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

Format:

Book

Conference/Event

Contributor:

Neumann, Frank (Mathematician), editor.

Pál, Ambrus, editor.

Conference Name:

LMS-CMI Research School on Homotopy Theory and Arithmetic Geometry -- Motivic and Diophantine Aspects (2018 : London, England)

Series:

Lecture notes in mathematics (Springer-Verlag) ; 2292.

Lecture notes in mathematics, 0075-8434 ; 2292

Language:

English

Subjects (All):

Homotopy theory--Congresses.

Homotopy theory.

Arithmetical algebraic geometry--Congresses.

Arithmetical algebraic geometry.

Genre:

Conference papers and proceedings.

Physical Description:

ix, 215 pages : illustrations ; 24 cm.

Place of Publication:

Cham : Springer, [2021]

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.

Contents:

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

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

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

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

Notes:

Includes bibliographical references and index.

5.2.5 A Snapshot of the Stable Motivic Homotopy Category.

Current copyright fee: GBP19.00 42\0.

ISBN:

9783030789763

3030789764

OCLC:

1252412052