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We will study algebraic geometry, one of the oldest and richest areas ofmathematics. During the 20th century, the theoretical and veryabstract side of the subject was prominent, but with the availabilityof computers, the computational roots have been reinvigorated.This course will develop the theory behind the computational tools.What is algebraic geometry? Think back to high-school algebra whereyou graphed polynomial equations and perhaps found the intersectionof plane curves defined by a line and a parabola or more generalcurves defined by polynomials.Now think about higher dimensional space and consider intersections ofhyper-surfaces defined by polynomial equations. Such objects are called algebraic sets or algebraic varieties.What is the dimension? How many components are there? What is thesimplest way to describe the intersection? These are somefundamental questions of algebraic geometry. The fundamental result in algebraic geometry is the algebra-geometry"dictionary" which gives a precise relationship between geometricalobjects and algebraic ones: between varieties in n-dimensionalspace and radicalideals in the polynomial ring in n variables.The fundamental tools in computational algebraic geometry are Grobnerbases for ideals and Buchberger's algorithm. Grobner bases are a generalization of the greatestcommon divisor of integers. Just as the Euclidean algorithm may be used tocompute the gcd, Buchberger's algorithm is used to compute a Grobnerbasis for an ideal.In the last few decades, numerous applications of algebraic geometryhave been discovered: in coding theory, cryptography, robotics, objectrecognition, engineering, genomics etc. Some links that show the scope of recent work are:The Society for Industrialand Applied Mathematics Activity Group on Algebraic Geometry,The SpecialSemester on Grobner Bases and Related Methods;The Thematic Year onApplications of Algebraic Geometry at the Institute for Mathematicsand Its Applications;and the work of BerndSturmfels. Powerful computational software has alsobeen developed. See for example Sage , Macaulay 2, Singular, and Magma.These computational tools are of great importance in application. Required MaterialsCox, Little, O'Shea Ideals, Varieties, and Algorithms: AnIntroduction to Computational Algebraic Geometry and CommutativeAlgebra 2nd Ed., Springer-Verlag, 1997, or 3rd edition 2007. 2b1af7f3a8