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

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.