Bücher der Reihe Springer Monographs in Mathematics

This book offers an up-to-date, comprehensive account of determinantal rings and varieties, presenting a multitude of methods used in their study, with tools from combinatorics, algebra, representation theory and geometry.After a concise introduction to Grobner and Sagbi bases, determinantal ideals are studied via the standard monomial theory and the straightening law. This opens the door for representation theoretic methods, such as the Robinson-Schensted-Knuth correspondence, which provide a description of the Grobner bases of determinantal ideals, yielding homological and enumerative theorems on determinantal rings. Sagbi bases then lead to the introduction of toric methods. In positive characteristic, the Frobenius functor is used to study properties of singularities, such as F-regularity and F-rationality. Castelnuovo-Mumford regularity, an important complexity measure in commutative algebra and algebraic geometry, is introduced in the general setting of a Noetherian base ring and then applied to powers and products of ideals. The remainder of the book focuses on algebraic geometry, where general vanishing results for the cohomology of line bundles on flag varieties are presented and used to obtain asymptotic values of the regularity of symbolic powers of determinantal ideals. In characteristic zero, the Borel-Weil-Bott theorem provides sharper results for GL-invariant ideals. The book concludes with a computation of cohomology with support in determinantal ideals and a survey of their free resolutions.Determinants, Grobner Bases and Cohomology provides a unique reference for the theory of determinantal ideals and varieties, as well as an introduction to the beautiful mathematics developed in their study. Accessible to graduate students with basic grounding in commutative algebra and algebraic geometry, it can be used alongside general texts to illustrate the theory with a particularly interesting and important class of varieties.

This monograph adopts an operational and functional analytic approach to the following problem: given a short exact sequence (group extension) 1 N G H 1 of finite groups, describe the irreducible representations of G by means of the structure of the group extension. This problem has attracted many mathematicians, including I. Schur, A.H. Clifford, and G. Mackey and, more recently, M. Isaacs, B. Huppert, Y.G. Berkovich & E.M. Zhmud, and J.M.G. Fell & R.S. Doran.The main topics are, on the one hand, Clifford Theory and the Little Group Method (of Mackey and Wigner) for induced representations, and, on the other hand, Kirillov's Orbit Method (for step-2 nilpotent groups of odd order) which establishes a natural and powerful correspondence between Lie rings and nilpotent groups. As an application, a detailed description is given of the representation theory of the alternating groups, of metacyclic, quaternionic, dihedral groups, and of the (finite) Heisenberg group. The Little Group Method may be applied if and only if a suitable unitary 2-cocycle (the Mackey obstruction) is trivial. To overcome this obstacle, (unitary) projective representations are introduced and corresponding Mackey and Clifford theories are developed. The commutant of an induced representation and the relative Hecke algebra is also examined. Finally, there is a comprehensive exposition of the theory of projective representations for finite Abelian groups which is applied to obtain a complete description of the irreducible representations of finite metabelian groups of odd order.

This monograph gives the reader an up-to-date account of the fine properties of real-valued functions and measures. The unifying theme of the book is the notion of nonmeasurability, from which one gets a full understanding of the structure of the subsets of the real line and the maps between them. The material covered in this book will be of interest to a wide audience of mathematicians, particularly to those working in the realm of real analysis, general topology, and probability theory. Set theorists interested in the foundations of real analysis will find a detailed discussion about the relationship between certain properties of the real numbers and the ZFC axioms, Martin's axiom, and the continuum hypothesis.

This book surveys the foundations of the theory of slice regular functions over the quaternions, introduced in 2006, and gives an overview of its generalizations and applications.As in the case of other interesting quaternionic function theories, the original motivations were the richness of the theory of holomorphic functions of one complex variable and the fact that quaternions form the only associative real division algebra with a finite dimension n>2. (Slice) regular functions quickly showed particularly appealing features and developed into a full-fledged theory, while finding applications to outstanding problems from other areas of mathematics. For instance, this class of functions includes polynomials and power series. The nature of the zero sets of regular functions is particularly interesting and strictly linked to an articulate algebraic structure, which allows several types of series expansion and the study of singularities. Integral representation formulas enrich the theory and are fundamental to the construction of a noncommutative functional calculus. Regular functions have a particularly nice differential topology and are useful tools for the construction and classification of quaternionic orthogonal complex structures, where they compensate for the scarcity of conformal maps in dimension four.This second, expanded edition additionally covers a new branch of the theory: the study of regular functions whose domains are not axially symmetric. The volume is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general.

This book furnishes a comprehensive treatment of differential graded Lie algebras, L-infinity algebras, and their use in deformation theory. We believe it is the first textbook devoted to this subject, although the first chapters are also covered in other sources with a different perspective.Deformation theory is an important subject in algebra and algebraic geometry, with an origin that dates back to Kodaira, Spencer, Kuranishi, Gerstenhaber, and Grothendieck. In the last 30 years, a new approach, based on ideas from rational homotopy theory, has made it possible not only to solve long-standing open problems, but also to clarify the general theory and to relate apparently different features. This approach works over a field of characteristic 0, and the central role is played by the notions of differential graded Lie algebra,  L-infinity algebra, and Maurer-Cartan equations.The book is written keeping in mind graduate students with a basic knowledge of homological algebra and complex algebraic geometry as utilized, for instance, in the book by K. Kodaira, Complex Manifolds and Deformation of Complex Structures. Although the main applications in this book concern deformation theory of complex manifolds, vector bundles, and holomorphic maps, the underlying algebraic theory also applies to a wider class of deformation problems, and it is a prerequisite for anyone interested in derived deformation theory. Researchers in algebra, algebraic geometry, algebraic topology, deformation theory,  and noncommutative geometry are the major targets for the book. 

This book is an easy-to-read reference providing a link between functional analysis and diffusion processes. More precisely, the book takes readers to a mathematical crossroads of functional analysis (macroscopic approach), partial differential equations (mesoscopic approach), and probability (microscopic approach) via the mathematics needed for the hard parts of diffusion processes. This work brings these three fields of analysis together and provides a profound stochastic insight (microscopic approach) into the study of elliptic boundary value problems.The author does a massive study of diffusion processes from a broad perspective and explains mathematical matters in a more easily readable way than one usually would find. The book is amply illustrated; 14 tables and 141 figures are provided with appropriate captions in such a fashion that readers can easily understand powerful techniques of functional analysis for the study of diffusion processes in probability.The scope of the author's work has been and continues to be powerful methods of functional analysis for future research of elliptic boundary value problems and Markov processes via semigroups. A broad spectrum of readers can appreciate easily and effectively the stochastic intuition that this book conveys.  Furthermore, the book will serve as a sound basis both for researchers and for graduate students in pure and applied mathematics who are interested in a modern version of the classical potential theory and Markov processes.For advanced undergraduates working in functional analysis, partial differential equations, and probability, it provides an effective opening to these three interrelated fields of analysis. Beginning graduate students and mathematicians in the field looking for a coherent overview will find the book to be a helpful beginning. This work will be a major influence in a very broad field of study for a long time.

This monograph is designed to be an in-depth introduction to domination in graphs. It focuses on three core concepts: domination, total domination, and independent domination. It contains major results on these foundational domination numbers, including a wide variety of in-depth proofs of selected results providing the reader with a toolbox of proof techniques used in domination theory. Additionally, the book is intended as an invaluable reference resource for a variety of readerships, namely, established researchers in the field of domination who want an updated, comprehensive coverage of domination theory; next, researchers in graph theory who wish to become acquainted with newer topics in domination, along with major developments in the field and some of the proof techniques used; and, graduate students with interests in graph theory, who might find the theory and many real-world applications of domination of interest for masters and doctoral thesis topics. The focusedcoverage also provides a good basis for seminars in domination theory or domination algorithms and complexity.The authors set out to provide the community with an updated and comprehensive treatment on the major topics in domination in graphs. And by Jove, they¿ve done it! In recent years, the authors have curated and published two contributed volumes: Topics in Domination in Graphs, © 2020 and Structures of Domination in Graphs, © 2021. This book rounds out the coverage entirely. The reader is assumed to be acquainted with the basic concepts of graph theory and has had some exposure to graph theory at an introductory level. As graph theory terminology sometimes varies, a glossary of terms and notation is provided at the end of the book.

This book introduces and discusses the self-adjoint extension problem for symmetric operators on Hilbert space. It presents the classical von Neumann and Krein¿Vishik¿Birman extension schemes both in their modern form and from a historical perspective, and provides a detailed analysis of a range of applications beyond the standard pedagogical examples (the latter are indexed in a final appendix for the reader¿s convenience).Self-adjointness of operators on Hilbert space representing quantum observables, in particular quantum Hamiltonians, is required to ensure real-valued energy levels, unitary evolution and, more generally, a self-consistent theory. Physical heuristics often produce candidate Hamiltonians that are only symmetric: their extension to suitably larger domains of self-adjointness, when possible, amounts to declaring additional physical states the operator must act on in order to have a consistent physics, and distinct self-adjoint extensions describe different physics.Realising observables self-adjointly is the first fundamental problem of quantum-mechanical modelling.The discussed applications concern models of topical relevance in modern mathematical physics currently receiving new or renewed interest, in particular from the point of view of classifying self-adjoint realisations of certain Hamiltonians and studying their spectral and scattering properties. The analysis also addresses intermediate technical questions such as characterising the corresponding operator closures and adjoints. Applications include hydrogenoid Hamiltonians, Dirac¿Coulomb Hamiltonians, models of geometric quantum confinement and transmission on degenerate Riemannian manifolds of Grushin type, and models of few-body quantum particles with zero-range interaction.Graduate students and non-expert readers will benefit from a preliminary mathematical chapter collecting all the necessary pre-requisites on symmetric and self-adjoint operators on Hilbert space (including the spectral theorem), and from a further appendix presenting the emergence from physical principles of the requirement of self-adjointness for observables in quantum mechanics.

The contents of this book was created by the authors as a simultaneous generalization of Witten zeta-functions, Mordell¿Tornheim multiple zeta-functions, and Euler¿Zagier multiple zeta-functions. Zeta-functions of root systems are defined by certain multiple series, given in terms of root systems. Therefore, they intrinsically have the action of associated Weyl groups. The exposition begins with a brief introduction to the theory of Lie algebras and root systems and then provides the definition of zeta-functions of root systems, explicit examples associated with various simple Lie algebras, meromorphic continuation and recursive analytic structure described by Dynkin diagrams, special values at integer points, functional relations, and the background given by the action of Weyl groups. In particular, an explicit form of Witten¿s volume formula is provided. It is shown that various relations among special values of Euler¿Zagier multiple zeta-functions¿which usually are called multiple zeta values (MZVs) and are quite important in connection with Zagier¿s conjecture¿are just special cases of various functional relations among zeta-functions of root systems. The authors further provide other applications to the theory of MZVs and also introduce generalizations with Dirichlet characters, and with certain congruence conditions. The book concludes with a brief description of other relevant topics.

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.

This book provides a comprehensive introduction to the Calculus of Variations and its use in modelling mechanics and physics problems. Presenting a geometric approach to the subject, it progressively guides the reader through this very active branch of mathematics, accompanying key statements with a huge variety of exercises, some of them solved. Stressing the need to overcome limitations of the initial point of view, and emphasising the interconnectivity of various branches of mathematics (algebra, analysis and geometry), the book includes some advanced material to challenge the most motivated students. Systematic, short historical notes provide details on the subject's odyssey, and how new tools have been developed over the last two centuries. This English translation updates a set of notes for a course first given at the Ecole polytechnique in 1987. It will be accessible to graduate students and advanced undergraduates.

The main body of this book consists of 106 numbered theorems and a dozen of examples of models of set theory. A large number of additional results is given in the exercises, which are scattered throughout the text. Most exer- cises are provided with an outline of proof in square brackets [ ], and the more difficult ones are indicated by an asterisk. I am greatly indebted to all those mathematicians, too numerous to men- tion by name, who in their letters, preprints, handwritten notes, lectures, seminars, and many conversations over the past decade shared with me their insight into this exciting subject. XI CONTENTS Preface xi PART I SETS Chapter 1 AXIOMATIC SET THEORY I. Axioms of Set Theory I 2. Ordinal Numbers 12 3. Cardinal Numbers 22 4. Real Numbers 29 5. The Axiom of Choice 38 6. Cardinal Arithmetic 42 7. Filters and Ideals. Closed Unbounded Sets 52 8. Singular Cardinals 61 9. The Axiom of Regularity 70 Appendix: Bernays-Godel Axiomatic Set Theory 76 Chapter 2 TRANSITIVE MODELS OF SET THEORY 10. Models of Set Theory 78 II. Transitive Models of ZF 87 12. Constructible Sets 99 13. Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis 108 14. The In Hierarchy of Classes, Relations, and Functions 114 15. Relative Constructibility and Ordinal Definability 126 PART II MORE SETS Chapter 3 FORCING AND GENERIC MODELS 16. Generic Models 137 17. Complete Boolean Algebras 144 18.

Author's Note: The material of this book has been reworked and expanded with a lot more detail and published in the author's 2014 book "Upper and Lower Bounds for Stochastic Processes" (Ergebnisse Vol. 60, ISBN 978-3-642-54074-5). That book is much easier to read and covers everything that is in "The Generic Chaining" book in a more detailed and comprehensible way. ************What is the maximum level a certain river is likely to reach over the next 25 years? (Having experienced three times a few feet of water in my house, I feel a keen personal interest in this question. ) There are many questions of the same nature: what is the likely magnitude of the strongest earthquake to occur during the life of a planned building, or the speed of the strongest wind a suspension bridge will have to stand? All these situations can be modeled in the same manner. The value X of the quantity of interest (be it water t level or speed of wind) at time t is a random variable. What can be said about the maximum value of X over a certain range of t? t A collection of random variables (X ), where t belongs to a certain index t set T, is called a stochastic process, and the topic of this book is the study of the supremum of certain stochastic processes, and more precisely to ?nd upper and lower bounds for the quantity EsupX . (0. 1) t t?T Since T might be uncountable, some care has to be taken to de?ne this quantity. For any reasonable de?nition of Esup X we have t t?T EsupX =sup{EsupX ; F?T,F ?nite} , (0. 2) t t t?T t?F an equality that we will take as the de?nition of the quantity Esup X . t t?T Thus, the crucial case for the estimation of the quantity (0.