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Topics in Algebraic Geometry and Geometric Modeling
Algebraic geometry and geometric modeling both deal with curves and surfaces generated by polynomial equations. Algebraic geometry investigates the theoretical properties of polynomial curves and surfaces; geometric modeling uses polynomial, piecewise polynomial, and rational curves and surfaces to build computer models of mechanical components and assemblies for industrial design and manufacture. The NSF sponsored the four-day ''Vilnius Workshop on Algebraic Geometry and Geometric Modeling'', which brought together some of the top experts in the two research communities to examine a wide range of topics of interest to both fields. This volume is an outgrowth of that workshop. Included are surveys, tutorials, and research papers. In addition, the editors have included a translation of Minding's 1841 paper, ''On the determination of the degree of an equations obtained by elimination'', which foreshadows the modern application of mixed volumes in algebraic geometry. The volume is suitable for mathematicians, computer scientists, and engineers interested in applications of algebraic geometry to geometric modeling.
Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering
This volume presents the proceedings from the research conference, Symbolic Computation: Solving Equations in Algebra, Analysis, and Engineering, held at Mount Holyoke College, USA. It provides an overview of contemporary research in symbolic computation as it applies to the solution of polynomial systems. The conference brought together pure and applied mathematicians, computer scientists, and engineers, who use symbolic computation to solve systems of equations or who develop the theoretical background and tools needed for this purpose. Within this general framework, the conference focused on several themes: systems of polynomials, systems of differential equations, noncommutative systems, and applications.
Algorithmic Number Theory
Self-organized criticality (SOC) has become a magic word in various scientific disciplines; it provides a framework for understanding complexity and scale invariance in systems showing irregular fluctuations. In the first 10 years after Per Bak and his co-workers presented their seminal idea, more than 2000 papers on this topic appeared. Seismology has been a field in earth sciences where the SOC concept has already deepened the understanding, but there seem to be much more examples in earth sciences where applying the SOC concept may be fruitful. After introducing the reader into the basics of fractals, chaos and SOC, the book presents established and new applications of SOC in earth sciences, namely earthquakes, forest fires, landslides and drainage networks.
Randomization, Relaxation, and Complexity in Polynomial Equation Solving
This volume corresponds to the Banff International Research Station Workshop on Randomization, Relaxation, and Complexity, held from February 28-March 5, 2010. It contains a sample of advanced algorithmic techniques underpinning the solution of systems of polynomial equations. The papers are written by leading experts in algorithmic algebraic geometry and examine core topics.
Geometric and Algorithmic Aspects of Computer-Aided Design and Manufacturing
Computer-Aided Design and Manufacturing (CAD/CAM) is concerned with all aspects of the process of designing, prototyping, manufacturing, inspecting, and maintaining complex geometric objects under computer control. As such, there is a natural synergy between this field and Computational Geometry (CG), which involves the design, analysis, implementation, and testing of efficient algorithms and data representation techniques for geometric entities such as points, polygons, polyhedra, curves, and surfaces. The DIMACS Center (Piscataway, NJ) sponsored a workshop to further promote the interaction between these two fields. Attendees from academia, research laboratories, and industry took part in the invited talks, contributed presentations, and informal discussions. This volume is an outgrowth of that meeting.
Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry
This book is the result of a meeting that took place at the University of Ghent (Belgium) on the relations between Hilbert's tenth problem, arithmetic, and algebraic geometry. Included are written articles detailing the lectures that were given as well as contributed papers on current topics of interest. The following areas are addressed: an historical overview of Hilbert's tenth problem, Hilbert's tenth problem for various rings and fields, model theory and local-global principles, including relations between model theory and algebraic groups and analytic geometry, conjectures in arithmetic geometry and the structure of diophantine sets, for example with Mazur's conjecture, Lang's conjecture, and Bücchi's problem, and results on the complexity of diophantine geometry, highlighting the relation to the theory of computation. The volume allows the reader to learn and compare different approaches (arithmetical, geometrical, topological, model-theoretical, and computational) to the general structural analysis of the set of solutions of polynomial equations. It would make a nice contribution to graduate and advanced graduate courses on logic, algebraic geometry, and number theory
Algebra, Geometry and Their Interactions
This volume's papers present work at the cutting edge of current research in algebraic geometry, commutative algebra, numerical analysis, and other related fields, with an emphasis on the breadth of these areas and the beneficial results obtained by the interactions between these fields. This collection of two survey articles and sixteen refereed research papers, written by experts in these fields, gives the reader a greater sense of some of the directions in which this research is moving, as well as a better idea of how these fields interact with each other and with other applied areas. The topics include blowup algebras, linkage theory, Hilbert functions, divisors, vector bundles, determinantal varieties, (square-free) monomial ideals, multiplicities and cohomological degrees, and computer vision.
Algorithms in Algebraic Geometry
In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric computation. The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its Applications on September 2006 is one tangible indication of the interest. This volume of articles captures some of the spirit of the IMA workshop.
Foundations of Computational Mathematics
This book contains a collection of articles corresponding to some of the talks delivered at the Foundations of Computational Mathematics conference held at IMPA in Rio de Janeiro in January 1997. Some ofthe others are published in the December 1996 issue of the Journal of Complexity. Both of these publications were available and distributed at the meeting. Even in this aspect we hope to have achieved a synthesis of the mathematics and computer science cultures as well as of the disciplines. The reaction to the Park City meeting on Mathematics of Numerical Analy sis: Real Number Algorithms which was chaired by Steve Smale and had around 275 participants, was very enthusiastic. At the suggesti...
