New Spaces in Mathematics: Formal and Conceptual Reflections, Volume 1

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Mathieu Anel, Gabriel Catren
Cambridge University Press, 2021 - Mathematics - 550 pages
"Mathematicians have long known that geometry is not absolute. Our understanding of what constitutes a "space" has driven, and been driven by, significant applications inside and outside of mathematics. The workshop "New Spaces in Mathematics and Physics," held in 2015 at the Institut Henri Poincaré, brought together many researchers to discuss various new notions of space in mathematics and theoretical physics, with particular attention given to often overlooked aspects of their subjects, conceptual reflections and historical perspectives. This volume and its companion New Spaces in Mathematics arise from their lectures and discussions. This volume covers a broad range of topics in mathematical physics, including noncommutative geometry, supergeometry, derived symplectic geometry, higher geometric quantization, intuitionistic quantum logic, problems with the continuum description of spacetime, twistor theory, loop quantum gravity, and geometry in string theory. Mathieu Anel is Assistant Professor at Carnegie Mellon University. His research interests include higher category theory, topos theory, and symplectic geometry. Gabriel Catren is Permanent Researcher at the French National Centre for Scientific Research (CNRS). His main areas of research are the foundations of classical and quantum mechanics, and the foundations of gauge theories (general relativity and Yang-Mills theories)"--
 

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Contents

A Brief History of Space
3
A Geometric Perspective for Describing
7
Summaries of the Chapters
12
Contents for New Spaces in Physics page vii
19
SUPERCOMMUTATIVE GEOMETRIES
21
Acknowledgments
22
Synthetic Differential Geometry
83
The Logic of Quantum Mechanics Revisited
85
Higher Prequantum Geometry
202
Spaces as InfinityGroupoids
258
Struggles with the Continuum
281
The Logic of Space
322
Quantum Geometry of Space
373
Sheaves and Functors of Points
407
Stacks
462
An Introduction to the Ideas
505

Supergeometry in Mathematics and Physics
114
Microlocal Analysis and Beyond
117
Topologie
155
Geometry in dgCategories
554
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About the author (2021)

Mathieu Anel is a Visiting Assistant Professor at Carnegie Mellon University. His research interests include higher category theory, algebraic topology, and topos theory.

Gabriel Catren is Permanent Researcher in philosophy of physics at the French National Centre for Scientific Research (CNRS). His research interests include the foundations of classical and quantum mechanics, and the foundations of gauge theories.

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