This paper offers a systematic approach to all mathematical theories with local/global behavior. To build objects with local and global aspects, on begins with a category of [script]C of allowed local structures, and somehow derives a category [script]C[superscript]gl of things which are 'locally' in [script]C. Some global objects, such as manifolds or schemes, can be represented as a sheaf of algebras on a topological base space; others, like algebraic spaces, are more technical. These theories share common structure--certain theorems on inverse limits, descent, and dependence on special class of morphism appear in all cases. Yet, classical proofs for universal properties proceed by case-by-case study. Separate examples require distinct arguments.

Requiring only familiarity with the terminology of categories, this book will interest algebraic geometers and students studying schemes for the first time. Feit translates the geometric intuition of local structure into a purely categorical format, filling a gap at the foundations of algebraic geometry. The main result is that, given an initial category ${\cal C $ of ''local'' objects and morphisms, there is a canonical enlargement of ${\cal C $ to a category ${\cal C {gl $ which contains all ''global'' objects whose local structure derives from ${\cal C $ and which is functorially equivalent to the traditional notion of ''global objects''. Using this approach, Feit unifies definitions for numerous technical objects of algebraic geometry, including schemes, Tate's rigid analytic spaces, and algebraic spaces.

The basic definitions and properties of vertex operator algebras, modules, intertwining operators and related concepts are presented, following a fundamental analogy with Lie algebra theory. The first steps in the development of the general theory are taken, and various natural and useful reformulations of the axioms are given. In particular, tensor products of algebras and modules, adjoint vertex operators and contragradient modules, adjoint intertwining operators and fusion rules are studied in greater depth. This paper lays the monodromy-free axiomatic foundation of the general theory of vertex operator algebras, modules and intertwining operators.

"July 1993, volume 104, number 497 (fourth of 6 numbers)."

The philosophy of the first part of this work is to understand (and classify) Kummer surfaces by studying (16, 6) configurations. Chapter 1 is devoted to classifying (16, 6) configurations and studying their manifold symmetries and the underlying questions about finite subgroups of [italic capitals]PGL4([italic]k). In chapter 2 we use this information to give a complete classification of Kummer surfaces together with explicit equations and the explicit description of their singularities.

This work studies abelian branched coverings of smooth complex projective surfaces from the topological viewpoint. Geometric information about the coverings (such as the first Betti numbers of a smooth model or intersections of embedded curves) is related to topological and combinatorial information about the base space and branch locus. Special attention is given to examples in which the base space is the complex projective plane and the branch locus is a configuration of lines.

This memoir consists of two independent papers. In the first, "The symplectic cobordism ring III" the classical Adams spectral sequence is used to study the symplectic cobordism ring [capital Greek]Omega[superscript]* [over] [subscript italic capital]S[subscript italic]p. In the second, "The symplectic Adams Novikov spectral sequence for spheres" we analyze the symplectic Adams-Novikov spectral sequence converging to the stable homotopy groups of spheres.