Includes bibliographical references and index.

"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 Physics arise from their lectures and discussions. This volume covers 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. 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)"--