The focus of this book is on providing students with insights into geometry that can help them understand deep learning from a unified perspective. Rather than describing deep learning as an implementation technique, as is usually the case in many existing deep learning books, here, deep learning is explained as an ultimate form of signal processing techniques that can be imagined. To support this claim, an overview of classical kernel machine learning approaches is presented, and their advantages and limitations are explained. Following a detailed explanation of the basic building blocks of deep neural networks from a biological and algorithmic point of view, the latest tools such as attention, normalization, Transformer, BERT, GPT-3, and others are described. Here, too, the focus is on the fact that in these heuristic approaches, there is an important, beautiful geometric structure behind the intuition that enables a systematic understanding. A unified geometric analysis to understand the working mechanism of deep learning from high-dimensional geometry is offered. Then, different forms of generative models like GAN, VAE, normalizing flows, optimal transport, and so on are described from a unified geometric perspective, showing that they actually come from statistical distance-minimization problems. Because this book contains up-to-date information from both a practical and theoretical point of view, it can be used as an advanced deep learning textbook in universities or as a reference source for researchers interested in acquiring the latest deep learning algorithms and their underlying principles. In addition, the book has been prepared for a codeshare course for both engineering and mathematics students, thus much of the content is interdisciplinary and will appeal to students from both disciplines.

This book provides an introduction to deformation quantization and its relation to quantum field theory, with a focus on the constructions of Kontsevich and Cattaneo & Felder. This subject originated from an attempt to understand the mathematical structure when passing from a commutative classical algebra of observables to a non-commutative quantum algebra of observables. Developing deformation quantization as a semi-classical limit of the expectation value for a certain observable with respect to a special sigma model, the book carefully describes the relationship between the involved algebraic and field-theoretic methods. The connection to quantum field theory leads to the study of important new field theories and to insights in other parts of mathematics such as symplectic and Poisson geometry, and integrable systems. Based on lectures given by the author at the University of Zurich, the book will be of interest to graduate students in mathematics or theoretical physics. Readers will be able to begin the first chapter after a basic course in Analysis, Linear Algebra and Topology, and references are provided for more advanced prerequisites.

This volume originated in talks given in Cortona at the conference "Geometric aspects of harmonic analysis" held in honor of the 70th birthday of Fulvio Ricci. It presents timely syntheses of several major fields of mathematics as well as original research articles contributed by some of the finest mathematicians working in these areas. The subjects dealt with are topics of current interest in closely interrelated areas of Fourier analysis, singular integral operators, oscillatory integral operators, partial differential equations, multilinear harmonic analysis, and several complex variables. The work is addressed to researchers in the field.

This is a volume originating from the Conference on Partial Differential Equations and Applications, which was held in Moscow in November 2018 in memory of professor Boris Sternin and attracted more than a hundred participants from eighteen countries. The conference was mainly dedicated to partial differential equations on manifolds and their applications in mathematical physics, geometry, topology, and complex analysis. The volume contains selected contributions by leading experts in these fields and presents the current state of the art in several areas of PDE. It will be of interest to researchers and graduate students specializing in partial differential equations, mathematical physics, topology, geometry, and their applications. The readers will benefit from the interplay between these various areas of mathematics.

What do the classification of algebraic surfaces, Weyl's dimension formula and maximal orders in central simple algebras have in common? All are related to a type of manifold called locally mixed symmetric spaces in this book. The presentation emphasizes geometric concepts and relations and gives each reader the "roter Faden", starting from the basics and proceeding towards quite advanced topics which lie at the intersection of differential and algebraic geometry, algebra and topology. Avoiding technicalities and assuming only a working knowledge of real Lie groups, the text provides a wealth of examples of symmetric spaces. The last two chapters deal with one particular case (Kuga fiber spaces) and a generalization (elliptic surfaces), both of which require some knowledge of algebraic geometry. Of interest to topologists, differential or algebraic geometers working in areas related to arithmetic groups, the book also offers an introduction to the ideas for non-experts.

This textbook provides a thorough introduction to the differential geometry of parametrized curves and surfaces, along with a wealth of applications to specific architectural elements. Geometric elements in architecture respond to practical, physical and aesthetic needs. Proper understanding of the mathematics underlying the geometry provides control over the construction. This book relates the classical mathematical theory of parametrized curves and surfaces to multiple applications in architecture. The presentation is mathematically complete with numerous figures and animations illustrating the theory, and special attention is given to some of the recent trends in the field. Solved exercises are provided to see the theory in practice. Intended as a textbook for lecture courses, Parametric Geometry of Curves and Surfaces is suitable for mathematically-inclined students in engineering, architecture and related fields, and can also serve as a textbook for traditional differential geometry courses to mathematics students. Researchers interested in the mathematics of architecture or computer-aided design will also value its combination of precise mathematics and architectural examples.

The book presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds and the most important structures on them. The authors' approach is that the source of all constructions in Riemannian geometry is a manifold that allows one to compute scalar products of tangent vectors. With this approach, the authors show that Riemannian geometry has a great influence to several fundamental areas of modern mathematics and its applications.In particular, Geometry is a bridge between pure mathematics and natural sciences, first of all physics. Fundamental laws of nature are formulated as relations between geometric fields describing various physical quantities. The study of global properties of geometric objects leads to the far-reaching development of topology, including topology and geometry of fiber bundles. Geometric theory of Hamiltonian systems, which describe many physical phenomena, led to the development of symplectic and Poisson geometry. Field theory and the multidimensional calculus of variations, presented in the book, unify mathematics with theoretical physics. Geometry of complex and algebraic manifolds unifies Riemannian geometry with modern complex analysis, as well as with algebra and number theory. Prerequisites for using the book include several basic undergraduate courses, such as advanced calculus, linear algebra, ordinary differential equations, and elements of topology.

The correspondence between Einstein metrics and their conformal boundaries has recently been the focus of great interest. This is particularly so in view of the relation with the physical theory of the AdS/CFT correspondence. In this book, this correspondence is seen in the wider context of asymptotically symmetric Einstein metrics, that is Einstein metrics whose curvature is asymptotic to that of a rank one symmetric space. There is an emphasis on the correspondence between Einstein metrics and geometric structures on their boundary at infinity: conformal structures, CR structures, and quaternionic contact structures introduced and studied in the book. Two new constructions of such Einstein metrics are given, using two different kinds of techniques: analytic methods to construct complete Einstein metrics, with a unified treatment of all rank one symmetric spaces, relying on harmonic analysis; algebraic methods (twistor theory) to construct local solutions of the Einstein equation near the boundary.

This concise textbook introduces the reader to advanced mathematical aspects of general relativity, covering topics like Penrose diagrams, causality theory, singularity theorems, the Cauchy problem for the Einstein equations, the positive mass theorem, and the laws of black hole thermodynamics. It emerged from lecture notes originally conceived for a one-semester course in Mathematical Relativity which has been taught at the Instituto Superior Tecnico (University of Lisbon, Portugal) since 2010 to Masters and Doctorate students in Mathematics and Physics. Mostly self-contained, and mathematically rigorous, this book can be appealing to graduate students in Mathematics or Physics seeking specialization in general relativity, geometry or partial differential equations. Prerequisites include proficiency in differential geometry and the basic principles of relativity. Readers who are familiar with special relativity and have taken a course either in Riemannian geometry (for students of Mathematics) or in general relativity (for those in Physics) can benefit from this book.

Deep connections exist between harmonic and applied analysis and the diverse yet connected topics of machine learning, data analysis, and imaging science. This volume explores these rapidly growing areas and features contributions presented at the second and third editions of the Summer Schools on Applied Harmonic Analysis, held at the University of Genova in 2017 and 2019. Each chapter offers an introduction to essential material and then demonstrates connections to more advanced research, with the aim of providing an accessible entrance for students and researchers. Topics covered include ill-posed problems; concentration inequalities; regularization and large-scale machine learning; unitarization of the radon transform on symmetric spaces; and proximal gradient methods for machine learning and imaging.

This book studies a class of monopoles defined by certain mild conditions, called periodic monopoles of generalized Cherkis-Kapustin (GCK) type. It presents a classification of the latter in terms of difference modules with parabolic structure, revealing a kind of Kobayashi-Hitchin correspondence between differential geometric objects and algebraic objects. It also clarifies the asymptotic behaviour of these monopoles around infinity. The theory of periodic monopoles of GCK type has applications to Yang-Mills theory in differential geometry and to the study of difference modules in dynamical algebraic geometry. A complete account of the theory is given, including major generalizations of results due to Charbonneau, Cherkis, Hurtubise, Kapustin, and others, and a new and original generalization of the nonabelian Hodge correspondence first studied by Corlette, Donaldson, Hitchin and Simpson. This work will be of interest to graduate students and researchers in differential and algebraic geometry, as well as in mathematical physics.

This volume presents lectures given at the Wisla 19 Summer School: Differential Geometry, Differential Equations, and Mathematical Physics, which took place from August 19 - 29th, 2019 in Wisla, Poland, and was organized by the Baltic Institute of Mathematics. The lectures were dedicated to symplectic and Poisson geometry, tractor calculus, and the integration of ordinary differential equations, and are included here as lecture notes comprising the first three chapters. Following this, chapters combine theoretical and applied perspectives to explore topics at the intersection of differential geometry, differential equations, and mathematical physics. Specific topics covered include: Parabolic geometry Geometric methods for solving PDEs in physics, mathematical biology, and mathematical finance Darcy and Euler flows of real gases Differential invariants for fluid and gas flow Differential Geometry, Differential Equations, and Mathematical Physics is ideal for graduate students and researchers working in these areas. A basic understanding of differential geometry is assumed.

This text is an enhanced, English version of the Russian edition, published in early 2021 and is appropriate for an introductory course in geometric control theory. The concise presentation provides an accessible treatment of the subject for advanced undergraduate and graduate students in theoretical and applied mathematics, as well as to experts in classic control theory for whom geometric methods may be introduced. Theory is accompanied by characteristic examples such as stopping a train, motion of mobile robot, Euler elasticae, Dido's problem, and rolling of the sphere on the plane. Quick foundations to some recent topics of interest like control on Lie groups and sub-Riemannian geometry are included. Prerequisites include only a basic knowledge of calculus, linear algebra, and ODEs; preliminary knowledge of control theory is not assumed. The applications problems-oriented approach discusses core subjects and encourages the reader to solve related challenges independently. Highly-motivated readers can acquire working knowledge of geometric control techniques and progress to studying control problems and more comprehensive books on their own. Selected sections provide exercises to assist in deeper understanding of the material. Controllability and optimal control problems are considered for nonlinear nonholonomic systems on smooth manifolds, in particular, on Lie groups. For the controllability problem, the following questions are considered: controllability of linear systems, local controllability of nonlinear systems, Nagano-Sussmann Orbit theorem, Rashevskii-Chow theorem, Krener's theorem. For the optimal control problem, Filippov's theorem is stated, invariant formulation of Pontryagin maximum principle on manifolds is given, second-order optimality conditions are discussed, and the sub-Riemannian problem is studied in detail. Pontryagin maximum principle is proved for sub-Riemannian problems, solution to the sub-Riemannian problems on the Heisenberg group, the group of motions of the plane, and the Engel group is described.

This monograph presents recent developments in comparison geometry and geometric analysis on Finsler manifolds. Generalizing the weighted Ricci curvature into the Finsler setting, the author systematically derives the fundamental geometric and analytic inequalities in the Finsler context. Relying only upon knowledge of differentiable manifolds, this treatment offers an accessible entry point to Finsler geometry for readers new to the area. Divided into three parts, the book begins by establishing the fundamentals of Finsler geometry, including Jacobi fields and curvature tensors, variation formulas for arc length, and some classical comparison theorems. Part II goes on to introduce the weighted Ricci curvature, nonlinear Laplacian, and nonlinear heat flow on Finsler manifolds. These tools allow the derivation of the Bochner-Weitzenboeck formula and the corresponding Bochner inequality, gradient estimates, Bakry-Ledoux's Gaussian isoperimetric inequality, and functional inequalities in the Finsler setting. Part III comprises advanced topics: a generalization of the classical Cheeger-Gromoll splitting theorem, the curvature-dimension condition, and the needle decomposition. Throughout, geometric descriptions illuminate the intuition behind the results, while exercises provide opportunities for active engagement. Comparison Finsler Geometry offers an ideal gateway to the study of Finsler manifolds for graduate students and researchers. Knowledge of differentiable manifold theory is assumed, along with the fundamentals of functional analysis. Familiarity with Riemannian geometry is not required, though readers with a background in the area will find their insights are readily transferrable.

This book is devoted to Killing vector fields and the one-parameter isometry groups of Riemannian manifolds generated by them. It also provides a detailed introduction to homogeneous geodesics, that is, geodesics that are integral curves of Killing vector fields, presenting both classical and modern results, some very recent, many of which are due to the authors. The main focus is on the class of Riemannian manifolds with homogeneous geodesics and on some of its important subclasses. To keep the exposition self-contained the book also includes useful general results not only on geodesic orbit manifolds, but also on smooth and Riemannian manifolds, Lie groups and Lie algebras, homogeneous Riemannian manifolds, and compact homogeneous Riemannian spaces. The intended audience is graduate students and researchers whose work involves differential geometry and transformation groups.

There are rich connections between classical analysis and number theory. For instance, analytic number theory contains many examples of asymptotic expressions derived from estimates for analytic functions, such as in the proof of the Prime Number Theorem. In combinatorial number theory, exact formulas for number-theoretic quantities are derived from relations between analytic functions. Elliptic functions, especially theta functions, are an important class of such functions in this context, which had been made clear already in Jacobi's "Fundamenta Nova". Theta functions are also classically connected with Riemann surfaces and with the modular group Gamma, which provide another path for insights into number theory. Hershel Farkas and Irwin Kra uncover combinatorial identities by means of the function theory on Riemann surfaces related to the principal congruence subgroups Gamma.

The volume consists of a set of surveys on geometry in the broad sense. The goal is to present a certain number of research topics in a non-technical and appealing manner. The topics surveyed include spherical geometry, the geometry of finite-dimensional normed spaces, metric geometry (Bishop-Gromov type inequalities in Gromov-hyperbolic spaces), convexity theory and inequalities involving volumes and mixed volumes of convex bodies, 4-dimensional topology, Teichmuller spaces and mapping class groups actions, translation surfaces and their dynamics, and complex higher-dimensional geometry. Several chapters are based on lectures given by their authors to middle-advanced level students and young researchers. The whole book is intended to be an introduction to current research trends in geometry.

This is an introduction to noncommutative geometry, with special emphasis on those cases where the structure algebra, which defines the geometry, is an algebra of matrices over the complex numbers. Applications to elementary particle physics are also discussed. This second edition is thoroughly revised and includes new material on reality conditions and linear connections plus examples from Jordanian deformations and quantum Euclidean spaces. Only some familiarity with ordinary differential geometry and the theory of fiber bundles is assumed, making this book accessible to graduate students and newcomers to this field.

During the last few years, the field of nonlinear problems has undergone great development. This book consisting of the updated Grundlehren volume 252 by the author and of a newly written part, deals with some important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved. Each problem is explained, up-to-date results are given and proofs are presented. Thus, the reader is given access, for each specific problem, to its present status of solution as well as to the most up-to-date methods for approaching it. The main objective of the book is to explain some methods and new techniques, and to apply them. It deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber.

This book provides a complete exposition of equidistribution and counting problems weighted by a potential function of common perpendicular geodesics in negatively curved manifolds and simplicial trees. Avoiding any compactness assumptions, the authors extend the theory of Patterson-Sullivan, Bowen-Margulis and Oh-Shah (skinning) measures to CAT(-1) spaces with potentials. The work presents a proof for the equidistribution of equidistant hypersurfaces to Gibbs measures, and the equidistribution of common perpendicular arcs between, for instance, closed geodesics. Using tools from ergodic theory (including coding by topological Markov shifts, and an appendix by Buzzi that relates weak Gibbs measures and equilibrium states for them), the authors further prove the variational principle and rate of mixing for the geodesic flow on metric and simplicial trees-again without the need for any compactness or torsionfree assumptions. In a series of applications, using the Bruhat-Tits trees over non-Archimedean local fields, the authors subsequently prove further important results: the Mertens formula and the equidistribution of Farey fractions in function fields, the equidistribution of quadratic irrationals over function fields in their completions, and asymptotic counting results of the representations by quadratic norm forms. One of the book's main benefits is that the authors provide explicit error terms throughout. Given its scope, it will be of interest to graduate students and researchers in a wide range of fields, for instance ergodic theory, dynamical systems, geometric group theory, discrete subgroups of locally compact groups, and the arithmetic of function fields.

This textbook is suitable for a one semester lecture course on differential geometry for students of mathematics or STEM disciplines with a working knowledge of analysis, linear algebra, complex analysis, and point set topology. The book treats the subject both from an extrinsic and an intrinsic view point.The first chapters give a historical overview of the field and contain an introduction to basic concepts such as manifolds and smooth maps, vector fields and flows, and Lie groups, leading up to the theorem of Frobenius. Subsequent chapters deal with the Levi-Civita connection, geodesics, the Riemann curvature tensor, a proof of the Cartan-Ambrose-Hicks theorem, as well as applications to flat spaces, symmetric spaces, and constant curvature manifolds. Also included are sections about manifolds with nonpositive sectional curvature, the Ricci tensor, the scalar curvature, and the Weyl tensor. An additional chapter goes beyond the scope of a one semester lecture course and deals with subjects such as conjugate points and the Morse index, the injectivity radius, the group of isometries and the Myers-Steenrod theorem, and Donaldson's differential geometric approach to Lie algebra theory.

This volume is a compilation of papers presented at the conference on differential geometry, in particular, minimal surfaces, real hypersurfaces of a non-flat complex space form, submanifolds of symmetric spaces and curve theory. It also contains new results or brief surveys in these areas. This volume provides fundamental knowledge to readers (such as differential geometers) who are interested in the theory of real hypersurfaces in a non-flat complex space form.

This book is devoted to geometric problems of foliation theory, in particular those related to extrinsic geometry, modern branch of Riemannian Geometry. The concept of mixed curvature is central to the discussion, and a version of the deep problem of the Ricci curvature for the case of mixed curvature of foliations is examined. The book is divided into five chapters that deal with integral and variation formulas and curvature and dynamics of foliations. Different approaches and methods (local and global, regular and singular) in solving the problems are described using integral and variation formulas, extrinsic geometric flows, generalizations of the Ricci and scalar curvatures, pseudo-Riemannian and metric-affine geometries, and 'computable' Finsler metrics. The book presents the state of the art in geometric and analytical theory of foliations as a continuation of the authors' life-long work in extrinsic geometry. It is designed for newcomers to the field as well as experienced geometers working in Riemannian geometry, foliation theory, differential topology, and a wide range of researchers in differential equations and their applications. It may also be a useful supplement to postgraduate level work and can inspire new interesting topics to explore.

This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincare that brought hyperbolic geometry into the mainstream of mathematics. A recognition of Beltrami comparable to that given the pioneering works of Bolyai and Lobachevsky seems long overdue - not only because Beltrami rescued hyperbolic geometry from oblivion by proving it to be logically consistent, but because he gave it a concrete meaning (a model) that made hyperbolic geometry part of ordinary mathematics. The models subsequently discovered by Klein and Poincare brought hyperbolic geometry even further down to earth and paved the way for the current explosion of activity in low-dimensional geometry and topology.By placing the works of these three mathematicians side by side and providing commentaries, this book gives the student, historian, or professional geometer a bird's-eye view of one of the great episodes in mathematics. The unified setting and historical context reveal the insights of Beltrami, Klein, and Poincare in their full brilliance.

The package of Gromov's pseudo-holomorphic curves is a major tool in global symplectic geometry and its applications, including mirror symmetry and Hamiltonian dynamics. The Kuranishi structure was introduced by two of the authors of the present volume in the mid-1990s to apply this machinery on general symplectic manifolds without assuming any specific restrictions. It was further amplified by this book's authors in their monograph Lagrangian Intersection Floer Theory and in many other publications of theirs and others. Answering popular demand, the authors now present the current book, in which they provide a detailed, self-contained explanation of the theory of Kuranishi structures. Part I discusses the theory on a single space equipped with Kuranishi structure, called a K-space, and its relevant basic package. First, the definition of a K-space and maps to the standard manifold are provided. Definitions are given for fiber products, differential forms, partitions of unity, and the notion of CF-perturbations on the K-space. Then, using CF-perturbations, the authors define the integration on K-space and the push-forward of differential forms, and generalize Stokes' formula and Fubini's theorem in this framework. Also, "virtual fundamental class" is defined, and its cobordism invariance is proved. Part II discusses the (compatible) system of K-spaces and the process of going from "geometry" to "homological algebra". Thorough explanations of the extension of given perturbations on the boundary to the interior are presented. Also explained is the process of taking the "homotopy limit" needed to handle a system of infinitely many moduli spaces. Having in mind the future application of these chain level constructions beyond those already known, an axiomatic approach is taken by listing the properties of the system of the relevant moduli spaces and then a self-contained account of the construction of the associated algebraic structures is given. This axiomatic approach makes the exposition contained here independent of previously published construction of relevant structures.