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The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the “tropical” approach to Gromov-Witten theory, and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Geometry studies piecewise-linear objects which appear as “degenerations” of the corresponding algebro-geometric objects.

Mirror symmetry. --- Tropical geometry. --- Geometry, Tropical --- Geometry, Algebraic --- Symmetry (Physics) --- Geometry, algebraic. --- Global differential geometry. --- Algebraic Geometry. --- Differential Geometry. --- Geometry, Differential --- Algebraic geometry --- Geometry --- Algebraic geometry. --- Differential geometry. --- Differential geometry

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The main focus of this volume is on the problem of describing the automorphism groups of affine and projective varieties, a classical subject in algebraic geometry where, in both cases, the automorphism group is often infinite dimensional. The collection covers a wide range of topics and is intended for researchers in the fields of classical algebraic geometry and birational geometry (Cremona groups) as well as affine geometry with an emphasis on algebraic group actions and automorphism groups. It presents original research and surveys and provides a valuable overview of the current state of the art in these topics. Bringing together specialists from projective, birational algebraic geometry and affine and complex algebraic geometry, including Mori theory and algebraic group actions, this book is the result of ensuing talks and discussions from the conference “Groups of Automorphisms in Birational and Affine Geometry” held in October 2012, at the CIRM, Levico Terme, Italy. The talks at the conference highlighted the close connections between the above-mentioned areas and promoted the exchange of knowledge and methods from adjacent fields.

Automorphisms --- Geometry, Affine --- Mathematics. --- Geometry, Algebraic. --- Math --- Science --- Algebraic geometry --- Geometry --- Affine geometry --- Geometry, Modern --- Group theory --- Symmetry (Mathematics) --- Geometry, algebraic. --- Algebraic Geometry. --- Algebraic geometry.

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Over the course of the last century it has become clear that both elementary particle physics and relativity theories are based on the notion of symmetries. These symmetries become manifest in that the "laws of nature" are invariant under spacetime transformations and/or gauge transformations. The consequences of these symmetries were analyzed as early as in 1918 by Emmy Noether on the level of action functionals. Her work did not receive due recognition for nearly half a century, but can today be understood as a recurring theme in classical mechanics, electrodynamics and special relativity, Yang-Mills type quantum field theories, and in general relativity. As a matter of fact, as shown in this monograph, many aspects of physics can be derived solely from symmetry considerations. This substantiates the statement of E.P.Wigner "... if we knew all the laws of nature, or the ultimate Law of nature, the invariance properties of these laws would not furnish us new information." Thanks to Wigner we now also understand the implications of quantum physics and symmetry considerations: Poincare invariance dictates both the characteristic properties of particles (mass, spin, ...) and the wave equations of spin 0, 1/2, 1, ... objects. Further, the work of C.N.Yang and R.Mills reveals the consequences of internal symmetries as exemplified in the symmetry group of elementary particle physics. Given this pivotal role of symmetries it is thus not surprising that current research in fundamental physics is to a great degree motivated and inspired by considerations of symmetry.The treatment of symmetries in this monograph ranges from classical physics to now well-established theories of fundamental interactions, to the latest research on unified theories and quantum gravity.

Symmetry (Physics) --- Invariance principles (Physics) --- Symmetry (Chemistry) --- Conservation laws (Physics) --- Physics --- Quantum theory. --- Theoretical, Mathematical and Computational Physics. --- Elementary Particles, Quantum Field Theory. --- Phase Transitions and Multiphase Systems. --- Classical and Quantum Gravitation, Relativity Theory. --- Mathematical Physics. --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Mechanics --- Thermodynamics --- Mathematical physics. --- Elementary particles (Physics). --- Quantum field theory. --- Phase transitions (Statistical physics). --- Gravitation. --- Field theory (Physics) --- Matter --- Antigravity --- Centrifugal force --- Relativity (Physics) --- Phase changes (Statistical physics) --- Phase transitions (Statistical physics) --- Phase rule and equilibrium --- Statistical physics --- Relativistic quantum field theory --- Quantum theory --- Elementary particles (Physics) --- High energy physics --- Nuclear particles --- Nucleons --- Nuclear physics --- Physical mathematics --- Properties --- Mathematics

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The principle aim of the book is to present a self-contained, modern account of similarity and symmetry methods, which are important mathematical tools for both physicists, engineers and applied mathematicians. The idea is to provide a balanced presentation of the mathematical techniques and applications of symmetry methods in mathematics, physics and engineering. That is why it includes recent developments and many examples in finding systematically conservation laws, local and nonlocal symmetries for ordinary and partial differential equations. The role of continuous symmetries in classical and quantum field theories is exposed at a technical level accessible even for non specialists. The importance of symmetries in continuum mechanics and mechanics of materials is highlighted through recent developments, such as the construction of constitutive models for various materials combining Lie symmetries with experimental data. As a whole this book is a unique collection of contributions from experts in the field, including specialists in the mathematical treatment of symmetries, researchers using symmetries from a fundamental, applied or numerical viewpoint. The book is a fascinating overview of symmetry methods aimed for graduate students in physics, mathematics and engineering, as well as researchers either willing to enter in the field or to capture recent developments and applications of symmetry methods in different scientific fields.

Symmetry (Mathematics) --- Engineering. --- Construction --- Industrial arts --- Technology --- Invariance (Mathematics) --- Group theory --- Automorphisms --- Mechanics. --- Mechanics, Applied. --- Computer science. --- Solid Mechanics. --- Classical Mechanics. --- Computational Science and Engineering. --- Informatics --- Science --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Computer mathematics. --- Computer mathematics --- Electronic data processing --- Mathematics --- Solids. --- Data processing. --- Solid state physics --- Transparent solids

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This thesis describes the thorough analysis of the rare B meson decay into ϕ K* on data taken by the Belle Collaboration at the B-meson-factory KEKB over 10 years. This reaction is very interesting, because it in principle allows the observation of CP-violation effects. In the Standard Model however, no CP violation in this reaction is expected. An observation of CP asymmetries thus immediately implies new physics. This thesis presents an amplitude analysis of this decay and the search for CP violation in detail and discusses methods to solve related problems: The quantification of multivariate dependence and the improvement of numeric evaluation speed of normalization integrals in amplitude analysis. In addition it provides an overview of the theory, experimental setup, (blind) statistical data analysis and estimation of systematic uncertainties.

Particles (Nuclear physics) --- Polarization (Nuclear physics) --- CP violation (Nuclear physics) --- Charge conjugation parity violation --- Violation, Charge conjugation parity --- Symmetry (Physics) --- Quantum theory. --- Elementary Particles, Quantum Field Theory. --- Theoretical, Mathematical and Computational Physics. --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Elementary particles (Physics). --- Quantum field theory. --- Mathematical physics. --- Physical mathematics --- Relativistic quantum field theory --- Field theory (Physics) --- Quantum theory --- Relativity (Physics) --- Elementary particles (Physics) --- High energy physics --- Nuclear particles --- Nucleons --- Nuclear physics --- Mathematics