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Supermanifolds (Mathematics).

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"The aim of the book is to present a precise and comprehensive introduction to the basic theory of derived functors, with an emphasis on sheaf cohomology and spectral sequences. It keeps the treatment as simple as possible, aiming at the same time to provide a number of examples, mainly from sheaf theory, and also from algebra. The first part of the book provides the foundational material: Chapter 1 deals with category theory and homological algebra. Chapter 2 is devoted to the development of the theory of derived functors, based on the notion of injective object. In particular, the universal properties of derived functors is stressed, with a view to make the proofs in the following chapters as simple and natural as possible. Chapter 3 provides a rather thorough introduction to sheaves, in a general topological setting. Chapter 4 introduces sheaf cohomology as a derived functor, and, after also defining Čech cohomology, develops a careful comparison between the two cohomologies which is a detailed analysis not easily available in the literature. This comparison is made using general, universal properties of derived functors. This chapter also establishes the relations with the de Rham and Dolbeault cohomologies. Chapter 5 offers a friendly approach to the rather intricate theory of spectral sequences by means of the theory of derived triangles, which is precise and relatively easy to grasp. It also includes several examples of specific spectral sequences. Readers will find exercises throughout the text, with additional exercises included at the end of each chapter"--

Functor theory --- Sheaf theory --- Spectral sequences (Mathematics)

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This book offers a review of the vibrant areas of geometric representation theory and gauge theory, which are characterized by a merging of traditional techniques in representation theory with the use of powerful tools from algebraic geometry, and with strong inputs from physics. The notes are based on lectures delivered at the CIME school "Geometric Representation Theory and Gauge Theory" held in Cetraro, Italy, in June 2018. They comprise three contributions, due to Alexander Braverman and Michael Finkelberg, Andrei Negut, and Alexei Oblomkov, respectively. Braverman and Finkelberg’s notes review the mathematical theory of the Coulomb branch of 3D N=4 quantum gauge theories. The purpose of Negut’s notes is to study moduli spaces of sheaves on a surface, as well as Hecke correspondences between them. Oblomkov's notes concern matrix factorizations and knot homology. This book will appeal to both mathematicians and theoretical physicists and will be a source of inspiration for PhD students and researchers.

Geometry, Algebraic. --- Algebra, Homological. --- Moduli theory. --- Gauge fields (Physics) --- Particles (Nuclear physics) --- Elementary particles (Physics) --- High energy physics --- Nuclear particles --- Nucleons --- Nuclear physics --- Fields, Gauge (Physics) --- Gage fields (Physics) --- Gauge theories (Physics) --- Field theory (Physics) --- Group theory --- Symmetry (Physics) --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Homological algebra --- Algebra, Abstract --- Homology theory --- Algebraic geometry --- Geometry --- Algebraic geometry. --- Physics. --- Category theory (Mathematics). --- Homological algebra. --- Elementary particles (Physics). --- Quantum field theory. --- Algebraic Geometry. --- Mathematical Methods in Physics. --- Category Theory, Homological Algebra. --- Elementary Particles, Quantum Field Theory. --- Relativistic quantum field theory --- Quantum theory --- Relativity (Physics) --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics