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This book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.
Algebraic geometry. --- Algebraic Geometry. --- Geometry, Enumerative. --- Logarithms. --- Tropical geometry.
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Geometry, Enumerative --- String models --- Geometry, Enumerative. --- String models. --- Géométrie énumérative --- Modèles des cordes vibrantes (Physique nucléaire) --- 512.7 --- 512.7 Algebraic geometry. Commutative rings and algebras --- Algebraic geometry. Commutative rings and algebras --- Models, String --- String theory --- Nuclear reactions
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Curves.. --- Magnetism.. --- Mathematical physics --- Physics --- Electricity --- Magnetics --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Mathematics --- Shapes
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Differential geometry. Global analysis --- Singularities (Mathematics) --- Curves. --- Singularités (Mathématiques) --- Courbes --- Curves --- Surface --- Surface area --- 512.7 --- Algebraic geometry. Commutative rings and algebras --- Singularities (Mathematics). --- 512.7 Algebraic geometry. Commutative rings and algebras --- Singularités (Mathématiques) --- Geometry, Algebraic --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Mathematics --- Shapes --- Géométrie --- Singularite --- Lemme de morse
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Starting in the middle of the 80s, there has been a growing and fruitful interaction between algebraic geometry and certain areas of theoretical high-energy physics, especially the various versions of string theory. Physical heuristics have provided inspiration for new mathematical definitions (such as that of Gromov-Witten invariants) leading in turn to the solution of problems in enumerative geometry. Conversely, the availability of mathematically rigorous definitions and theorems has benefited the physics research by providing the required evidence in fields where experimental testing seems problematic. The aim of this volume, a result of the CIME Summer School held in Cetraro, Italy, in 2005, is to cover part of the most recent and interesting findings in this subject.
Geometry, Enumerative --- String models --- Algebra. --- Geometry, algebraic. --- Global differential geometry. --- Quantum theory. --- Algebraic Geometry. --- Differential Geometry. --- Quantum Physics. --- Algebraic geometry --- Geometry --- Mathematics --- Mathematical analysis --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Geometry, Differential --- Algebraic geometry. --- Differential geometry. --- Quantum physics. --- Differential geometry --- Enumerative invariants --- String theory --- Geometry, Algebraic
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Differential geometry. Global analysis --- Geometry, Differential --- Curves --- Surfaces --- Géométrie différentielle --- Courbes --- 514.752 --- Curved surfaces --- Geometry --- Shapes --- Differential geometry --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Enumerative --- Mathematics --- Differential geometry in Euclidean and pseudo-Euclidean spaces --- Geometry, Differential. --- Curves. --- Surfaces. --- 514.752 Differential geometry in Euclidean and pseudo-Euclidean spaces --- Géométrie différentielle
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Geometry, Differential --- Curves --- Surfaces --- Geometry, Affine --- Géométrie différentielle --- Courbes --- Géométrie affine --- 514.75 --- Curved surfaces --- Geometry --- Shapes --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Enumerative --- Mathematics --- Differential geometry --- Differential geometry in spaces with fundamental groups --- 514.75 Differential geometry in spaces with fundamental groups --- Géométrie différentielle --- Géométrie affine
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This book presents systematic research results on curved shock wave-curved compression surface applied to the compression surface design of supersonic–hypersonic inlet, which is a brand new inlet design. The concept of supersonic inlet curved compression discussed originated from the author’s research at the Deutsches Zentrum fur Luft- und Raumfahrt (DLR SM-ES) in the early 1990s. This book introduces the research history, working characteristics, performance calculation and aerodynamic configuration design method of this compression mode in detail. It also describes method of estimating the minimum drag in inlet and drag reduction effect of curved compression and proposes a new index for evaluating unit area compression efficiency of the inlet. Further, it reviews the relevant recent research on curved compression. As such it is a valuable resource for students, researchers and scientists in the fields of hypersonic propulsion and aeronautics.
Fluids. --- Fluid mechanics. --- Aerospace engineering. --- Astronautics. --- Fluid- and Aerodynamics. --- Engineering Fluid Dynamics. --- Aerospace Technology and Astronautics. --- Space sciences --- Aeronautics --- Astrodynamics --- Space flight --- Space vehicles --- Aeronautical engineering --- Astronautics --- Engineering --- Hydromechanics --- Continuum mechanics --- Hydraulics --- Mechanics --- Physics --- Hydrostatics --- Permeability --- Aerodynamics, Hypersonic. --- Aerodynamics of hypersonic flight --- Hypersonic aerodynamics --- Hypersonic speeds --- Hypersonics --- Aerodynamics, Supersonic --- Mach number --- Sound pressure --- Curves. --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Mathematics --- Shapes
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This volume presents the invited lectures of the workshop "Infinite Dimensional Algebras and Quantum Integrable Systems'' held in July 2003 at the University of Algarve, Faro, Portugal, as a satellite workshop of the XIV. International Congress on Mathematical Physics. Recent developments in the theory of infinite dimensional algebras and their applications to quantum integrable systems are reviewed by some of the leading experts in the field. The volume will be of interest to a broad audience from graduate students to researchers in mathematical physics and related fields. Contributors: E. Frenkel O.A. Castro-Alvaredo and A. Fring V.G. Kac and M. Wakimoto A. Gerasimov, S. Kharchev and D. Lebedev H.E. Boos, V.E. Korepin and F.A. Smirnov Kanehisa Takasaki Takashi Takebe L.A. Takhtajan and Lee-Peng Teo V. Tarasov.
Lie algebras --- Lie superalgebras --- Curves --- Functions of several complex variables --- Quantum theory --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Mathematics --- Shapes --- Superalgebras --- Algebra. --- Quantum theory. --- Mathematical physics. --- Topological Groups. --- Matrix theory. --- Systems theory. --- Quantum Physics. --- Mathematical Methods in Physics. --- Topological Groups, Lie Groups. --- Linear and Multilinear Algebras, Matrix Theory. --- Systems Theory, Control. --- System theory. --- Groups, Topological --- Continuous groups --- Physical mathematics --- Physics --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Mechanics --- Thermodynamics --- Mathematical analysis --- Systems, Theory of --- Systems science --- Science --- Philosophy --- Quantum physics. --- Physics. --- Topological groups. --- Lie groups. --- Groups, Lie --- Symmetric spaces --- Topological groups --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics
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This book is an elementary introduction to stable maps and quantum cohomology, starting with an introduction to stable pointed curves, and culminating with a proof of the associativity of the quantum product. The viewpoint is mostly that of enumerative geometry, and the red thread of the exposition is the problem of counting rational plane curves. Kontsevich's formula is initially established in the framework of classical enumerative geometry, then as a statement about reconstruction for Gromov–Witten invariants, and finally, using generating functions, as a special case of the associativity of the quantum product. Emphasis is given throughout the exposition to examples, heuristic discussions, and simple applications of the basic tools to best convey the intuition behind the subject. The book demystifies these new quantum techniques by showing how they fit into classical algebraic geometry. Some familiarity with basic algebraic geometry and elementary intersection theory is assumed. Each chapter concludes with some historical comments and an outline of key topics and themes as a guide for further study, followed by a collection of exercises that complement the material covered and reinforce computational skills. As such, the book is ideal for self-study, as a text for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory. The book will prove equally useful to graduate students in the classroom setting as to researchers in geometry and physics who wish to learn about the subject.
Geometry, Enumerative. --- Quantum theory. --- Homology theory. --- Curves, Plane. --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Higher plane curves --- Plane curves --- Geometry, algebraic. --- K-theory. --- Mathematical physics. --- Algebraic topology. --- Geometry. --- Mathematics. --- Algebraic Geometry. --- K-Theory. --- Mathematical Methods in Physics. --- Algebraic Topology. --- Applications of Mathematics. --- Math --- Science --- Mathematics --- Euclid's Elements --- Physical mathematics --- Homology theory --- Algebraic geometry --- Geometry --- Topology --- Geometry, Algebraic. --- Algebraic geometry. --- Physics. --- Applied mathematics. --- Engineering mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics
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