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The selected contributions in this volume originated at the Sundance conference, which was devoted to discussions of current work in the area of free resolutions

Based on a course given by the author, which focuses on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group. This book is devoted to rationality and rigidity criteria and their application in realizing certain groups as Galois groups of regular extensions of Q(T).

This book presents several basic techniques in algebraic geometry, group representations, number theory, ¿-adic and standard cohomology, and modular forms. It explores how NX(p) varies with p when the family (X) of polynomial equations is fixed. The text examines the size and congruence properties of NX(p) and describes the ways in which it is computed. Along with covering open problems and offering simple, illustrative examples, the author presents various theorems, including the Chebotarev density theorem and the prime number theorem.

This volume touches upon some significant themes that have arisen in the fields of operator algebras, low-dimensional topology and mathematical physics. The intention of this collection of papers is to reveal that these seemingly diverse fields are now inextricably interconnected.