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This book is about the foundations of geometric symmetry, namely, Lie groups and differential geometry. Although this is a classical subject about which hundreds of books have been written, this book takes a new and innovative approach. The main idea is to replace the Maurer–Cartan form with absolute parallelism and its curvature. Unlike the classical approach, where the model is fixed beforehand by the Maurer–Cartan form, this new approach is model-free, and also revisits the foundational concepts of differential geometry, such as covariant differentiation, from a different perspective.

The disc embedding theorem provides a detailed proof of the eponymous theorem in 4-manifold topology. The theorem, due to Michael Freedman, underpins virtually all of our understanding of 4-manifolds in the topological category. Most famously, this includes the 4-dimensional topological Poincaré conjecture. Combined with the concurrent work of Simon Donaldson, the theorem reveals a remarkable disparity between the topological and smooth categories for 4-manifolds. A thorough exposition of Freedman’s proof of the disc embedding theorem is given, with many new details. A self-contained account of decomposition space theory, a beautiful but outmoded branch of topology that produces non-differentiable homeomorphisms between manifolds, is provided. Techniques from decomposition space theory are used to show that an object produced by an infinite, iterative process, which we call a skyscraper, is homeomorphic to a thickened disc, relative to its boundary. A stand-alone interlude explains the disc embedding theorem’s key role in smoothing theory, the existence of exotic smooth structures on Euclidean space, and all known homeomorphism classifications of 4-manifolds via surgery theory and the s-cobordism theorem. The book is written to be accessible to graduate students working on 4-manifolds, as well as researchers in related areas. It contains over a hundred professionally rendered figures.

One of the subjects of functional analysis is classification of Banach spaces depending on various properties of the unit ball. The need of such considerations comes from a number of applications to problems of mathematical analysis. The list of subjects contains: differential calculus in normed spaces, approximation theory, weak topologies and reflexivity, general theory of convexity and convex functions, metric fixed point theory, and others. The aim of this book is to present basic facts from this field. It is addressed to advanced undergraduate and graduate students interested in the subject. For some it may result in further interest, a continuation and deepening of their study of the subject. It may be also useful for instructors running courses on functional analysis, supervising diploma theses or essays on various levels.

This book illustrates how different methods of finite group theory including representation theory, cohomology theory, combinatorial group theory, and local analysis, are combined to construct one of the last of the sporadic finite simple groups — the fourth Janko group J₄. This book's approach is based on analysis of group amalgams and the geometry of the complexes of these amalgams with emphasis on the underlying theory.

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.

Harmonic morphisms are maps which preserve Laplace's equation. More explicitly, a map between Riemannian manifolds is called a harmonic morphism if its composition with any locally defined harmonic function on the codomain is a harmonic function on the domain; it thus ‘pulls back’ germs of harmonic functions to germs of harmonic functions. Harmonic morphisms can be characterized as harmonic maps satisfying a condition dual to weak conformality called ‘horizontal weak conformality’ or ‘semiconformality’. Examples include harmonic functions, conformal mappings in the plane, holomorphic mappings with values in a Riemann surface, and certain submersions arising from Killing fields and geodesic fields. The study of harmonic morphisms involves many different branches of mathematics: the book includes discussion on aspects of the theory of foliations, polynomials induced by Clifford systems and orthogonal multiplications, twistor and mini-twistor spaces, and Hermitian structures. Relations with topology are discussed, including Seifert fibre spaces and circle actions, also relations with isoparametric functions and the Beltrami fields equation of hydrodynamics.

This book presents the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles the basic concepts, theory, and methodologies. A major trend in modern mathematics, inspired largely by physics, is toward ‘noncommutative’ or ‘quantized’ phenomena. In functional analysis, this has appeared notably under the name of ‘operator spaces’, which is a variant of Banach spaces which is particularly appropriate for solving problems concerning spaces or algebras of operators on Hilbert space arising in ‘noncommutative mathematics’. The category of operator spaces includes operator algebras, selfadjoint (that is, C*-algebras) or otherwise. Also, most of the important modules over operator algebras are operator spaces. A common treatment of the subjects of C*-algebras, nonselfadjoint operator algebras, and modules over such algebras (such as Hilbert C*-modules), together under the umbrella of operator space theory, is the main topic of the book. A general theory of operator algebras and their modules naturally develops out of the operator space methodology. Indeed, operator space theory is a sensitive enough medium to reflect accurately many important noncommutative phenomena. Using recent advances in the field, the book shows how the underlying operator space structure captures, very precisely, the profound relations between the algebraic and the functional analytic structures involved. The rich interplay between spectral theory, operator theory, C*-algebra and von Neumann algebra techniques, and the influx of important ideas from related disciplines, such as pure algebra, Banach space theory, Banach algebras, and abstract function theory is highlighted. Each chapter ends with a section of notes containing additional information.

This book gives an account of those parts of the analysis of closed linear operators acting in Banach or Hilbert spaces that are relevant to spectral problems involving differential operators, and makes applications to such questions. After the exposition of the abstract theory in the first four chapters, Sobolev spaces are introduced and their main properties established. The remaining seven chapters are largely concerned with second-order elliptic differential operators and related boundary-value problems. Particular attention is paid to the spectrum of the Schrödinger operator. Its original form contains material of lasting importance that is relatively unaffected by advances in the theory since 1987, when the book was first published. The present edition differs from the old by virtue of the correction of minor errors and improvements of various proofs. In addition, it contains Notes at the ends of most chapters, intended to give the reader some idea of recent developments together with additional references that enable more detailed accounts to be accessed.

This book is devoted to a general study of geometric theories from a topos-theoretic perspective. After recalling the necessary topos-theoretic preliminaries, it presents the main methodology it uses to extract ‘concrete’ information on theories from properties of their classifying toposes—the ‘bridge’ technique. As a first implementation of this methodology, a duality is established between the subtoposes of the classifying topos of a geometric theory and the geometric theory extensions (also called ‘quotients’) of the theory. Many concepts of elementary topos theory which apply to the lattice of subtoposes of a given topos are then transferred via this duality into the context of geometric theories. A second very general implementation of the ‘bridge’ technique is the investigation of the class of theories of presheaf type (i.e. classified by a presheaf topos). After establishing a number of preliminary results on flat functors in relation to classifying toposes, the book carries out a systematic investigation of this class resulting in a number of general results and a characterization theorem allowing one to test whether a given theory is of presheaf type by considering its models in arbitrary Grothendieck toposes. Expansions of geometric theories and faithful interpretations of theories of presheaf type are also investigated. As geometric theories can always be written (in many ways) as quotients of presheaf type theories, the study of quotients of a given theory of presheaf type is undertaken. Lastly, the book presents a number of applications in different fields of mathematics of the theory it develops.

This book provides a comprehensive account of the theory of infinite soluble groups, from its foundations up to research level. Topics covered include: polycyclic groups, Cernikov groups, Mal’cev completions, soluble linear groups, P. Hall’s theory of finitely generated soluble groups, soluble groups with finite rank, soluble groups whose abelian subgroups satisfy finiteness conditions, simple modules over polycyclic groups, the Jategaonkar-Roseblade theorem, centrality in finitely generated soluble groups and the Lennox-Roseblade theorem, algorithmic problems for polycyclic and metabelian groups, cohomological topics including groups with finite (co)homological dimension and vanishing theorems, finitely presented soluble groups, constructible soluble groups, the Bieri-Strebel invariant, subnormality, and soluble groups.