New Spaces in Mathematics: Formal and Conceptual ReflectionsMathieu Anel, Gabriel Catren Cambridge University Press, 2021 - Broj stranica: 550 "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)"-- |
Sadržaj
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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New Spaces in Mathematics: Volume 1: Formal and Conceptual Reflections Mathieu Anel,Gabriel Catren Ograničeni pregled - 2021 |
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action affine algebraic geometry algebraic topology applications associated assume axioms bundle called classical classifying closed cohomology colimits commutative complex condition connected consider construction continuous corresponding covering defined definition denote derived descent diagram diffeological space diffeology discrete domains elements embedding equivalent etale example exists extended fact fiber field finite formal frame functions functor fundamental geometry give given groupoid higher homotopy homotopy type idea important introduced Intuitively isomorphism limits locally logic logoi logos manifold Math mathematics means models moduli morphism natural Note notion objects open subsets operations particular paths points poset precisely present problem projection prove quotient Recall relation result ring satisfying schemes sense Sh(X sheaf sheaves simplicial simply smooth stacks structure symplectic theorem topoi topological space topos type theory unique universal values