Homotopy theory and arithmetic geometry : motivic and diophantine aspects, LMS-CMI Research School, London, July 2018 / / edited by Frank Neumann and Ambrus Pál

• Record Nr.: UNISA996466408503316
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2021]; ©2021
• ISBN: 3-030-78977-2
• Descrizione fisica: 1 online resource (223 pages)
• Collana: Lecture Notes in Mathematics ; ; v.2292
• Disciplina: 514.24
• Soggetti: Arithmetical algebraic geometry; Homotopy theory; Teoria de l'homotopia; Geometria algebraica aritmètica; Congressos; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Includes index.
• Nota di contenuto: 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 -- 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 ζ-Functions -- 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 -- 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 -- 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 -- 5.2.5 A Snapshot of the Stable Motivic Homotopy Category -- Stable Representablity of Algebraic K-theory -- Milnor-Witt K-theory -- 5.3 Concrete A1-weak Equivalences -- 5.3.1 Constructing A1-weak Equivalences of Smooth Schemes -- 5.3.2 A1-weak Equivalences vs. Weak Equivalences -- 5.3.3 Cancellation Questions and A1-weak Equivalences -- 5.3.4 Danielewski Surfaces and Generalizations -- 5.3.5 Building Quasi-Affine A1-contractible Varieties -- Unipotent Quotients -- Other Quasi-Affine A1-contractible Varieties -- 5.4 Further Computations in A1-homotopy Theory -- 5.4.1 A1-homotopy Sheaves -- Basic Definitions -- A1-rigid Varieties Embed into H(k) -- 5.4.2 A1-connectedness and Geometry -- A1-connectedness and Rationality Properties -- 5.4.3 A1-homotopy Sheaves Spheres and Brouwer Degree -- 5.4.4 A1-homotopy at Infinity.; One-point Compactifications -- Stable End Spaces -- 5.5 Cancellation Questions and A1-contractibility -- 5.5.1 The Biregular Cancellation Problem -- 5.5.2 A1-contractibility vs Topological Contractibility -- Affine Lines on Topologically Contractible Surfaces -- Chow Groups and Vector Bundles on Topologically Contractible Surfaces -- 5.5.3 Cancellation Problems and the Russell Cubic -- The Russell Cubic and Equivariant K-theory -- Higher Chow Groups and Stable A1-contractibility -- 5.5.4 A1-contractibility of the Koras-Russell Threefold -- 5.5.5 Koras-Russell Fiber Bundles -- References -- Index.

A philosophy of the possible : modalities in thought and culture / / by Mikhail Epstein ; translated from the Russian by Vern W. McGee and Marina Eskina

• Record Nr.: UNINA9910795312203321
• Autore: Epstein Mikhail
• Pubbl/distr/stampa: Leiden ; ; Boston : , : Brill Rodopi, , [2019]
• ISBN: 90-04-39834-1
• Descrizione fisica: 1 online resource (379 pages)
• Collana: Value inquiry book series, , 0929-8436 ; ; volume 333
• Disciplina: 123
• Soggetti: Possibility
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Front Matter -- Copyright Page -- Dedication -- Preface -- Introduction: Fundamental Concepts of the Theory of the Possible -- The Possible in Philosophy -- Criticism and Activism -- Philosophy and Reality -- Change of Modalities in the History of Philosophy -- Philosophy as Possibilistic Thinking -- The Area of the Thinkable: the Value of Thinking in Itself -- Theory, Utopia, and Hypothesis -- Catharsis of Thinking -- Personified Thinking -- Possible and Impossible: Aporia of Thinking -- Language, Thinking, and Signifiability -- Universals as Potentials: Conceptualism -- From the General to the Concrete and Universal -- Multiplication of Entities -- Philosophy as Parody and Grotesque -- The Fate of Metaphysics: from Deconstruction to Possibilization -- Introduction to Part 2 -- Reverse Metaphysics: Critique and Deconstruction -- Beyond Being and Nothingness: the Feeling of the Possible -- A Worldview, Not a Point of View: “A Net with No Knots” -- The Possible in Jean Derrida -- The Metaphysics of Deconstruction: the Main Terms -- The Radical Nature of Difference: Profit and Transcendence -- Center and Structure -- Reverse Metaphysics: the Other, the Play, and the Writing -- Différance and the Tao -- Construction and Possibilization -- From Deconstruction to Construction -- Construction and Creativity -- De- and Con- -- Potentiation as Method: Eros of Thinking -- What Is “the Interesting”? Proposed Criteria -- Small Metaphysics: the Unique -- The Worlds of the Possible -- Introduction to Part 3 -- Society -- Culture -- ; Ethics -- Psychology -- Religion -- Conclusion -- Back Matter -- To Be Able, to Be, and to Know: a System of Modalities -- Index of Names.
• Sommario/riassunto: In this book, Mikhail Epstein offers a systematic theory of modalities (the actual, possible, and necessary), as applied to the discourse of philosophy in its post-Kantian and especially post-Derridean perspectives. He relies on his own experience of living in the USSR and the US, dominated respectively by imperative and possibilist modalities. Possibilism assumes that a thing or event acquires meaning only in the context of its multiple possibilities, inviting counterfactual and conditional modes of description. The author focuses on the creative potentials of possibilistic thinking and its heuristic value. The book demonstrates the range of modal approaches to society, culture, ethics, and language, and outlines potentiology as a new philosophical discipline interacting with ontology and epistemology.