New Spaces in Mathematics: Volume 1
Formal and Conceptual Reflections

In this graduate-level book, leading researchers explore various new notions of 'space' in mathematics.

Mathieu Anel (Edited by), Gabriel Catren (Edited by)

9781108490634, Cambridge University Press

Hardback, published 1 April 2021

550 pages, 20 b/w illus. 30 tables
23.4 x 15.6 x 3.7 cm, 0.94 kg

'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, Choice Magazine

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.

Introduction Mathieu Anel and Gabriel Catren
Part I. Differential geometry: 1. An Introduction to diffeology Patrick Iglesias-Zemmour
2. New methods for old spaces: synthetic differential geometry Anders Kock
3. Microlocal analysis and beyond Pierre Schapira
Part II. Topology and algebraic topology: 4. Topo-logie Mathieu Anel and André Joyal
5. Spaces as infinity-groupoids Timothy Porter
6. Homotopy type theory: the logic of space Michael Shulman
Part III. Algebraic geometry: 7. Sheaves and functors of points Michel Vaquié
8. Stacks Nicole Mestrano and Carlos Simpson
9. The geometry of ambiguity: an introduction to the ideas of derived geometry Mathieu Anel
10. Geometry in dg categories Maxim Kontsevich.

Subject Areas: Mathematical physics [PHU], Philosophy of science [PDA], Topology [PBP], Geometry [PBM]