New Spaces in Mathematics: Formal and Conceptual Reflections

doi:10.1017/9781108854429

After the development of manifolds and algebraic varieties in the previous century, mathematicians and physicists have continued to advance concepts of space. This book and its companion explore various new notions of space, including both formal and conceptual points of view, as presented by leading experts at the New Spaces in Mathematics and Physics workshop held at the Institut Henri Poincaré in 2015.The chapters in this volume cover a broad range of topics in mathematics, including diffeologies, synthetic differential geometry, microlocal analysis, topos theory, infinity-groupoids, homotopy type theory, category-theoretic methods in geometry, stacks, derived geometry, and noncommutative geometry. It is addressed primarily to mathematicians and mathematical physicists, but also to historians and philosophers of these disciplines.

‘The essays are self-contained, providing motivation to read selectively. Examples in each chapter illustrate the usefulness of these new notions of space … Recommended.’

M. Clay Source: Choice Magazine

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Contents for New Spaces in Physics
pp vii-viii
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Introduction
pp 1-28
- By Mathieu Anel, Gabriel Catren
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Part I - Differential Geometry
pp 29-30
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1 - An Introduction to Diffeology
pp 31-82
- By Patrick Iglesias-Zemmour
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2 - New Methods forOld Spaces: Synthetic Differential Geometry
pp 83-116
- By Anders Kock
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3 - Microlocal Analysis and Beyond
pp 117-152
- By Pierre Schapira
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Part II - Topology and Algebraic Topology
pp 153-154
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4 - Topo-logie
pp 155-257
- By Mathieu Anel, André Joyal
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5 - Spaces as Infinity-Groupoids
pp 258-321
- By Timothy Porter
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6 - Homotopy Type Theory: The Logic of Space
pp 322-404
- By Michael Shulman
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Part III - Algebraic Geometry
pp 405-406
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7 - Sheaves and Functors of Points
pp 407-461
- By Michel Vaquié
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8 - Stacks
pp 462-504
- By Nicole Mestrano, Carlos Simpson
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9 - The Geometry of Ambiguity: An Introduction to the Ideas of Derived Geometry
pp 505-553
- By Mathieu Anel
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10 - Geometry in dg-Categories
pp 554-592
- By Maxim Kontsevich
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New Spaces in Mathematics

Formal and Conceptual Reflections

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Contents for New Spaces in Physics
pp vii-viii

Introduction
pp 1-28

Part I - Differential Geometry
pp 29-30

1 - An Introduction to Diffeology
pp 31-82

2 - New Methods forOld Spaces: Synthetic Differential Geometry
pp 83-116

3 - Microlocal Analysis and Beyond
pp 117-152

Part II - Topology and Algebraic Topology
pp 153-154

4 - Topo-logie
pp 155-257

5 - Spaces as Infinity-Groupoids
pp 258-321

6 - Homotopy Type Theory: The Logic of Space
pp 322-404

Part III - Algebraic Geometry
pp 405-406

7 - Sheaves and Functors of Points
pp 407-461

8 - Stacks
pp 462-504

9 - The Geometry of Ambiguity: An Introduction to the Ideas of Derived Geometry
pp 505-553

10 - Geometry in dg-Categories
pp 554-592

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New Spaces in Mathematics

Formal and Conceptual Reflections

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