Lectures on Morse Homology, by Augustin Banyaga and David Hurtubise
This book presents in great detail all the results one needs to prove the Morse Homology Theorem using classical techniques from algebraic topology and homotopy theory. Most of these results can be found scattered throughout the literature dating from the mid to late 1900’s in some form or other, but often the results are proved in different contexts with a multitude of different notations and different goals. This book collects all these results together into a single reference with complete and detailed proofs. The core material in this book includes CW-complexes, Morse theory, hyperbolic dynamical systems (the Lamba-Lemma, the Stable/Unstable Manifold Theorem), transversality theory, the Morse-Smale-Witten boundary operator, and Conley index theory. More advanced topics include Morse theory on Grassmann manifolds and Lie groups, and an overview of Floer homology theories. With the stress on completeness and by its elementary approach to Morse homology, this book is suitable as a textbook for a graduate level course, or as a reference for working mathematicians and physicists.
Dimension theory of general spaces, by A. R. Pears
A complete and self-contained account of the dimension theory of general topological spaces, with particular emphasis on the dimensional properties of non-metrizable spaces. It makes the subject accessible to beginning graduate students and will also serve as a reference work for general topologists.
Dimension Theory, by Witold Hurewicz and Henry Wallman
The standard treatise on classical dimension theory. Dimension theory is that area of topology concerned with giving a precise mathematical meaning to the concept of the dimension of a space. An active area of reasearch in the early 20th century, but one that has fallen into disuse in topology, dimension theory has experienced a revitalization due to connections with fractals and dynamical systems, but none of those developments are in this 1948 book. Instead,this book is primarily used as a reference today for its proof of Brouwer’s Theorem on the Invariance of Domain.
Homotopy theoretic methods in Group Cohomology, by William G. Dwyer and Hans-Werner Henn
This 89-pages-book looks at group cohomology with tools that come from homotopy theory. These tools give both decomposition theorems (which rely on homotopy colimits to obtain a description of the cohomology of a group in terms of the cohomology of suitable subgroups) and global structure theorems (which exploit the action of the ring of topological cohomology operations). The approach is expository and thus suitable for graduate students and others who would like an introduction to the subject that organizes and adds to the relevant literature and leads to the frontier of current research. The book should also be interesting to anyone who wishes to learn some of the machinery of homotopy theory (simplicial sets, homotopy colimits, Lannes’ T-functor, the theory of unstable modules over the Steenrod algebra) by seeing how it is used in a practical setting.
A compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab, by William E. Schiesser and Graham W. Griffiths
A Compendium of Partial Differential Equation Models presents numerical methods and associated computer codes in Matlab for the solution of a spectrum of models expressed as partial differential equations (PDEs), one of the mostly widely used forms of mathematics in science and engineering. The authors focus on the method of lines (MOL), a well-established numerical procedure for all major classes of PDEs in which the boundary value partial derivatives are approximated algebraically by finite differences. This reduces the PDEs to ordinary differential equations (ODEs) and thus makes the computer code easy to understand, implement, and modify.
Pseudodifferential operators and nonlinear PDEs, by Michael Taylor
For the past 25 years the theory of pseudodifferential operators has played an important role in many exciting and deep investigations into linear PDE. Over the past decade, this tool has also begun to yield interesting results in nonlinear PDE. This book is devoted to a summary and reconsideration of some uses of pseudodifferential operator techniques in nonlinear PDE.
A history of Non-Euclidean Geometry, by Boris A. Rosenfeld
This book is an investigation of the mathematical and philosophical factors underlying the discovery of the concept of noneuclidean geometries, and the subsequent extension of the concept of space. Chapters one through five are devoted to the evolution of the concept of space, leading up to chapter six which describes the discovery of noneuclidean geometry, and the corresponding broadening of the concept of space. The author goes on to discuss concepts such as multidimensional spaces and curvature, and transformation groups. The book ends with a chapter describing the applications of nonassociative algebras to geometry.
Topology of 4-manifolds, by Michael H. Freedman and Frank Quinn
This book introduces the reader to a fascinating branch of topology and has the clearest proof of the 4-dimensional Poincare conjecture. In addition, the authors do not hesitate to employ diagrams as needed to illustrate the main points and to assist the reader in visualizing 4-dimensional objects. The authors give a fine discussion as to the reasons why four dimensions is harder to deal with topologically than dimensions five or greater, this being essentially due to the behavior of 2-dimensional disks: mapping 2-disks into 3-manifolds results (generically) with 1-dimensional self-intersections; in 4-dimensions the intersections are isolated points, and in 5 dimensions or more the 2-disks can be embedded.
Tensors in Image Processing and Computer Vision, by Santiago Aja-Fernández, Rodrigo de Luis García, Dacheng Tao and Xuelong Li
Tensor signal processing is an emerging field with important applications to computer vision and image processing. This book presents the state of the art in this new branch of signal processing, offering a great deal of research and discussions by leading experts in the area.
Perspectives on the history of Mathematical Logic, by Thomas Drucker
This volume offers insights into the development of mathematical logic over the last century. Arising from a special session of the history of logic at an American Mathematical Society meeting, the chapters explore technical innovations, the philosophical consequences of work during the period, and the historical and social context in which the logicians worked.
The concentration-compactness principle in the Calculus of Variations, by Pierre-Louis Lions
The author of this paper was the first to give a complete solution with proof to the Boltzmann equation. He was awarded the Fields Medal in 1994.
The geometry of Topological Stability, by Andrew du Plessis and Terry Wall
In presenting a detailed study of the geometry and topology of numerous classes of “generic” singularities, Geometry of Topological Stability bridges the gap between algebraic calculations and continuity arguments to detail the necessary and sufficient conditions for a C (infinity) to be C0-stable. Throughout, the authors masterfully examine this important subject using results culled from a broad range of mathematical disciplines, including geometric topology, stratification theory, algebraic geometry, and commutative algebra. Packed with original research, much of which is presented here for the first time, the book will be welcomed by students and researchers interested in singularity theory and related areas.
Spectra of graphs, by Andries E. Brouwer and Willem H. Haemers
An introduction to the interesting subject of graph spectra: the analysis of the relation between the properties of a (maybe directed, with loops and multiedges) graph and the eigendecomposition of its adjacency matrix.
And in Physics:
Bose-Condensed gases at finite temperatures, by Allan Griffin, Tetsuro Nikuni and Eugene Zaremba
The discovery of Bose Einstein condensation (BEC) in trapped ultracold atomic gases in 1995 has led to an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first treatment of BEC at finite temperatures, this book presents a thorough account of the theory of two-component dynamics and nonequilibrium behaviour in superfluid Bose gases.
Brownian movement and molecular reality, by Jean Perrin
Early studies by Einstein and Perrin provided some of the first evidence for the existence of molecules. Perrin, a Nobel Laureate, wrote this classic to explain his measurements of displaced particles of a resin suspended in water. It introduced the concept of Avogadro’s number, along with other groundbreaking work. 1910 edition.
Contextual approach to Quantum Formalism, by Andrei Khrennikov
The aim of this book is to show that the probabilistic formalisms of classical statistical mechanics and quantum mechanics can be unified on the basis of a general contextual probabilistic model. By taking into account the dependence of (classical) probabilities on contexts (i.e. complexes of physical conditions), one can reproduce all distinct features of quantum probabilities such as the interference of probabilities and the violation of Bell’s inequality. Moreover, by starting with a formula for the interference of probabilities (which generalizes the well known classical formula of total probability), one can construct the representation of contextual probabilities by complex probability amplitudes or, in the abstract formalism, by normalized vectors of the complex Hilbert space or its hyperbolic generalization. Thus the Hilbert space representation of probabilities can be naturally derived from classical probabilistic assumptions.
Many-Body problems and Quantum Field Theory: An introduction, by Philippe A. Martin and Francois Rothen
The book gives an introduction to the concepts and methods of many-body problems and quantum fields for graduate students and researchers. The formalism is developed in close conjunction with the description of a number of physical systems: cohesion and dielectric properties of the electron gas, superconductivity, superfluidity, nuclear matter and nucleon pairing, matter and radiation, interaction of fields by particle exchange and mass generation. Emphasis is put on analogies between the various systems.
Introduction to Statistical Physics, by Silvio Salinas
A text for graduate-level students, covering the statistical basis of thermodynamics, with examples from solid-state physics. Reviews statistical methods and classical thermodynamics, moving into statistical mechanics, with discussion of quantum statistical mechanics. Includes coverage of blackbody radiation, phonons, and magnons.
Electronic properties of materials, by Rolf E. Hummel
This book on electrical, optical, magnetic and thermal properties of materials differs from other introductory texts in solid state physics. First, it is written for engineers, particularly materials and electrical engineers, who want to gain a fundamental understanding of semiconductor devices, magnetic materials, lasers, alloys, and so forth. Second, it stresses concepts rather than mathematical formalism. Third, it is not an encyclopedia: The topics are restricted to material considered to be essential and which can be covered in one 15-week semester. The book is divided into five parts. The first part, “Fundamentals of Electron Theory,” introduces the essential quantum mechanical concepts needed for understanding materials science; the other parts may be read independently of each other.
Complex Variables: A physical approach with applications and MATLAB tutorials, by Steven G. Krantz
From the algebraic properties of a complete number field, to the analytic properties imposed by the Cauchy integral formula, to the geometric qualities originating from conformality, Complex Variables: A Physical Approach with Applications and MATLAB explores all facets of this subject, with particular emphasis on using theory in practice. The first five chapters encompass the core material of the book. These chapters cover fundamental concepts, holomorphic and harmonic functions, Cauchy theory and its applications, and isolated singularities. Subsequent chapters discuss the argument principle, geometric theory, and conformal mapping, followed by a more advanced discussion of harmonic functions. The author also presents a detailed glimpse of how complex variables are used in the real world, with chapters on Fourier and Laplace transforms as well as partial differential equations and boundary value problems. The final chapter explores computer tools
Quantum Field Theory: A tourist guide for mathematicians, by Gerald B. Folland
Quantum field theory has been a great success for physics, but it is difficult for mathematicians to learn because it is mathematically incomplete. Folland, who is a mathematician, has spent considerable time digesting the physical theory and sorting out the mathematical issues in it. Fortunately for mathematicians, Folland is a gifted expositor. The purpose of this book is to present the elements of quantum field theory, with the goal of understanding the behavior of elementary particles rather than building formal mathematical structures, in a form that will be comprehensible to mathematicians. Rigorous definitions and arguments are presented as far as they are available, but the text proceeds on a more informal level when necessary, with due care in identifying the difficulties. The book begins with a review of classical physics and quantum mechanics, then proceeds through the construction of free quantum fields to the perturbation-theoretic development of interacting field theory and renormalization theory, with emphasis on quantum electrodynamics. The final two chapters present the functional integral approach and the elements of gauge field theory, including the Salam-Weinberg model of electromagnetic and weak interactions.
Musimathics, volume 1: The mathematical foundations of Music, by Gareth Loy
In this volume, Loy presents the materials of music (notes, intervals, and scales); the physical properties of music (frequency, amplitude, duration, and timbre); the perception of music and sound (how we hear); and music composition. Musimathics is carefully structured so that new topics depend strictly on topics already presented, carrying the reader progressively from basic subjects to more advanced ones.
Path integral methods, by T. Kashiwa, Y. Ohnuki and M. Suzuki
Path integrals are useful in quantum as well as statistical mechanics, providing more intuitive and easier methods of handling equations than other treatments. This book addresses fundamental issues in the field and covers more recent topics, such as the lattice fermion problem in quantum theory and the Monte Carlo method in statistical mechanics.
Quantum Mechanics, by Ernest S. Abers
A comprehensive book that intends to cover all subjects of Quantum Mechanics with a modern, quite rigorous mathematical exposition.
Quantum: The Quantum Theory of Particles, Fields and Cosmology, by Edgar Elbaz
This book gives a new insight into the interpretation of quantum mechanics (stochastic, integral paths, decoherence), a completely new treatment of angular momentum (graphical spin algebra) and an introduction to Fermion fields (Dirac equation) and Boson fields (e.m. and Higgs) as well as an introduction to QED (quantum electrodynamics), supersymmetry and quantum cosmology.
Special Relativity and Quantum Mechanics, by Francis Halpern
Special Relativity and Quantum Mechanics in just 138 pages.