Integer points in polyhedra : geometry, number theory, algebra, optimization : proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Integer Points in Polyhedra, Geometry, Number Theory, Algebra, Optimization, June 11-15, 2006, Snowbird, Utah

This is a self-contained exposition of several core aspects of the theory of rational polyhedra with a view towards algorithmic applications to efficient counting of integer points, a problem arising in many areas of pure and applied mathematics. The approach is based on the consistent development and application of the apparatus of generating functions and the algebra of polyhedra. Topics range from classical, such as the Euler characteristic, continued fractions, Ehrhart polynomial, Minkowski Convex Body Theorem, and the Lenstra– Lenstra–Lovász lattice reduction algorithm, to recent advances such as the Berline–Vergne local formula. The text is intended for graduate students and researchers. Prerequisites are a modest background in linear algebra and analysis as well as some general mathematical maturity. Numerous figures, exercises of varying degree of difficulty as well as references to the literature and publicly available software make the text suitable for a graduate course. "The book starts with an introduction to Geometric Invariant Theory (GIT). The fundamental results of Hilbert and Mumford are exposed as well as more recent topics such as the instability flag, the finiteness of the number of quotients, and the variation of quotients." "In the second part, GIT is applied to solve the classification problem of decorated principal bundles on a compact Riemann surface. The solution is a quasi-projective moduli scheme which parameterizes those objects that satisfy a semistability condition originating from gauge theory. The moduli space is equipped with a generalized Hitchin map." "Via the universal Kobayashi-Hitchin correspondence, these moduli spaces are related to moduli spaces of solutions of certain vortex type equations. Potential applications include the study of representation spaces of the fundamental group of compact Riemann surfaces." "The book concludes with a brief discussion of generalizations of these findings to higher dimensional base varieties, positive characteristic, and parabolic bundles. The text is fairly self-contained and features numerous examples and exercises. It addresses students and researchers with a working knowledge of elementary algebraic geometry." The algebra of polyhedra Linear transformations and polyhedra The structure of polyhedra Polarity Tangent cones : decompositions modulo polyhedra with lines Open polyhedra The exponential valuation Computing volumes Lattices, bases, and parallelepipeds The Minkowski convex body theorem Reduced basis Exponential sums and generating functions Totally unimodular polytopes Decomposing a 2-dimensional cone into unimodular cones via continued fractions Decomposing a rational cone of an arbitrary dimension into unimodular cones Efficient counting of integer points in rational polytopes The polynomial behavior of the number of integer points in polytopes A valuation on rational cones A "local" formula for the number of integer points in a polytope.