Springer, 2018. — 120 p. — (Trends in Mathematics 09: Research Perspectives CRM Barcelona). — ISBN 978-3-030-00026-4.This volume collects Extended Conference Abstracts originated at the workshop “Positivity and Valuations”, held at the Centre de Recerca Matemàtica in February 2016. This workshop brought together a variety of researchers, some of them experts on valuations, others interested in their use in the study of positivity in Algebraic Geometry. Valuation theory was initiated by Kürschák for treating the theory of p-adic fields more than a century ago; it has been flourishing ever since, with deep connections to algebraic number theory, algebraic geometry and the theory of ordered fields. Much of algebraic number theory can be better understood by using valuation theoretic methods, and the same principle applies to the resolution of singularities or the structure of singularities as realized by Zariski and Abhyankar. Contents Newton–Okounkov Bodies of Exceptional Curve Plane Valuations Non-positive at Infinity Sufficient Conditions for the Finite Generation of Valuation Semigroups From Convex Geometry of Certain Valuations to Positivity Aspects in Algebraic Geometry Non-positive at Infinity Valuations Very General Monomial Valuations on P2 and a Nagata Type Conjecture Valuations on Equicharacteristic Complete Noetherian Local Domains Desingularization by charðXÞ-Alterations Semigroup and Poincaré Series for Divisorial Valuations Computing Multiplier Ideals in Smooth Surfaces Notes on Divisors Computing MLD’s and LCT’s On the Containment Hierarchy for Simplicial Ideals The Universal Zeta Function for Curve Singularities and its Relation with Global Zeta Functions Algebraic Volumes of Divisors On Hirzebruch Type Inequalities and Applications On the Completion of Normal Toric Schemes Over Rank One Valuation Rings Duality on Value Semigroups Notes on Local Positivity and Newton–Okounkov Bodies Newton–Okounkov Bodies and Reified Valuations of Higher Rank
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