There are many proposed aims for scientific inquiry - to explain or predict events, to confirm or falsify hypotheses, or to find hypotheses that cohere with our other beliefs in some logical or probabilistic sense. This book is devoted to a different proposal - that the logical structure of the scientist's method should guarantee eventual arrival at the truth, given the scientist's background assumptions. Interest in this methodological property, called "logical reliability", stems from formal learning theory, which draws its insights not from the theory of probability, but from the theory of computability. Kelly first offers an accessible explanation of formal learning theory, then goes on to develop and explore a systematic framework in which various standard learning-theoretic results can be seen as special cases of simpler and more general considerations. Finally, Kelly clarifies the relationship between the resulting framework and other standard issues in the philosophy of science, such as probability, causation, and relativism. Extensively illustrated with figures by the author, The Logic of Reliable Inquiry assumes only introductory knowledge of basic logic and computability theory. It is a major contribution to the literature and will be essential reading for scientists, statiticians, psychologists, linguists, logicians, and philosophers.

Dirk van Dalen's biography studies the fascinating life of the famous Dutch mathematician and philosopher Luitzen Egbertus Jan Brouwer. Brouwer belonged to a special class of genius; complex and often controversial and gifted with a deep intuition, he had an unparalleled access to the secrets and intricacies of mathematics. Most mathematicians remember L.E.J. Brouwer from his scientific breakthroughs in the young subject of topology and for the famous Brouwer fixed point theorem. Brouwer's main interest, however, was in the foundation of mathematics which led him to introduce, and then consolidate, constructive methods under the name 'intuitionism'. This made him one of the main protagonists in the 'foundation crisis' of mathematics. As a confirmed internationalist, he also got entangled in the interbellum struggle for the ending of the boycott of German and Austrian scientists. This time during the twentieth century was turbulent; nationalist resentment and friction between formalism and intuitionism led to the Mathematische Annalen conflict ('The war of the frogs and the mice'). It was here that Brouwer played a pivotal role. The present biography is an updated revision of the earlier two volume biography in one single book. It appeals to mathematicians and anybody interested in the history of mathematics in the first half of the twentieth century.

Topological Phases of Matter are an exceptionally dynamic field of research: several of the most exciting recent experimental discoveries and conceptual advances in modern physics have originated in this field. These have generated new, topological, notions of order, interactions and excitations. This text provides an accessible, unified and comprehensive introduction to the phenomena surrounding topological matter, with detailed expositions of the underlying theoretical tools and conceptual framework, alongside accounts of the central experimental breakthroughs. Among the systems covered are topological insulators, magnets, semimetals, and superconductors. The emergence of new particles with remarkable properties such as fractional charge and statistics is discussed alongside possible applications such as fault-tolerant topological quantum computing. Suitable as a textbook for graduate or advanced undergraduate students, or as a reference for more experienced researchers, the book assumes little prior background, providing self-contained introductions to topics as varied as phase transitions, superconductivity, and localisation.

This textbook is an alternative to a classical introductory book in point-set topology. The approach, however, is radically different from the classical one. It is based on convergence rather than on open and closed sets. Convergence of filters is a natural generalization of the basic and well-known concept of convergence of sequences, so that convergence theory is more natural and intuitive to many, perhaps most, students than classical topology. On the other hand, the framework of convergence is easier, more powerful and far-reaching which highlights a need for a theory of convergence in various branches of analysis.Convergence theory for filters is gradually introduced and systematically developed. Topological spaces are presented as a special subclass of convergence spaces of particular interest, but a large part of the material usually developed in a topology textbook is treated in the larger realm of convergence spaces.

This textbook is an alternative to a classical introductory book in point-set topology. The approach, however, is radically different from the classical one. It is based on convergence rather than on open and closed sets. Convergence of filters is a natural generalization of the basic and well-known concept of convergence of sequences, so that convergence theory is more natural and intuitive to many, perhaps most, students than classical topology. On the other hand, the framework of convergence is easier, more powerful and far-reaching which highlights a need for a theory of convergence in various branches of analysis.Convergence theory for filters is gradually introduced and systematically developed. Topological spaces are presented as a special subclass of convergence spaces of particular interest, but a large part of the material usually developed in a topology textbook is treated in the larger realm of convergence spaces.

This book is part of the series of three books arise from lectures organized by Hitoshi Murakami at RIMS, Kyoto University in the summer of 2001. The lecture series was aimed at a broad audience that included many graduate students. Its purpose lay in familiarizing the audience with the basics of 3-manifold theory and introducing some topics of current research. The first portion of the lecture series was devoted to standard topics in the theory of 3-manifolds. The middle portion was devoted to a brief study of Heegaard splittings and generalized Heegaard splittings.In the standard schematic diagram for generalized Heegaard splittings, Heegaard splittings are stacked on top of each other in a linear fashion. This can cause confusion in those cases in which generalized Heegaard splittings possess interesting connectivity properties. Fork complexes were invented in an effort to illuminate some of the more subtle issues arising in the study of generalized Heegaard splittings.

This book provides an introduction to the beautiful and deep subject of filling Dehn surfaces in the study of topological 3-manifolds. This book presents, for the first time in English and with all the details, the results from the PhD thesis of the first author, together with some more recent results in the subject. It also presents some key ideas on how these techniques could be used on other subjects.Representing 3-Manifolds by Filling Dehn Surfaces is mostly self-contained requiring only basic knowledge on topology and homotopy theory. The complete and detailed proofs are illustrated with a set of more than 600 spectacular pictures, in the tradition of low-dimensional topology books. It is a basic reference for researchers in the area, but it can also be used as an advanced textbook for graduate students or even for adventurous undergraduates in mathematics. The book uses topological and combinatorial tools developed throughout the twentieth century making the volume a trip along the history of low-dimensional topology.

The book presents surveys describing recent developments in most of the primary subfields of
General Topology and its applications to Algebra and Analysis during the last decade. It follows freely
the previous edition (North Holland, 1992), Open Problems in Topology (North Holland, 1990) and
Handbook of Set-Theoretic Topology (North Holland, 1984). The book was prepared in
connection with the Prague Topological Symposium, held in 2001. During the last 10 years the focus
in General Topology changed and therefore the selection of topics differs slightly from those
chosen in 1992. The following areas experienced significant developments: Topological Groups, Function Spaces, Dimension Theory, Hyperspaces, Selections, Geometric Topology (including
Infinite-Dimensional Topology and the Geometry of Banach Spaces). Of course, not every important topic
could be included in this book.

Except surveys, the book contains several historical essays written by such eminent topologists as:
R.D. Anderson, W.W. Comfort, M. Henriksen, S. Mardeŝić, J. Nagata, M.E. Rudin, J.M. Smirnov (several reminiscences of L. Vietoris are added). In addition to extensive author and subject indexes, a list of all problems and questions posed in this book are added.

List of all authors of surveys:
A. Arhangel'skii, J. Baker and K. Kunen, H. Bennett and D. Lutzer, J. Dijkstra and J. van Mill, A. Dow, E. Glasner, G. Godefroy, G. Gruenhage, N. Hindman and D. Strauss, L. Hola and J. Pelant, K. Kawamura, H.-P. Kuenzi, W. Marciszewski, K. Martin and M. Mislove and M. Reed, R. Pol and H. Torunczyk, D. Repovs and P. Semenov, D. Shakhmatov, S. Solecki, M. Tkachenko.

Handbook of Convex Geometry, Volume B offers a survey of convex geometry and its many ramifications and connections with other fields of mathematics, including convexity, lattices, crystallography, and convex functions. The selection first offers information on the geometry of numbers, lattice points, and packing and covering with convex sets. Discussions focus on packing in non-Euclidean spaces, problems in the Euclidean plane, general convex bodies, computational complexity of lattice point problem, centrally symmetric convex bodies, reduction theory, and lattices and the space of lattices. The text then examines finite packing and covering and tilings, including plane tilings, monohedral tilings, bin packing, and sausage problems. The manuscript takes a look at valuations and dissections, geometric crystallography, convexity and differential geometry, and convex functions. Topics include differentiability, inequalities, uniqueness theorems for convex hypersurfaces, mixed discriminants and mixed volumes, differential geometric characterization of convexity, reduction of quadratic forms, and finite groups of symmetry operations. The selection is a dependable source of data for mathematicians and researchers interested in convex geometry.

Handbook of Convex Geometry, Volume A offers a survey of convex geometry and its many ramifications and relations with other areas of mathematics, including convexity, geometric inequalities, and convex sets. The selection first offers information on the history of convexity, characterizations of convex sets, and mixed volumes. Topics include elementary convexity, equality in the Aleksandrov-Fenchel inequality, mixed surface area measures, characteristic properties of convex sets in analysis and differential geometry, and extensions of the notion of a convex set. The text then reviews the standard isoperimetric theorem and stability of geometric inequalities. The manuscript takes a look at selected affine isoperimetric inequalities, extremum problems for convex discs and polyhedra, and rigidity. Discussions focus on include infinitesimal and static rigidity related to surfaces, isoperimetric problem for convex polyhedral, bounds for the volume of a convex polyhedron, curvature image inequality, Busemann intersection inequality and its relatives, and Petty projection inequality. The book then tackles geometric algorithms, convexity and discrete optimization, mathematical programming and convex geometry, and the combinatorial aspects of convex polytopes. The selection is a valuable source of data for mathematicians and researchers interested in convex geometry.

This monograph presents an approachable proof of Mirzakhani's curve counting theorem, both for simple and non-simple curves. Designed to welcome readers to the area, the presentation builds intuition with elementary examples before progressing to rigorous proofs. This approach illuminates new and established results alike, and produces versatile tools for studying the geometry of hyperbolic surfaces, Teichmuller theory, and mapping class groups. Beginning with the preliminaries of curves and arcs on surfaces, the authors go on to present the theory of geodesic currents in detail. Highlights include a treatment of cusped surfaces and surfaces with boundary, along with a comprehensive discussion of the action of the mapping class group on the space of geodesic currents. A user-friendly account of train tracks follows, providing the foundation for radallas, an immersed variation. From here, the authors apply these tools to great effect, offering simplified proofs of existing results and a new, more general proof of Mirzakhani's curve counting theorem. Further applications include counting square-tiled surfaces and mapping class group orbits, and investigating random geometric structures. Mirzakhani's Curve Counting and Geodesic Currents introduces readers to powerful counting techniques for the study of surfaces. Ideal for graduate students and researchers new to the area, the pedagogical approach, conversational style, and illuminating illustrations bring this exciting field to life. Exercises offer opportunities to engage with the material throughout. Basic familiarity with 2-dimensional topology and hyperbolic geometry, measured laminations, and the mapping class group is assumed.

This book is an introduction to manifolds at the beginning graduate level, and accessible to any student who has completed a solid undergraduate degree in mathematics. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields.

Although this second edition has the same basic structure as the first edition, it has been extensively revised and clarified; not a single page has been left untouched. The major changes include a new introduction to CW complexes (replacing most of the material on simplicial complexes in Chapter 5); expanded treatments of manifolds with boundary, local compactness, group actions, and proper maps; and a new section on paracompactness."

The study of triangulations of topological spaces has always been at the root of geometric topology. Among the most studied triangulations are piecewise linear triangulations of high-dimensional topological manifolds. Their study culminated in the late 1960s-early 1970s in a complete classification in the work of Kirby and Siebenmann. It is this classification that we discuss in this book, including the celebrated Hauptvermutung and Triangulation Conjecture.The goal of this book is to provide a readable and well-organized exposition of the subject, which would be suitable for advanced graduate students in topology. An exposition like this is currently lacking.

The aim of this book is to serve both as an introduction to profinite groups and as a reference for specialists in some areas of the theory. The book is reasonably self-contained. Profinite groups are Galois groups. As such they are of interest in algebraic number theory. Much of recent research on abstract infinite groups is related to profinite groups because residually finite groups are naturally embedded in a profinite group. In addition to basic facts about general profinite groups, the book emphasizes free constructions (particularly free profinite groups and the structure of their subgroups). Homology and cohomology is described with a minimum of prerequisites.

This second edition contains three new appendices dealing with a new characterization of free profinite groups, presentations of pro-p groups and a new conceptually simpler approach to the proof of some classical subgroup theorems. Throughout the text there are additions in the form of new results, improved proofs, typographical corrections, and an enlarged bibliography. The list of open questions has been updated; comments and references have been added about those previously open problems that have been solved after the first edition appeared.

This book is intended as an introduction to classical Fourier analysis, Fourier series, and the Fourier transform. The topics are developed slowly for the reader who has never seen them before, with a preference for clarity of exposition in stating and proving results. More recent developments, such as the discrete and fast Fourier transforms and wavelets, are covered in the last two chapters. The first three, short, chapters present requisite background material, and these could be read as a short course in functional analysis. The text includes many historical notes to place the material in a cultural and mathematical context; from the fact that Jean Baptiste Joseph Fourier was the nineteenth, but not the last, child in his family to the impact that Fourier series have had on the evolution of the concept of the integral.

The volume is focused on the basic calculation skills of various knot invariants defined from topology and geometry. It presents the detailed Hecke algebra and braid representation to illustrate the original Jones polynomial (rather than the algebraic formal definition many other books and research articles use) and provides self-contained proofs of the Tait conjecture (one of the big achievements from the Jones invariant). It also presents explicit computations to the Casson-Lin invariant via braid representations.With the approach of an explicit computational point of view on knot invariants, this user-friendly volume will benefit readers to easily understand low-dimensional topology from examples and computations, rather than only knowing terminologies and theorems.

This is the first book to provide a systematic explanation of both the problems of symplectic topology, and analytical details and techniques in applying the machinery embedded in the Floer theory as a whole. It provides a self-contained exposition of all foundational materials of Floer theory and its applications to various problems arising in symplectic topology. The author gives complete analytic details assuming the reader's knowledge of basic elliptic theory of (first-order) partial differential equations, second-year graduate differential geometry and first-year algebraic topology. He motivates various constructions appearing in Floer theory from the historical context of Lagrange Hamilton's variational principle and Hamiltonian mechanics. He also provides 100 exercises so that readers can test their understanding. The book is a comprehensive resource suitable for experts and newcomers alike."

Spaces of constant curvature, i.e. Euclidean space, the sphere, and Loba chevskij space, occupy a special place in geometry. They are most accessible to our geometric intuition, making it possible to develop elementary geometry in a way very similar to that used to create the geometry we learned at school. However, since its basic notions can be interpreted in different ways, this geometry can be applied to objects other than the conventional physical space, the original source of our geometric intuition. Euclidean geometry has for a long time been deeply rooted in the human mind. The same is true of spherical geometry, since a sphere can naturally be embedded into a Euclidean space. Lobachevskij geometry, which in the first fifty years after its discovery had been regarded only as a logically feasible by-product appearing in the investigation of the foundations of geometry, has even now, despite the fact that it has found its use in numerous applications, preserved a kind of exotic and even romantic element. This may probably be explained by the permanent cultural and historical impact which the proof of the independence of the Fifth Postulate had on human thought."

The book explains concepts and ideas of mathematics and physics that are relevant for advanced students and researchers of condensed matter physics. With this aim, a brief intuitive introduction to many-body theory is given as a powerful qualitative tool for understanding complex systems. The important emergent concept of a quasiparticle is then introduced as a way to reduce a many-body problem to a single particle quantum problem. Examples of quasiparticles in graphene, superconductors, superfluids and in a topological insulator on a superconductor are discussed.The mathematical idea of self-adjoint extension, which allows short distance information to be included in an effective long distance theory through boundary conditions, is introduced through simple examples and then applied extensively to analyse and predict new physical consequences for graphene.The mathematical discipline of topology is introduced in an intuitive way and is then combined with the methods of differential geometry to show how the emergence of gapless states can be understood. Practical ways of carrying out topological calculations are described.

The techniques and concepts of modern algebra are introduced for their natural role in the study of projectile geometry; groups appear as automorphism groups of configurations, division rings appear in the study of Desargues' theorem and the study of the independence of the seven axioms given for projectile geometry.

Harmonic maps are generalisations of the concept of geodesics. They encompass many fundamental examples in differential geometry and have recently become of widespread use in many areas of mathematics and mathematical physics. This is an accessible introduction to some of the fundamental connections between differential geometry, Lie groups, and integrable Hamiltonian systems. The specific goal of the book is to show how the theory of loop groups can be used to study harmonic maps. By concentrating on the main ideas and examples, the author leads up to topics of current research. The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists.

The notion of an ( ,1)-category has become widely used in homotopy theory, category theory, and in a number of applications. There are many different approaches to this structure, all of them equivalent, and each with its corresponding homotopy theory. This book provides a relatively self-contained source of the definitions of the different models, the model structure (homotopy theory) of each, and the equivalences between the models. While most of the current literature focusses on how to extend category theory in this context, and centers in particular on the quasi-category model, this book offers a balanced treatment of the appropriate model structures for simplicial categories, Segal categories, complete Segal spaces, quasi-categories, and relative categories, all from a homotopy-theoretic perspective. Introductory chapters provide background in both homotopy and category theory and contain many references to the literature, thus making the book accessible to graduates and to researchers in related areas.

This research monograph is a detailed account with complete proofs of rational homotopy theory for general non-simply connected spaces, based on the minimal models introduced by Sullivan in his original seminal article. Much of the content consists of new results, including generalizations of known results in the simply connected case. The monograph also includes an expanded version of recently published results about the growth and structure of the rational homotopy groups of finite dimensional CW complexes, and concludes with a number of open questions.This monograph is a sequel to the book Rational Homotopy Theory [RHT], published by Springer in 2001, but is self-contained except only that some results from [RHT] are simply quoted without proof.

Far from being separate entities, many social and engineering systems can be considered as complex network systems (CNSs) associated with closely linked interactions with neighbouring entities such as the Internet and power grids. Roughly speaking, a CNS refers to a networking system consisting of lots of interactional individuals, exhibiting fascinating collective behaviour that cannot always be anticipated from the inherent properties of the individuals themselves. As one of the most fundamental examples of cooperative behaviour, consensus within CNSs (or the synchronization of complex networks) has gained considerable attention from various fields of research, including systems science, control theory and electrical engineering. This book mainly studies consensus of CNSs with dynamics topologies - unlike most existing books that have focused on consensus control and analysis for CNSs under a fixed topology. As most practical networks have limited communication ability, switching graphs can be used to characterize real-world communication topologies, leading to a wider range of practical applications. This book provides some novel multiple Lyapunov functions (MLFs), good candidates for analysing the consensus of CNSs with directed switching topologies, while each chapter provides detailed theoretical analyses according to the stability theory of switched systems. Moreover, numerical simulations are provided to validate the theoretical results. Both professional researchers and laypeople will benefit from this book.