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 Poincare 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.

Throughout recent history, the theory of knot invariants has been a fascinating melting pot of ideas and scientific cultures, blending mathematics and physics, geometry, topology and algebra, gauge theory, and quantum gravity. The 2013 Seminaire de Mathematiques Superieures in Montreal presented an opportunity for the next generation of scientists to learn in one place about the various perspectives on knot homology, from the mathematical background to the most recent developments, and provided an access point to the relevant parts of theoretical physics as well. This volume presents a cross-section of topics covered at that summer school and will be a valuable resource for graduate students and researchers wishing to learn about this rapidly growing field.

Homotopy is a basic discipline of mathematics having fundamental and various applications to important fields of mathematics. The Journal has a wide scope which ranges from homotopical algebra and algebraic number theory and functional analysis. Diverse algebraic, geometric, topological and categorical structures are closely related to homotopy and the influence of homotopy is found in many fundamental areas of mathematics such as general algebra, algebraic topology, algebraic geometry, category theory, differential geometry, computer science, K-theory, functional analysis, Galois theory ad in physical sciences as well. The J. Homotopy and Related Structures intends to develop its vision on the determining role of homotopy in mathematics. the aim of the Journal is to show the importance, merit and diversity of homotopy in mathematical sciences. The J. Homotopy and Related structures is primarily concerned with publishing carefully refereed significant and original research papers. However a limited number of carefully selected survey and expository papers are also included, and special issues devoted to Proceedings of meetings in the field as well as to Festschrifts.