Algebraic Equations and Applied Group Theory
Physics and Chemistry laws (equations) have behind them a use of
symmetry (group theory). They appear by the device of preserving an
outcome under some action. Three methods from my research: "Monodromy
Method," "Galois Stratification" and "Modular Towers" used modern group
theory to reformulate, and solve, long-outstanding problems. These
methods correspond roughly to the division into three parts of my
They started by inverting this group/equation relation. The paper "Variables separated equations and simple groups" shows how this changed our thinking about equations and their relation to functions.
One dramatic influence was in its formulation (and solution) of the "genus 0 problem,"
by Bob Guralnick (USC) and John Thompson (Fields Medalist: Cambridge and U.
of Florida). That story also documents how other researchers managed to
learn the techniques behind my methods.