Katz (mathematics, Bar Ilan U.) presents the systolic geometry of manifolds and polyhedra, starting with two classical inequalities and then proceeding to recent results, including a proof of Gromov's filling area conjecture in a hyperelliptic setting. Along the way he describes the literature and historical progression of the study, the work of Gauss, the global geometry of surfaces, inequalities of Loewner and Pu, systolic applications of integral geometry, surfaces at a basic level, Loewner's hyperelliptic surfaces, an optimal inequality for CAT(0) metrics, volume entropy and asymptotic upper bounds. He then moves to systolic geometry and topology in n dimensions, covering systoles and their category, Gromov's optimal stable systolic inequality and those dependent on Massey products, cup products and stable systoles, dual-critical lattices and systoles, generalized degree and Loewner-type inequalities, higher inequalities of the Loewner-Gromov type, Systolic inequalities for certain norms and Four-manifold systolic asymtotics. Annotation ©2007 Book News, Inc., Portland, OR (booknews.com)