| Description |
1 online resource (ix, 215 pages) : illustrations |
| Series |
Lecture notes in mathematics ; v. 2292 |
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Lecture notes in mathematics (Springer-Verlag) ; v. 2292.
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| Contents |
2.3.2 Sketch of Proof for Theorem 4 -- 2.3.3 A1-Milnor Numbers -- 2.3.4 An Arithmetic Count of the Lines on a Smooth Cubic Surface -- 2.3.5 An Arithmetic Count of the Lines Meeting 4Lines in Space -- Notation Guide -- References -- 3 Cohomological Methods in Intersection Theory -- 3.1 Introduction -- 3.2 Étale Motives -- 3.2.1 The h-topology -- 3.2.2 Construction of Motives, After Voevodsky -- 3.2.3 Functoriality -- 3.2.4 Representability Theorems -- 3.3 Finiteness and Euler Characteristic -- 3.3.1 Locally Constructible Motives -- 3.3.2 Integrality of Traces and Rationality of -Functions |
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Intro -- Preface -- Contents -- 1 Homotopy Theory and Arithmetic Geometry-Motivic and Diophantine Aspects: An Introduction -- 1.1 Overview of Themes -- 1.2 Summaries of Individual Contributions -- References -- 2 An Introduction to A1-Enumerative Geometry -- 2.1 Introduction -- 2.2 Preliminaries -- 2.2.1 Enriching the Topological Degree -- 2.2.2 The Grothendieck-Witt Ring -- 2.2.3 Lannes' Formula -- 2.2.4 The Unstable Motivic Homotopy Category -- 2.2.5 Colimits -- 2.2.6 Purity -- 2.3 A1-enumerative Geometry -- 2.3.1 The Eisenbud-Khimshiashvili-Levine Signature Formula |
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3.3.3 Grothendieck-Verdier Duality -- 3.3.4 Generic Base Change: A Motivic Variation on Deligne's Proof -- 3.4 Characteristic Classes -- 3.4.1 Künneth Formula -- 3.4.2 Grothendieck-Lefschetz Formula -- References -- 4 Étale Homotopy and Obstructions to Rational Points -- 4.1 Introduction -- 4.2 ∞-Categories -- 4.2.1 Motivation -- 4.2.2 Quasi-Categories -- 4.2.3 ∞-Groupoids and the Homotopy Hypothesis -- 4.2.4 Quasi-Categories from Topological Categories -- 4.2.5 ∞-Category Theory -- 4.2.6 The Homotopy Category -- 4.2.7 ∞-Categories and Homological Algebra -- 4.2.8 Stable ∞-Categories |
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4.2.9 Localization -- 4.3 ∞-Topoi -- 4.3.1 Definitions -- 4.3.2 The Shape of an ∞-Topos -- 4.4 Obstruction Theory -- 4.4.1 Obstruction Theory for Homotopy Types -- 4.4.2 For ∞-Topoi and Linear(ized) Versions -- 4.5 Étale Homotopy and Rational Points -- 4.5.1 The étale ∞-Topos -- 4.5.2 Rational Points -- 4.5.3 The Local-to-Global Principle -- 4.6 Galois Theory and Embedding Problems -- 4.6.1 Topoi and Embedding Problems -- References -- 5 A1-homotopy Theory and Contractible Varieties: A Survey -- 5.1 Introduction: Topological and Algebro-Geometric Motivations |
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5.1.1 Open Contractible Manifolds -- 5.1.2 Contractible Algebraic Varieties -- 5.2 A User's Guide to A1-homotopy Theory -- 5.2.1 Brief Topological Motivation -- 5.2.2 Homotopy Functors in Algebraic Geometry -- 5.2.3 The Unstable A1-homotopy Category: Construction -- Spaces -- Nisnevich and cdh Distinguished Squares -- Localization -- 5.2.4 The Unstable A1-homotopy Category: Basic Properties -- Motivic Spheres -- Representability Statements -- Representability of Chow Groups -- The Purity Isomorphism -- Comparison of Nisnevich and cdh-local A1-weak Equivalences |
| Summary |
This book provides an introduction to state-of-the-art applications of homotopy theory to arithmetic geometry. The contributions to this volume are based on original lectures by leading researchers at the LMS-CMI Research School on Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects and the Nelder Fellow Lecturer Series, which both took place at Imperial College London in the summer of 2018. The contribution by Brazelton, based on the lectures by Wickelgren, provides an introduction to arithmetic enumerative geometry, the notes of Cisinski present motivic sheaves and new cohomological methods for intersection theory, and Schlanks contribution gives an overview of the use of etale homotopy theory for obstructions to the existence of rational points on algebraic varieties. Finally, the article by Asok and stvr, based in part on the Nelder Fellow lecture series by stvr, gives a survey of the interplay between motivic homotopy theory and affine algebraic geometry, with a focus on contractible algebraic varieties. Now a major trend in arithmetic geometry, this volume offers a detailed guide to the fascinating circle of recent applications of homotopy theory to number theory. It will be invaluable to research students entering the field, as well as postdoctoral and more established researchers |
| Notes |
5.2.5 A Snapshot of the Stable Motivic Homotopy Category |
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Online resource; title from PDF title page (SpringerLink, viewed October 7, 2021) |
| Subject |
Homotopy theory -- Congresses
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Arithmetical algebraic geometry -- Congresses
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Homotopía -- Congresos
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Geometría algebraica aritmética -- Congresos
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Arithmetical algebraic geometry
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Homotopy theory
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| Genre/Form |
proceedings (reports)
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Conference papers and proceedings
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Conference papers and proceedings.
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Actes de congrès.
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| Form |
Electronic book
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| Author |
Neumann, Frank (Mathematician)
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Pál, Ambrus.
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LMS-CMI Research School on Homotopy Theory and Arithmetic Geometry -- Motivic and Diophantine Aspects (2018 : London, England)
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| ISBN |
9783030789770 |
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3030789772 |
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