Geometry, Topology and Operator Algebras

This book offers a comprehensive exploration of contemporary intersections between geometry, topology, and theoretical physics, emphasizing their mathematical foundations and applications. Originating from lectures presented by experts during two summer schools held in Villa de Leyva, Colombia, the book reflects the synergy between global analysis, operator algebras, and their role in modern physics.

The chapters present state-of-the-art developments on a wide range of topics: the geometry and topology of foliations, affine manifolds, C*-algebras, and the pseudo-differential calculus of boundary value problems. These are enriched by applications to the theory of topological quantum matter.

The book is suitable for graduate students and researchers, offering detailed introductions to advanced topics such as the longitudinal index theorem for foliations, the geometry of the Poincaré half-space in a C*-algebra, and mathematical frameworks for topological matter. With a balance of foundational material and novel insights, it serves as both a learning resource and a reference for advanced studies at the intersection of mathematics and physics.

Based on lecture notes by experts in their fields Includes the necessary background material for the more advanced topics Hints at new promising applications and emerging areas of research

Auteur

Alexander Cardona is an Associate Professor in the Mathematics Department at Universidad de los Andes, Colombia. His research activity concerns mathematical tools that are motivated by physical problems. He studies differential geometry and global analysis, and their applications to physical models in which the underlying topology is important.

Andrés F. Reyes-Lega is an Associate Professor in the Physics Department at Universidad de los Andes, Colombia. His research contributions include studies on the algebraic structures underpinning quantum entanglement, the interplay between geometry, topology, and physics and mathematical aspects of quantum field theory.

Contenu
Geometric structures, invariants and their uses in physics by A. Cardona and A.F. Reyes-Lega.- . Lectures on the Euler characteristic of affine manifolds by Camilo Arias-Abad and Sebastian Velez-Vasquez.- Elliptic Curves by Philip Candelas.- The arithmetic of Calabi-Yau varieties, by Xenia de la Ossa.- Foliations and operator algebras by Georges Skandalis.- Pseudo-differential operators on groups and nonharmonic analysis, by Michael Ruzhansky.- Mathematical Foundations of Topological Matter, by Manuel Asorey.