The geometric distribution of ranks and Selmer groups over function fields

Bhargava, Kane, Lenstra, Poonen, and Rains proposed heuristics for the distribution of arithmetic data relating to elliptic curves, such as their ranks, Selmer groups, and Tate-Shafarevich groups.  As a special case of their heuristics, they obtain the minimalist conjecture, which predicts that $50\%$ of elliptic curves have rank $0$ and $50\%$ of elliptic curves have rank $1$.  After surveying these conjectures, we will explain joint work with Tony Feng and Eric Rains, verifying many of these conjectures over function fields of the form $\mathbb F_q(t)$, after taking a certain large $q$ limit.