketch a proof of the classical result, and then we will explore why a dynamical analogue of this result has proven to be so elusive. |