Abstract:
 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 abelian 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 $a_1+a_2+a_3,a_2+a_3+a_4,\ldots,a_{n2}+a_{n1}+a_n,a_{n1}+a_n+a_1,a_n+a_1+a_2$ are pairwise distinct. The speaker has proved this for any torsionfree abelian group $G$, but it is even open for cyclic groups of prime orders.
