Prelude: Arithmetic Fundamental Groups and Noncommutative Algebra

From number theory to string theory, analyzing algebraic relations in two variables still dominates how we view laws governing relations between quantities. An algebraic relation between two variables defines a nonsingular projective curve. Our understanding starts with moduli of curves. From, however, cryptography to Hamiltonian mechanics, we command complicated data through a key data variable. We’re human; we come to complicated issues through specific compelling interests. That data variable eventually drags us into deeper, less personal territory. Abel, Galois and Riemann knew that; though some versions of algebraic geometry from the late 1970s lost it. A choice data variable (function to the Riemann sphere) to extract information brings the tool of finite group theory. Giving such relations structure, and calling for advanced tools and interpretations, is the study of their moduli — the main goal of the papers of this volume. Applications in this volume are to analyzing properties of the absolute Galois group GQ (Part I) and to describing systems of relations over a finite field (Part II). Such applications have us looking at finite group theory and related algebra (Parts III and IV)