Abstract: The minimalist conjecture predicts that, in quadratic twist families of abelian varieties, half have rank 0 and half have rank 1. This fits into the larger picture of the Bhargava-Kane-Lenstra-Poonen-Rains heuristics, which predict the distribution of Selmer groups of these abelian varieties. In joint work with Jordan Ellenberg, we prove a version of these heuristics: over function fields over the finite field $\mathbb{F_q}$, we show that the above heuristics are correct to within an error term in $q$, which goes to 0 as $q$ grows. The main inputs are a new homological stability theorem in topology for a generalized version of Hurwitz spaces and an expression of average sizes of Selmer groups in terms of the number of rational points on these Hurwitz spaces over finite fields.

Abstract: Mutation is the engine of evolution. By considering only single point mutations (SNPs) on DNA sequences, we see a natural group theoretic model of mutations acting on the set of nucleotides. In this talk, we will investigate the implications of this structure for synonymous mutations (mutations that do not change the encoded amino acids) and how this affects the notion of distance between two genetic sequences.