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Designed for intermediate graduate studies, this text will broaden students' core knowledge of differential geometry providing foundational material to relevant topics in classical differential geometry. The method of moving frames, a natural means for discovering and proving important results, provides the basis of treatment for topics discussed. Its application in many areas helps to connect the various geometries and to uncover many deep relationships, such as the Lawson correspondence. The nearly 300 problems and exercises range from simple applications to open problems. Exercises are embedded in the text as essential parts of the exposition. Problems are collected at the end of each chapter; solutions to select problems are given at the end of the book. Mathematica®, Matlab™, and Xfig are used to illustrate selected concepts and results. The careful selection of results serves to show the reader how to prove the most important theorems in the subject, which may become the foundation of future progress. The book pursues significant results beyond the standard topics of an introductory differential geometry course. A sample of these results includes the Willmore functional, the classification of cyclides of Dupin, the Bonnet problem, constant mean curvature immersions, isothermic immersions, and the duality between minimal surfaces in Euclidean space and constant mean curvature surfaces in hyperbolic space. The book concludes with Lie sphere geometry and its spectacular result that all cyclides of Dupin are Lie sphere equivalent. The exposition is restricted to curves and surfaces in order to emphasize the geometric interpretation of invariants and other constructions. Working in low dimensions helps students develop a strong geometric intuition. Aspiring geometers will acquire a working knowledge of curves and surfaces in classical geometries. Students will learn the invariants of conformal geometry and how these relate to the invariants of Euclidean, spherical, and hyperbolic geometry. They will learn the fundamentals of Lie sphere geometry, which require the notion of Legendre immersions of a contact structure. Prerequisites include a completed one semester standard course on manifold theory.

Geometry --- Mathematics --- Physical Sciences & Mathematics --- Mathematics. --- Convex geometry. --- Discrete geometry. --- Differential geometry. --- Differential Geometry. --- Convex and Discrete Geometry. --- Differential geometry --- Combinatorial geometry --- Math --- Science --- Global differential geometry. --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Geometry, Differential --- Discrete mathematics --- Surfaces. --- Curved surfaces --- Shapes --- Convex geometry .

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Featuring the work of twenty-three internationally-recognized experts, this volume explores the trace formula, spectra of locally symmetric spaces, p-adic families, and other recent techniques from harmonic analysis and representation theory. Each peer-reviewed submission in this volume, based on the Simons Foundation symposium on families of automorphic forms and the trace formula held in Puerto Rico in January-February 2014, is the product of intensive research collaboration by the participants over the course of the seven-day workshop. The goal of each session in the symposium was to bring together researchers with diverse specialties in order to identify key difficulties as well as fruitful approaches being explored in the field. The respective themes were counting cohomological forms, p-adic trace formulas, Hecke fields, slopes of modular forms, and orbital integrals.

Mathematics. --- Harmonic analysis. --- Convex geometry. --- Discrete geometry. --- Number theory. --- Topology. --- Abstract Harmonic Analysis. --- Convex and Discrete Geometry. --- Number Theory. --- Automorphic forms --- Automorphic functions --- Forms (Mathematics) --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Number study --- Numbers, Theory of --- Algebra --- Discrete mathematics --- Convex geometry . --- Combinatorial geometry

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This is a collection of some easily-formulated problems that remain open in the study of the geometry and analysis of Banach spaces. Assuming the reader has a working familiarity with the basic results of Banach space theory, the authors focus on concepts of basic linear geometry, convexity, approximation, optimization, differentiability, renormings, weak compact generating, Schauder bases and biorthogonal systems, fixed points, topology and nonlinear geometry. The main purpose of this work is to help convince young researchers in Functional Analysis that the theory of Banach spaces is a fertile field of research, full of interesting open problems. Inside the Banach space area, the text should help expose young researchers to the depth and breadth of the work that remains, and to provide the perspective necessary to choose a direction for further study. Some of the problems presented herein are longstanding open problems, some are recent, some are more important and some are only "local" problems. Some would require new ideas, while others may be resolved with only a subtle combination of known facts. Regardless of their origin or longevity, each of these problems documents the need for further research in this area.

Mathematics. --- Approximation theory. --- Functional analysis. --- Measure theory. --- Convex geometry. --- Discrete geometry. --- Algebraic topology. --- Functional Analysis. --- Approximations and Expansions. --- Measure and Integration. --- Convex and Discrete Geometry. --- Algebraic Topology. --- Banach spaces. --- Geometry. --- Mathematics --- Euclid's Elements --- Functions of complex variables --- Generalized spaces --- Topology --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Math --- Science --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Discrete mathematics --- Convex geometry . --- Geometry --- Combinatorial geometry --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems

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This volume comprises eight papers delivered at the RIMS International Conference "Mathematical Challenges in a New Phase of Materials Science", Kyoto, August 4–8, 2014. The contributions address subjects in defect dynamics, negatively curved carbon crystal, topological analysis of di-block copolymers, persistence modules, and fracture dynamics. These papers highlight the strong interaction between mathematics and materials science and also reflect the activity of WPI-AIMR at Tohoku University, in which collaborations between mathematicians and experimentalists are actively ongoing.

Mathematics. --- Dynamics. --- Ergodic theory. --- Partial differential equations. --- Convex geometry. --- Discrete geometry. --- Physics. --- Partial Differential Equations. --- Mathematical Methods in Physics. --- Dynamical Systems and Ergodic Theory. --- Convex and Discrete Geometry. --- Materials science --- Mathematics --- Material science --- Physical sciences --- Differential equations, partial. --- Mathematical physics. --- Differentiable dynamical systems. --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Physical mathematics --- Physics --- Partial differential equations --- Discrete mathematics --- Convex geometry . --- Geometry --- Combinatorial geometry --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Natural philosophy --- Philosophy, Natural --- Dynamics

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This book constitutes the thoroughly refereed post-conference proceedings of the 18th Japanese Conference on Discrete and Computational Geometry and Graphs, JDCDGG 2015, held in Kyoto, Japan, in September 2015. The total of 25 papers included in this volume was carefully reviewed and selected from 64 submissions. The papers feature advances made in the field of computational geometry and focus on emerging technologies, new methodology and applications, graph theory and dynamics. This proceedings are dedicated to Naoki Katoh on the occasion of his retirement from Kyoto University.

Computer science. --- Data structures (Computer science). --- Algorithms. --- Computer science --- Computer graphics. --- Convex geometry. --- Discrete geometry. --- Computer Science. --- Computer Graphics. --- Discrete Mathematics in Computer Science. --- Algorithm Analysis and Problem Complexity. --- Data Structures. --- Convex and Discrete Geometry. --- Mathematics. --- Automatic drafting --- Graphic data processing --- Graphics, Computer --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Algorism --- Information structures (Computer science) --- Structures, Data (Computer science) --- Structures, Information (Computer science) --- Informatics --- Mathematics --- Geometry --- Combinatorial geometry --- Computer art --- Graphic arts --- Engineering graphics --- Image processing --- Algebra --- Arithmetic --- File organization (Computer science) --- Abstract data types (Computer science) --- Science --- Digital techniques --- Foundations --- Computational complexity. --- Computer software. --- Data structures (Computer science) --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Software, Computer --- Computer systems --- Complexity, Computational --- Machine theory --- Data structures (Computer scienc. --- Discrete geometry --- Computer science—Mathematics. --- Convex geometry . --- Discrete mathematics. --- Artificial intelligence—Data processing. --- Data Science. --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis