Topology, K. Janich

This volume of the Encyclopaedia consists of two independent parts. The first contains a survey of results related to the concept of compactness in general topology. It highlights the role that compactness plays in many areas of general topology. The second part is devoted to homology and cohomology theories of general spaces. Special emphasis is placed on the method of sheaf theory as a unified approach to constructions of such theories. Both authors have succeeded in presenting a wealth of material that is of interest to students and researchers in the area of topology. Each part illustrates deep connections between important mathematical concepts. Both parts reflect a certain new way of looking at well known facts by establishing interesting relationships between specialized results belonging to diverse areas of mathematics.

Riemannian Topology and Geometric Structures on Manifolds results from a similarly entitled conference held at the University of New Mexico in Albuquerque. The various contributions to this volume discuss recent advances in the areas of positive sectional curvature, Kähler and Sasaki geometry, and their interrelation to mathematical physics, notably M and superstring theory. Focusing on these fundamental ideas, this collection presents articles with original results, and plausible problems of interest for future research.

This text deals with A1-homotopy theory over a base field, i.e., with the natural homotopy theory associated to the category of smooth varieties over a field in which the affine line is imposed to be contractible. It is a natural sequel to the foundational paper on A1-homotopy theory written together with V. Voevodsky. Inspired by classical results in algebraic topology, we present new techniques, new results and applications related to the properties and computations of A1-homotopy sheaves, A1-homology sheaves, and sheaves with generalized transfers, as well as to algebraic vector bundles over affine smooth varieties.

Riemannian Topology and Geometric Structures on Manifolds results from a similarly entitled conference held at the University of New Mexico in Albuquerque. The various contributions to this volume discuss recent advances in the areas of positive sectional curvature, Kähler and Sasaki geometry, and their interrelation to mathematical physics, notably M and superstring theory. Focusing on these fundamental ideas, this collection presents articles with original results, and plausible problems of interest for future research.

This book contains a clear exposition of two contemporary topics in modern differential geometry:

This book contains a clear exposition of two contemporary topics in modern differential geometry:

Topology is one of the fundamental tools in relating entities. Topology naturally finds application in all fields of engineering, physical sciences, life sciences, social sciences, medicine, business and even the arts. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. Circa 1750, Euler stated the polyhedron formula, V − E + F = 2 (where V, E, and F respectively indicate the number of vertices, edges, and faces of the polyhedron), which may be regarded as the first theorem, signaling the birth of topology. Subjects included in topology are algebraic topology and graph theory. A related branch to graph theory is molecular topology (with concerns of either chemical or biological structure).