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Wigner's theorem is a fundamental part of the mathematical formulation of quantum mechanics. The theorem characterizes unitary and anti-unitary operators as symmetries of quantum mechanical systems, and is a key result when relating preserver problems to quantum mechanics. At the heart of this book is a geometric approach to Wigner-type theorems, unifying both classical and more recent results. Readers are initiated in a wide range of topics from geometric transformations of Grassmannians to lattices of closed subspaces, before moving on to a discussion of applications. An introduction to all the key aspects of the basic theory is included as are plenty of examples, making this book a useful resource for beginning graduate students and non-experts, as well as a helpful reference for specialist researchers.

Hilbert space. --- Grassmann manifolds. --- Geometry, Projective. --- Quantum theory --- Mathematics.

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Topology --- Differential geometry. Global analysis --- Grassmann manifolds --- Gauss maps --- Piecewise linear topology --- Differential topology --- 514.745 --- Calculus of exterior forms. Grassman algebra --- Differential topology. --- Gauss maps. --- Grassmann manifolds. --- Piecewise linear topology. --- 514.745 Calculus of exterior forms. Grassman algebra

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Buildings are combinatorial constructions successfully exploited to study groups of various types. The vertex set of a building can be naturally decomposed into subsets called Grassmannians. The book contains both classical and more recent results on Grassmannians of buildings of classical types. It gives a modern interpretation of some classical results from the geometry of linear groups. The presented methods are applied to some geometric constructions non-related to buildings - Grassmannians of infinite-dimensional vector spaces and the sets of conjugate linear involutions. The book is self

Grassmann manifolds. --- Buildings (Group theory) --- Theory of buildings (Group theory) --- Tits's theory of buildings (Group theory) --- Linear algebraic groups --- Grassmannians --- Differential topology --- Manifolds (Mathematics) --- Architecture --- Mathematics.

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This work offers a contribution in the geometric form of the theory of several complex variables. Since complex Grassmann manifolds serve as classifying spaces of complex vector bundles, the cohomology structure of a complex Grassmann manifold is of importance for the construction of Chern classes of complex vector bundles. The cohomology ring of a Grassmannian is therefore of interest in topology, differential geometry, algebraic geometry, and complex analysis. Wilhelm Stoll treats certain aspects of the complex analysis point of view.This work originated with questions in value distribution theory. Here analytic sets and differential forms rather than the corresponding homology and cohomology classes are considered. On the Grassmann manifold, the cohomology ring is isomorphic to the ring of differential forms invariant under the unitary group, and each cohomology class is determined by a family of analytic sets.

Algebraic geometry --- Differential geometry. Global analysis --- Grassmann manifolds --- Differential forms. --- Grassmann manifolds. --- Invariants. --- Geometry, Differential. --- Géométrie différentielle. --- Differential invariants. --- Invariants différentiels. --- Forms, Differential --- Continuous groups --- Geometry, Differential --- Grassmannians --- Differential topology --- Manifolds (Mathematics) --- Calculation. --- Cohomology ring. --- Cohomology. --- Complex space. --- Cotangent bundle. --- Diagram (category theory). --- Exterior algebra. --- Grassmannian. --- Holomorphic vector bundle. --- Manifold. --- Regular map (graph theory). --- Remainder. --- Representation theorem. --- Schubert variety. --- Sesquilinear form. --- Theorem. --- Vector bundle. --- Vector space. --- Géométrie différentielle. --- Invariants différentiels.

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This text presents the mathematical concepts of Grassmann variables and the method of supersymmetry to a broad audience of physicists interested in applying these tools to disordered and critical systems, as well as related topics in statistical physics. Based on many courses and seminars held by the author, one of the pioneers in this field, the reader is given a systematic and tutorial introduction to the subject matter. The algebra and analysis of Grassmann variables is presented in part I. The mathematics of these variables is applied to a random matrix model, path integrals for fermions, dimer models and the Ising model in two dimensions. Supermathematics - the use of commuting and anticommuting variables on an equal footing - is the subject of part II. The properties of supervectors and supermatrices, which contain both commuting and Grassmann components, are treated in great detail, including the derivation of integral theorems. In part III, supersymmetric physical models are considered. While supersymmetry was first introduced in elementary particle physics as exact symmetry between bosons and fermions, the formal introduction of anticommuting spacetime components, can be extended to problems of statistical physics, and, since it connects states with equal energies, has also found its way into quantum mechanics. Several models are considered in the applications, after which the representation of the random matrix model by the nonlinear sigma-model, the determination of the density of states and the level correlation are derived. Eventually, the mobility edge behavior is discussed and a short account of the ten symmetry classes of disorder, two-dimensional disordered models, and superbosonization is given.

Physics - General --- Physics --- Physical Sciences & Mathematics --- Mathematical physics. --- Statistical physics. --- Mathematical Methods in Physics. --- Mathematical Physics. --- Complex Systems. --- Mathematical Applications in the Physical Sciences. --- Statistical Physics and Dynamical Systems. --- Physical mathematics --- Mathematical statistics --- Mathematics --- Statistical methods --- Grassmann manifolds. --- Grassmannians --- Differential topology --- Manifolds (Mathematics) --- Physics. --- Dynamical systems. --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics