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