This book contains articles corresponding to some of the talks delivered at the Foundations of Computational Mathematics (FoCM) conference at IMPA in Rio de Janeiro in January 1997. FoCM brings together a novel constellation of subjects in which the computational process itself and the foundational mathematical underpinnings of algorithms are the objects of study. The conference was organized around nine workshops: systems of algebraic equations and computational algebraic geometry, homotopy methods and real machines, information based complexity, numerical linear algebra, approximation and PDE's, optimization, differential equations and dynamical systems, relations to computer science and vision and related computational tools. The book gives the reader an idea of the state of the art in this emerging discipline.

1. Computing Roadmaps of Semi-Algebraic Sets on a Variety(Extended Abstract) 2. Essentially Smooth Lipschitz Functions: Compositiions and Chain Rules 3. Junction Detection and Filtering 4. Recognition in Hierarchical Models 5. Continuity Σ-Algebras(Extended Abstract) 6. Condition Number Analysis for Sparse Polynomial Systems 7. Residues in the Torus and Toric Varietes 8. Piecewise Smooth Orthonormal Factors 9. Algorithms for Computing Finite Semigroups 10. Extended Grzegorczyk Hierarchy in the BSS Model of Computability 11. Affine-Invariant Symmetry Sets...

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