Determinants, Permutations and Additive Combinatorics

In this talk we will introduce some new problems and related progress on determinants involving Legendre symbols, circular permutations and additive combinatorics. For example, we conjecture that for any finite subset $A$ of an additive cyclic group $G$ with $|A|=n>3$ there is a circular permutation $a_1,\ldots,a_n$ of the elements of $A$ such that all the $n$ sums
$$a_1+a_2+a_3, a_2+a_3+a_4,\ldots, a_{n-2}+a_{n-1}+a_n, a_{n-1}+a_n+a_1, a_n+a_1+a_2$$
are pairwise distinct. The speaker has proved this when $G$ is the infinite cyclic group $\mathbb Z$.