Problems and Tricks -- Number Theory -- Some Applications of Elementary Number Theory -- Ideas and Theories -- Induction and Recursion -- Arithmetic of algebraic numbers -- Arithmetic of algebraic varieties -- Zeta Functions and Modular Forms -- Fermat’s Last Theorem and Families of Modular Forms -- Analogies and Visions -- Introductory survey to part III: motivations and description -- Arake…
Stochastic Methods in Statistical Physics -- Coarse-Graining Techniques for (Random) Kac Models -- Euclidean Gibbs Measures of Quantum Crystals: Existence, Uniqueness and a Priori Estimates -- Some Jump Processes in Quantum Field Theory -- Gibbs Measures on Brownian Paths: Theory and Applications -- Spectral Theory for Nonstationary Random Potentials -- A Survey of Rigorous Results on Random Sc…
Gaudin Model and Opers -- Integrable Models with Unstable Particles -- Quantum Reduction in the Twisted Case -- Representation Theory and Quantum Integrability -- Connecting Lattice and Relativistic Models via Conformal Field Theory -- Elliptic Spectral Parameter and Infinite-Dimensional Grassmann Variety -- Trigonometric Degeneration and Orbifold Wess-Zumino-Witten Model. II -- Weil-Petersson …
Introductory Chapter -- Laplace Operators on Riemannian Manifolds -- Lagrangian Formalism on Riemannian Manifolds -- Harmonic Maps from a Lagrangian Viewpoint -- Conservation Theorems -- Hamiltonian Formalism -- Hamilton-Jacobi Theory -- Minimal Hypersurfaces -- Radially Symmetric Spaces -- Fundamental Solutions for Heat Operators with Potentials -- Fundamental Solutions for Elliptic Operators …
Summary -- Background -- Fundamental Theorem -- Trudinger-Moser Inequality -- The Green’s Function -- Equilibrium States -- Blowup Analysis for Stationary Solutions -- Multiple Existence -- Dynamical Equivalence -- Formation of Collapses -- Finiteness of Blowup Points -- Concentration Lemma -- Weak Solution -- Hyperparabolicity -- Quantized Blowup Mechanism -- Theory of Dual Variation.This bo…
Hyperbolicity and Beyond -- One-Dimensional Dynamics -- Homoclinic Tangencies -- Hénon-like Dynamics -- Non-Critical Dynamics and Hyperbolicity -- Heterodimensional Cycles and Blenders -- Robust Transitivity -- Stable Ergodicity -- Robust Singular Dynamics -- Generic Diffeomorphisms -- SRB Measures and Gibbs States -- Lyapunov Exponents.In broad terms, the goal of dynamics is to describe the l…
Measures and Transforms -- Selfadjointness and Spectrum -- Criteria for Identifying the Spectrum -- Operators of Interest -- Applications.Themainobjectiveofthisbookistogiveacollectionofcriteriaavailablein the spectral theory of selfadjoint operators, and to identify the spectrum and its components in the Lebesgue decomposition. Many of these criteria were published in several articles in di?ere…
1+1 Dimensional Integrable Systems -- 2+1 Dimensional Integrable Systems -- N + 1 Dimensional Integrable Systems -- Surfaces of Constant Curvature, Bäcklund Congruences and Darboux Transformation -- Darboux Transformation and Harmonic Map -- Generalized Self-Dual Yang-Mills Equations and Yang-Mills-Higgs Equations -- Two Dimensional Toda Equations and Laplace Sequences of Surfaces in Projectiv…
Translation Groups -- Euclidean and Hyperbolic Geometry -- Sphere Geometries of Möbius and Lie -- Lorentz Transformations.This book is based on real inner product spaces X of arbitrary (finite or infinite) dimension greater than or equal to 2. With natural properties of (general) translations and general distances of X, euclidean and hyperbolic geometries are characterized. For these spaces X …
Dirac structures, momentum maps, and quasi-Poisson manifolds -- Construction of Ricci-type connections by reduction and induction -- A mathematical model for geomagnetic reversals -- Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization -- Thompson’s conjecture for real semisimple Lie groups -- The Weinstein conjecture and theorems of nearby and almost exist…