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. Language: English