This volume contains the proceedings of an AMS special session held at the 1999 Joint Mathematics Meetings in San Antonio. The participants were an international group of researchers studying singularities from algebraic and analytic viewpoints. The contributed papers contain original results as well as some expository and historical material. This volume is dedicated to Oscar Zariski, on the one hundredth anniversary of his birth.

Topics include the role of valuation theory in algebraic geometry with recent applications to the structure of morphisms; algorithmic approaches to resolution of equisingular surface singularities and locally toric varieties; weak subintegral closures of ideals and Rees valuations; constructions of universal weakly subintegral extensions of rings; direct-sum decompositions of finitely generated modules; construction and examples of resolution graphs of surface singularities; Jacobians of meromorphic curves; investigation of spectral numbers of curve singularities using Puiseux pairs; Gröbner basis calculations of Hochschild homology for hypersurfaces with isolated singularities; and the theory of characteristic classes of singular spaces--a brief history with conjectures and open problems.

Readership

Graduate students and research mathematicians interested in singularities, algebraic geometry, analytic geometry or commutative algebra with an interest in singularities.

Table of Contents

• S. S. Abhyankar and A. Assi -- Factoring the Jacobian
• M. A. Vitulli -- Weak normalization and weak subintegral closure
• L. G. Roberts -- Integral dependence and weak subintegrality
• R. Wiegand -- Singularities and direct-sum decompositions
• S. D. Cutkosky -- Valuations in algebra and geometry
• C. Ban and L. J. McEwan -- Simultaneous resolution of equisingular quasi-ordinary singularities
• C. G. Melles and P. Milman -- Single-step combinatorial resolution via coherent sheaves of ideals
• A. Némethi -- Resolution graphs of some surface singularities, I. (Cyclic coverings)
• A. Némethi and Szilárd -- Resolution graphs of some surface singularities, II. (Generalized Iomdin series)
• L. J. McEwan -- Inequalities for spectral distributions of curve singularities
• R. I. Michler -- Isolated singularities with large Hochschild homology
• J.-P. Brasselet -- Milnor classes via polar varieties