The Mordell conjecture, famously proven by Faltings, that an algebraic curve of genus greater than one has only finitely many rational points admits a geometric reformulation which then naturally generalizes to higher dimensional varieties giving the Mordell-Lang conjecture: if $A$ is an abelian variety over the complex numbers, $\Gamma \leq A(\mathbb{C})$ is a finite rank subgroup, and $X \subseteq A$ is an algebraic subvariety, then $\Gamma \cap X(\mathbb{C})$ is a finite union of cosets of subgroups of $\Gamma$. The Mordell-Lang conjecture and related conjectures, such as the Manin-Mumford and Bogomolov conjectures, were proven already in the 1980s and 1990s and then the problem shifted to that of finding more effective descriptions and bounds for the intersections $\Gamma \cap X(\mathbb{C})$. Earlier work of such people as Bombieri, Faltings, Mumford, Rémond, Vojta, Ullmo, and Zhang, amongst others, has produced some effective bounds that depend on the arithmetic of the problem, usually formulated in terms of various heights. The main theorems discussed in this lecture give bounds depending entirely on geometric data, such as the dimensions and degrees of $X$ and $A$ and the rank of $\Gamma$. Interestingly, the new results are based on refinements of the arithmetic height inequalities already appearing in the earlier work together with a study of the so-called Betti map which takes into account the real analytic geometry of universal families of abelian varieties.
This is a report on work of several mathematicians including, but not
limited to, Cantat, Dimitrov, Gao, Ge, Habegger, Kühne, Masser, Xie, Yuan, Zannier, and Zhang.
