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The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009-2010 academic year. The notes by Laurent Berger provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory. The notes by Gebhard Böckle offer a comprehensive course on Galois deformation theory, starting from the foundational results of Mazur and discussing in detail the theory of pseudo-representations and their deformations, local deformations at places l ≠ p and local deformations at p which are flat. In the last section,the results of Böckle and Kisin on presentations of global deformation rings over local ones are discussed. The notes by Mladen Dimitrov present the basics of the arithmetic theory of Hilbert modular forms and varieties, with an emphasis on the study of the images of the attached Galois representations, on modularity lifting theorems over totally real number fields, and on the cohomology of Hilbert modular varieties with integral coefficients.  The notes by Lassina Dembélé and John Voight describe methods for performing explicit computations in spaces of Hilbert modular forms. These methods depend on the Jacquet-Langlands correspondence and on computations in spaces of quaternionic modular forms, both for the case of definite and indefinite quaternion algebras. Several examples are given, and applications to modularity of Galois representations are discussed.  The notes by Tim Dokchitser describe the proof, obtained by the author in a joint project with Vladimir Dokchitser, of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the Tate-Shafarevich group. The statement of the Birch and Swinnerton-Dyer conjecture is included, as well as a detailed study of local and global root numbers of elliptic curves and their classification.

Galois theory --- Curves, Elliptic --- Hilbert modular surfaces --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Modular surfaces, Hilbert --- Elliptic curves --- Mathematics. --- Algebra. --- Algebraic geometry. --- Number theory. --- Number Theory. --- Algebraic Geometry. --- Galois theory. --- Curves, Elliptic. --- Hilbert modular surfaces. --- Forms, Modular --- Surfaces --- Equations, Theory of --- Group theory --- Number theory --- Curves, Algebraic --- Geometry, algebraic. --- Mathematical analysis --- Algebraic geometry --- Geometry --- Number study --- Numbers, Theory of

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The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry. Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media, or for capillary phenomena. Further, as most techniques used in the theory of CMC surfaces not only involve geometric methods but also PDE and complex analysis, the theory is also of great interest for many other mathematical fields.   While minimal surfaces and CMC surfaces in general have already been treated in the literature, the present work is the first to present a comprehensive study of “compact surfaces with boundaries,” narrowing its focus to a geometric view. Basic issues include the discussion whether the symmetries of the curve inherit to the surface; the possible values of the mean curvature, area and volume; stability; the circular boundary case; and the existence of the Plateau problem in the non-parametric case. The exposition provides an outlook on recent research but also a set of techniques that allows the results to be expanded to other ambient spaces. Throughout the text, numerous illustrations clarify the results and their proofs.   The book is intended for graduate students and researchers in the field of differential geometry and especially theory of surfaces, including geometric analysis and geometric PDEs. It guides readers up to the state-of-the-art of the theory and introduces them to interesting open problems.

Fluid dynamics. --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Curves on surfaces. --- Geometry, Differential. --- Differential geometry --- Surfaces, Curves on --- Mathematics. --- Partial differential equations. --- Geometry. --- Differential geometry. --- Differential Geometry. --- Partial Differential Equations. --- Global differential geometry. --- Differential equations, partial. --- Euclid's Elements --- Partial differential equations --- Geometry, Differential --- Surfaces of constant curvature. --- Curves, Algebraic. --- Boundary value problems.

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The aim of the book is to study some aspects of geometric evolutions, such as mean curvature flow and anisotropic mean curvature flow of hypersurfaces. We analyze the origin of such flows and their geometric and variational nature. Some of the most important aspects of mean curvature flow are described, such as the comparison principle and its use in the definition of suitable weak solutions. The anisotropic evolutions, which can be considered as a generalization of mean curvature flow, are studied from the view point of Finsler geometry. Concerning singular perturbations, we discuss the convergence of the Allen–Cahn (or Ginsburg–Landau) type equations to (possibly anisotropic) mean curvature flow before the onset of singularities in the limit problem. We study such kinds of asymptotic problems also in the static case, showing convergence to prescribed curvature-type problems.

Mathematics --- Physical Sciences & Mathematics --- Geometry --- Mathematics. --- Geometry. --- Euclid's Elements --- Math --- Science --- Curvature. --- Flows (Differentiable dynamical systems) --- Differentiable dynamical systems --- Calculus --- Curves --- Surfaces

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This book contains a detailed account of the result of the author's recent Annals paper and JAMS paper on arithmetic invariant, including μ-invariant, L-invariant, and similar topics.   This book can be regarded as an introductory text to the author's previous book p-Adic Automorphic Forms on Shimura Varieties.  Written as a down-to-earth introduction to Shimura varieties, this text includes many examples and applications of the theory that provide motivation for the reader.  Since it is limited to modular curves and the corresponding Shimura varieties, this book is not only a great resource for experts in the field, but it is also accessible to advanced graduate students studying number theory.  Key topics include non-triviality of arithmetic invariants and special values of L-functions; elliptic curves over complex and p-adic fields; Hecke algebras; scheme theory; elliptic and modular curves over rings; and Shimura curves.

Curves, Elliptic. --- Invariants. --- Number theory. --- Curves, Elliptic --- Invariants --- Number theory --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Elliptic functions. --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Number study --- Numbers, Theory of --- Mathematics. --- Algebraic geometry. --- Number Theory. --- Algebraic Geometry. --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Geometry, algebraic. --- Algebraic geometry --- Geometry

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Introduction to Global Optimization Exploiting Space-Filling Curves provides an overview of classical and new results pertaining to the usage of space-filling curves in global optimization. The authors look at a family of derivative-free numerical algorithms applying space-filling curves to reduce the dimensionality of the global optimization problem; along with a number of unconventional ideas, such as adaptive strategies for estimating Lipschitz constant, balancing global and local information to accelerate the search. Convergence conditions of the described algorithms are studied in depth and theoretical considerations are illustrated through numerical examples. This work also contains a code for implementing space-filling curves that can be used for constructing new global optimization algorithms. Basic ideas from this text can be applied to a number of problems including problems with multiextremal and partially defined constraints and non-redundant parallel computations can be organized. Professors, students, researchers, engineers, and other professionals in the fields of pure mathematics, nonlinear sciences studying fractals, operations research, management science, industrial and applied mathematics, computer science, engineering, economics, and the environmental sciences will find this title useful . .

Mathematical optimization --- Nonconvex programming --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Curves --- Curves on surfaces --- Mathematical models. --- Surfaces, Curves on --- Mathematics. --- Algebraic geometry. --- Computer software. --- Numerical analysis. --- Operations research. --- Management science. --- Manifolds (Mathematics). --- Complex manifolds. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Operations Research, Management Science. --- Mathematical Software. --- Numerical Analysis. --- Algebraic Geometry. --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Shapes --- Cell aggregation --- Geometry, algebraic. --- Algebraic geometry --- Mathematical analysis --- Software, Computer --- Computer systems --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation --- Quantitative business analysis --- Management --- Problem solving --- Operations research --- Statistical decision --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- Analytic spaces --- Manifolds (Mathematics) --- Topology --- Mathematical optimization. --- Nonconvex programming.