Holden, Helge creator Holden, Helge Piene, Ragni text Cham, Switzerland Springer 2019 monographic eng

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xi, 774p. 24cm. The book presents the winners of the Abel Prize in mathematics for the period 2013?17: Pierre Deligne (2013); Yakov G. Sinai (2014); John Nash Jr. and Louis Nirenberg (2015); Sir Andrew Wiles (2016); and Yves Meyer (2017). The profiles feature autobiographical information as well as a scholarly description of each mathematician?s work. In addition, each profile contains a Curriculum Vitae, a complete bibliography, and the full citation from the prize committee. The book also includes photos for the period 2003?2017 showing many of the additional activities connected with the Abel Prize. As an added feature, video interviews with the Laureates as well as videos from the prize ceremony are provided at an accompanying website (http://extras.springer.com/). This book follows on The Abel Prize: 2003-2007. The First Five Years (Springer, 2010) and The Abel Prize 2008-2012 (Springer 2014), which profile the work of the previous Abel Prize winners Intro; Preface; Contents; Part I 2013 Pierre Deligne; Mathematical Autobiography; Definition of Mixed Hodge Structures; Definition of Shimura Varieties; Morphisms Between Motives; Conjectures on Critical Zeta Values; Relations Between Multizeta Values; Pierre Deligne: A Poet of Arithmetic Geometry; 1 Foundational Work: Topology, Homological Algebra, Étale Cohomology; 1.1 General Topology; 1.2 Spectral Sequences; Degeneration and Decomposition in the Derived Category; Décalage of Filtrations; 1.3 Cohomological Descent; 1.4 Duality and Finiteness Theorems in Étale Cohomology; Global Duality Derived FunctorsDiagram Compatibilities; The Functors Rf! and Rf!; Picard Stacks and Geometric Class Field Theory; Symmetric Künneth Formula; Finiteness; 2 Algebraic Stacks; 2.1 Deligne-Mumford Stacks; 2.2 Moduli of Curves of Genus ≥2; 2.3 Moduli of Elliptic Curves; Generalized Elliptic Curves and Compactifications; Reduction mod p; 3 Differential Equations, de Rham Cohomology; 3.1 The Canonical Extension and Hilbert's 21st Problem; The Curve Case; Higher Dimension: The Riemann-Hilbert Correspondence; 3.2 Betti-de Rham Comparison Theorems; 3.3 Crystalline Cohomology; Discontinuous Crystals Liftings of K3 Surfaces, Canonical CoordinatesThe de Rham-Witt Complex; 3.4 Irregular Connections; 3.5 Monodromy of the Hypergeometric Equation, Lattices; 4 Hodge Theory; 4.1 Hodge I; 4.2 Hodge II and Hodge III; Homological Algebra Infrastructure; Mixed Hodge Theory; The Fixed Part and Semisimplicity Theorems; 1-Motives; The du Bois Complex; Hodge Theory and Rational Homotopy; 4.3 Shimura Varieties; Axiomatization of Shimura Varieties; Canonical Models; 4.4 Absolute Hodge Cycles; 4.5 Deligne Cohomology; Link with the Hodge Conjecture and Intermediate Jacobians; Link with the Tame Symbol Link with Mixed Hodge Structures and Regulators4.6 Liftings mod p2 and Hodge Degeneration; 4.7 The Hodge Locus; 5 The Weil Conjectures; 5.1 The Zeta Function of a Variety Over a Finite Field; Basic Definitions; Statement of the Weil Conjectures; Grothendieck's Trace Formula; 5.2 A False Good Plan: Grothendieck's Standard Conjectures; 5.3 Partial Results Using Hodge Theory; K3 Surfaces; Complete Intersections of Hodge Level ≤1; 5.4 Integrality and Independence of; Integrality; Independence of l; 5.5 Weil I; 5.6 Weil II; Mixed Sheaves, Statement of the Main Theorem; Ingredients of the Proof First ApplicationsExponential Sums; 5.7 The -Adic Fourier Transform; Definition and First Properties; Laumon's Contribution and Applications; 5.8 Perverse Sheaves; t-Structures; Perverse Sheaves in the Étale Setting; The Purity and Decomposition Theorems; Consequences Over C; 5.9 Recent Results; A Finiteness Theorem; Counting Lisse -Adic Sheaves; 6 Modular and Automorphic Forms; 6.1 Construction of -Adic Representations; 6.2 The Weil-Deligne Group; 6.3 Local Constants of L-Functions; Construction of Local Constants; The Case of Function Fields; Additional Results / Helge Holden Mathematics Mathematics -- Awards Electronic Data Processing Informatique Mathématiques -- Prix et récompenses Mathematical Physics 510.9 HOL 9783319990279 :Technical Bureau India Pvt. Ltd. ;E/261, Shastri Nagar Delhi 220330 20220901120017.0