David Bryant Mumford Books

This volume is the first of three in a series surveying the theory of theta functions. Based on lectures given by the author at the Tata Institute of Fundamental Research in Bombay, these volumes constitute a systematic exposition of theta functions, beginning with their historical roots as analytic functions in one variable (Volume I), touching on some of the beautiful ways they can be used to describe moduli spaces (Volume II), and culminating in a methodical comparison of theta functions in analysis, algebraic geometry, and representation theory (Volume III).

Mumford's famous Red Book gives a simple readable account of the basic objects of algebraic geometry, preserving as much as possible their geometric flavor and integrating this with the tools of commutative algebra. It is aimed at graduate students or mathematicians in other fields wishing to learn quickly what algebraic geometry is all about. This new edition also includes an overview of the theory of curves, their moduli spaces and their Jacobians, one of the most exciting fields within algebraic geometry. The book is aimed at graduate students and professors seeking to learn i) the concept of „scheme“ as part of their study of algebraic geometry and ii) an overview of moduli problems for curves and of the use of theta functions to study these.

Focusing on algebraic geometry, this collection features 30 influential papers by David Mumford, organized into three major areas. Each section includes a summary highlighting key results and their impact, contributed by experts like David Gieseker and George Kempf. This compilation serves as a valuable resource for researchers, graduate students, mathematical physicists, and historians, providing insights into Mumford's significant contributions and their ongoing relevance in the field.

Mumford is a well-known mathematician and winner of the Fields Medal, the highest honor available in mathematics. Many of these papers are currently unavailable, and the commentaries by Gieseker, Lange, Viehweg and Kempf are being published here for the first time.

In the 20th century, algebraic geometry experienced three distinct phases. From 1900 to 1930, under the influence of Castelnuovo, Enriques, and Severi, the field expanded significantly, particularly in the study of surfaces, paralleling the earlier advancements in curves. This era established a comprehensive theory of surfaces, exploring connections between synthetic algebro-geometric techniques and topological and analytic methods. However, the abundance of tools and appealing geometric insights sometimes led to shortcuts in proofs and neglect of detailed analysis, particularly in special cases, a common challenge in geometry. The second phase, from 1930 to 1960, saw leaders like Zariski, Weil, and later Grothendieck launch a major initiative to integrate commutative algebra into algebraic geometry. This effort aimed to develop a unified language for discussing projective varieties across different fields, including characteristic p fields and complex numbers. The overarching goal, rooted in Kronecker's vision, was to create a geometry that formally encompassed both arithmetic and projective geometry.