Word equations in simple groups and polynomial equations in simple algebras

TL;DR

This paper surveys recent advances in understanding word maps on simple groups and polynomial maps on simple algebras, highlighting parallels and proposing new open problems and solutions.

Contribution

It draws connections between the theories of word maps and polynomial maps, introducing new perspectives and open questions in the study of simple algebraic structures.

Findings

Identifies parallels between word maps and polynomial maps

Proposes new open problems in the field

Suggests new methods for solving existing problems

Abstract

We give a brief survey of recent results on word maps on simple groups and polynomial maps on simple associative and Lie algebras. Our focus is on parallelism between these theories, allowing one to state many new open problems and giving new ways for solving older ones.