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Algebra, Homological. --- Mirror symmetry.

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Homology theory --- Homologie --- K-theory. --- K-théorie --- Algèbre homologique --- Algebra, Homological --- Differential topology.

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Algebra, Homological --- Representations of algebras --- Homological algebra --- Algebra, Abstract --- Homology theory --- Algebra

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This book is an exposition of recent progress on the Donaldson–Thomas (DT) theory. The DT invariant was introduced by R. Thomas in 1998 as a virtual counting of stable coherent sheaves on Calabi–Yau 3-folds. Later, it turned out that the DT invariants have many interesting properties and appear in several contexts such as the Gromov–Witten/Donaldson–Thomas conjecture on curve-counting theories, wall-crossing in derived categories with respect to Bridgeland stability conditions, BPS state counting in string theory, and others. Recently, a deeper structure of the moduli spaces of coherent sheaves on Calabi–Yau 3-folds was found through derived algebraic geometry. These moduli spaces admit shifted symplectic structures and the associated d-critical structures, which lead to refined versions of DT invariants such as cohomological DT invariants. The idea of cohomological DT invariants led to a mathematical definition of the Gopakumar–Vafa invariant, which was first proposed by Gopakumar–Vafa in 1998, but its precise mathematical definition has not been available until recently. This book surveys the recent progress on DT invariants and related topics, with a focus on applications to curve-counting theories.

Invariants --- Aplicacions (Matemàtica) --- Algebra, Homological. --- Algebra --- Study and teaching. --- Homological algebra --- Algebra, Abstract --- Homology theory --- Mathematical physics. --- Algebraic geometry. --- Mathematical Physics. --- Algebraic Geometry. --- Category Theory, Homological Algebra. --- Algebraic geometry --- Geometry --- Physical mathematics --- Physics --- Mathematics

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This book is the first systematic treatment of this area so far scattered in a vast number of articles. As in classical topology, concrete problems require restricting the (generalized point-free) spaces by various conditions playing the roles of classical separation axioms. These are typically formulated in the language of points; but in the point-free context one has either suitable translations, parallels, or satisfactory replacements. The interrelations of separation type conditions, their merits, advantages and disadvantages, and consequences are discussed. Highlights of the book include a treatment of the merits and consequences of subfitness, various approaches to the Hausdorff's axiom, and normality type axioms. Global treatment of the separation conditions put them in a new perspective, and, a.o., gave some of them unexpected importance. The text contains a lot of quite recent results; the reader will see the directions the area is taking, and may find inspiration for her/his further work. The book will be of use for researchers already active in the area, but also for those interested in this growing field (sometimes even penetrating into some parts of theoretical computer science), for graduate and PhD students, and others. For the reader's convenience, the text is supplemented with an Appendix containing necessary background on posets, frames and locales.

Algebra. --- Ordered algebraic structures. --- Category theory (Mathematics). --- Homological algebra. --- Topology. --- Order, Lattices, Ordered Algebraic Structures. --- Category Theory, Homological Algebra. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Algebraic structures, Ordered --- Structures, Ordered algebraic --- Algebra --- Mathematics --- Mathematical analysis