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Adopting an elegant geometrical approach, this advanced pedagogical text describes deep and intuitive methods for understanding the subtle logic of supersymmetry while avoiding lengthy computations. The book describes how complex results and formulae obtained using other approaches can be significantly simplified when translated to a geometric setting. Introductory chapters describe geometric structures in field theory in the general case, while detailed later chapters address specific structures such as parallel tensor fields, G-structures, and isometry groups. The relationship between structures in supergravity and periodic maps of algebraic manifolds, Kodaira-Spencer theory, modularity, and the arithmetic properties of supergravity are also addressed. Relevant geometric concepts are introduced and described in detail, providing a self-contained toolkit of useful techniques, formulae and constructions. Covering all the material necessary for the application of supersymmetric field theories to fundamental physical questions, this is an outstanding resource for graduate students and researchers in theoretical physics.

Supersymmetry. --- Geometrical constructions. --- Duality (Nuclear physics) --- Duality theory (Mathematics)

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Introducing Stone-Priestley duality theory and its applications to logic and theoretical computer science, this book equips graduate students and researchers with the theoretical background necessary for reading and understanding current research in the area. After giving a thorough introduction to the algebraic, topological, logical, and categorical aspects of the theory, the book covers two advanced applications in computer science, namely in domain theory and automata theory. These topics are at the forefront of active research seeking to unify semantic methods with more algorithmic topics in finite model theory. Frequent exercises punctuate the text, with hints and references provided.

Lattices, Distributive --- Duality theory (Mathematics)

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This undergraduate text takes the reader along the trail of light from Newton's particles to Einstein's relativity. Like the best detective stories, it presents clues and encourages the reader to draw conclusions before the answers are revealed. The first seven chapters describe how light behaves, develop Newton's particle theory, introduce waves and an electromagnetic wave theory of light, discover the photon, and culminate in the wave-particle duality. The book then goes on to develop the special theory of relativity, showing how time dilation and length contraction are consequences of the two simple principles on which the theory is founded. An extensive chapter derives the equation E = mc2 clearly from first principles and then explores its consequences and the misconceptions surrounding it. That most famous of issues arising from special relativity - the aging of the twins - is treated simply but compellingly.

Wave-particle duality. --- Special relativity (Physics)

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The calculus of variations has been an active area of mathematics for over 300 years. Its main use is to find stable critical points of functions for the solution of problems. To find unstable values, new approaches (Morse theory and min-max methods) were developed, and these are still being refined to overcome difficulties when applied to the theory of partial differential equations. Here, Professor Ghoussoub describes a point of view that may help when dealing with such problems. Building upon min-max methods, he systematically develops a general theory that can be applied in a variety of situations. In so doing he also presents a whole array of duality and perturbation methods. The prerequisites for following this book are relatively few; an appendix sketching certain methods in analysis makes the book reasonably self-contained. Consequently, it should be accessible to all mathematicians, pure or applied, economists and engineers working in nonlinear analysis or optimization.

Critical point theory (Mathematical analysis) --- Duality theory (Mathematics) --- Perturbation (Mathematics)

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The author introduces a notion of hyperbolic groupoids, generalizing the notion of a Gromov hyperbolic group. Examples of hyperbolic groupoids include actions of Gromov hyperbolic groups on their boundaries, pseudogroups generated by expanding self-coverings, natural pseudogroups acting on leaves of stable (or unstable) foliation of an Anosov diffeomorphism, etc. The author describes a duality theory for hyperbolic groupoids. He shows that for every hyperbolic groupoid mathfrak{G} there is a naturally defined dual groupoid mathfrak{G}^op acting on the Gromov boundary of a Cayley graph of mathfrak{G}. The groupoid mathfrak{G}^op is also hyperbolic and such that (mathfrak{G}^op)^op is equivalent to mathfrak{G}. Several classes of examples of hyperbolic groupoids and their applications are discussed.

Hyperbolic groups. --- Groupoids. --- Group theory. --- Duality theory (Mathematics)