Zurich Lectures in Advanced Mathematics 2008; 200 pp; softcover Volume: 9 ISBN10: 3037190523 ISBN13: 9783037190524 List Price: US$44 Member Price: US$35.20 Order Code: EMSZLEC/9
 This is a selfcontained 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 LenstraLenstraLovász lattice reduction algorithm, to recent advances such as the BerlineVergne 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. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in geometry and topology. Table of Contents  Introduction
 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 2dimensional 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
 Bibliography
 Index
