Beyond the Abelian World: New Frontiers in Zero-Sum Theory

Zero-sum theory is a central topic in additive combinatorics that studies the structure of sequences over finite groups and the conditions guaranteeing the existence of zero-sum subsequences. Fundamental parameters in this area include the Davenport constant and the Erdős–Ginzburg–Ziv constant, which measure the threshold lengths forcing zero-sum behavior. These invariants originated in the study of non-unique factorizations in algebraic number theory, but determining their exact values remains a challenging problem even for many finite abelian groups. In this talk, I will discuss recent progress on zero-sum problems in finite non-abelian groups. In particular, I will highlight how combinatorial techniques developed for abelian groups can be adapted—or fail—to extend to the non-abelian setting, and how new phenomena arise due to the lack of commutativity. I will also present several results that reveal surprising connections between classical zero-sum invariants of abelian groups and their analogues for non-abelian groups, pointing toward a broader combinatorial framework for zero-sum theory.

This talk was originally presented at the INMS 2026 Conference, held in Antalya, Turkey, from August 24–28, 2026. The conference travel was provided by the Office of the Dean, Graduate School and Research, Ahmedabad University.