Mathematical Surveys and Monographs, Volume: 187
2013; approx. 310 pp; hardcover
ISBN-13: 978-0-8218-9152-0
Expected publication date is March 31, 2013.
The book describes how functional inequalities are often manifestations of natural mathematical structures and physical phenomena, and how a few general principles validate large classes of analytic/geometric inequalities, old and new. This point of view leads to "systematic" approaches for proving the most basic inequalities, but also for improving them, and for devising new ones--sometimes at will--and often on demand. These general principles also offer novel ways for estimating best constants and for deciding whether these are attained in appropriate function spaces.
As such, improvements of Hardy and Hardy-Rellich type inequalities involving radially symmetric weights are variational manifestations of Sturm's theory on the oscillatory behavior of certain ordinary differential equations. On the other hand, most geometric inequalities, including those of Sobolev and Log-Sobolev type, are simply expressions of the convexity of certain free energy functionals along the geodesics on the Wasserstein manifold of probability measures equipped with the optimal mass transport metric. Caffarelli-Kohn-Nirenberg and Hardy-Rellich-Sobolev type inequalities are then obtained by interpolating the above two classes of inequalities via the classical ones of Holder. The subtle Moser-Onofri-Aubin inequalities on the two-dimensional sphere are connected to Liouville type theorems for planar mean field equations.
Graduate students and research mathematicians interested in analysis, calculus of variations, and PDEs.
Hardy type inequalities
Bessel pairs and Sturm's oscillation theory
The classical Hardy inequality and its improvements
Improved Hardy inequality with boundary singularity
Weighted Hardy inequalities
The Hardy inequality and second order nonlinear eigenvalue problems
Hardy-Rellich type inequalities
Improved Hardy-Rellich inequalities on H^2_0(Omega)
Weighted Hardy-Rellich inequalities on H^2(Omega)cap H^1_0(Omega)
Critical dimensions for 4^{textrm{th}} order nonlinear eigenvalue problems
Hardy inequalities for general elliptic operators
General Hardy inequalities
Improved Hardy inequalities for general elliptic operators
Regularity and stability of solutions in non-self-adjoint problems
Mass transport and optimal geometric inequalities
A general comparison principle for interacting gases
Optimal Euclidean Sobolev inequalities
Geometric inequalities
Hardy-Rellich-Sobolev inequalities
The Hardy-Sobolev inequalities
Domain curvature and best constants in the Hardy-Sobolev inequalities
Aubin-Moser-Onofri inequalities
Log-Sobolev inequalities on the real line
Trudinger-Moser-Onofri inequality on mathbb{S}^2
Optimal Aubin-Moser-Onofri inequality on mathbb{S}^2
Bibliography
2013; approx. 262 pp; softcover
ISBN-13: 978-0-8218-9492-7
Expected publication date is April 18, 2013.
There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and nonrigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such "folklore mathematics". But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog.
The articles, essays, and notes in this book are derived from the author's mathematical blog in 2010. It contains a broad selection of mathematical expositions, commentary, and self-contained technical notes in many areas of mathematics, such as logic, group theory, analysis, and partial differential equations. The topics range from the foundations of mathematics to discussions of recent mathematical breakthroughs.
Lecture notes from the author's courses that appeared on the blog have been published separately in the Graduate Studies in Mathematics series.
Graduate students and research mathematicians interested in analysis, logic and foundations, PDEs, algebra, and general topics related to mathematics.
Logic and foundations
Group theory
Analysis
Nonstandard analysis
Partial differential equations
Miscellaneous
Bibliography
Index
University Lecture Series, Volume: 61
2013; 149 pp; softcover
ISBN-13: 978-0-8218-9434-7
Expected publication date is April 20, 2013.
The theory of motives was created by Grothendieck in the 1960s as he searched for a universal cohomology theory for algebraic varieties. The theory of pure motives is well established as far as the construction is concerned. Pure motives are expected to have a number of additional properties predicted by Grothendieck's standard conjectures, but these conjectures remain wide open. The theory for mixed motives is still incomplete.
This book deals primarily with the theory of pure motives. The exposition begins with the fundamentals: Grothendieck's construction of the category of pure motives and examples. Next, the standard conjectures and the famous theorem of Jannsen on the category of the numerical motives are discussed. Following this, the important theory of finite dimensionality is covered. The concept of Chow-Kunneth decomposition is introduced, with discussion of the known results and the related conjectures, in particular the conjectures of Bloch-Beilinson type. We finish with a chapter on relative motives and a chapter giving a short introduction to Voevodsky's theory of mixed motives.
Graduate students and research mathematicians interested in algebraic cycles and motives.
Algebraic cycles and equivalence relations
Survey of some of the main results on Chow groups
Proof of the theorem of Voisin-Voevodsky
Motives: Construction and first properties
On Grothendieck's standard conjectures
Finite dimensionality of motives
Properties of finite dimensional motives
Chow-Kunneth decomposition; The Picard and Albanese motive
Chow-Kunneth decomposition in a special case
On the conjectural Bloch-Beilinson filtration
Relative Chow-Kunneth decomposition
Surfaces fibered over a curve
Beyond pure motives
The category of motivic complexes
Bibliography
Index of notation
Index
Pure and Applied Undergraduate Texts, Volume: 21
2013; approx. 474 pp; hardcover
ISBN-13: 978-0-8218-8478-2
Expected publication date is May 11, 2013.
The story of geometry is the story of mathematics itself: Euclidean geometry was the first branch of mathematics to be systematically studied and placed on a firm logical foundation, and it is the prototype for the axiomatic method that lies at the foundation of modern mathematics. It has been taught to students for more than two millennia as a model of logical thought.
This book tells the story of how the axiomatic method has progressed from Euclid's time to ours, as a way of understanding what mathematics is, how we read and evaluate mathematical arguments, and why mathematics has achieved the level of certainty it has. It is designed primarily for advanced undergraduates who plan to teach secondary school geometry, but it should also provide something of interest to anyone who wishes to understand geometry and the axiomatic method better. It introduces a modern, rigorous, axiomatic treatment of Euclidean and (to a lesser extent) non-Euclidean geometries, offering students ample opportunities to practice reading and writing proofs while at the same time developing most of the concrete geometric relationships that secondary teachers will need to know in the classroom.
Undergraduate students interested in geometry and secondary mathematics teaching.
Euclid
Incidence geometry
Axioms for plane geometry
Angles
Triangles
Models of neutral geometry
Perpendicular and parallel lines
Polygons
Quadrilaterals
The Euclidean parallel postulate
Area
Similarity
Right triangles
Circles
Circumference and circular area
Compass and straightedge constructions
The parallel postulate revisited
Introduction to hyperbolic geometry
Parallel lines in hyperbolic geometry
Epilogue: Where do we go from here?
Hilbert's axioms
Birkhoff's postulates
The SMSG postulates
The postulates used in this book
The language of mathematics
Proofs
Sets and functions
Properties of the real numbers
Rigid motions: Another approach
References
Index
Graduate Studies in Mathematics, Volume: 145
2013; approx. 642 pp; hardcover
ISBN-13: 978-0-8218-9132-2
Expected publication date is May 18, 2013.
Informally, K-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebraic and geometric questions. Algebraic K-theory, which is the main character of this book, deals mainly with studying the structure of rings. However, it turns out that even working in a purely algebraic context, one requires techniques from homotopy theory to construct the higher K-groups and to perform computations. The resulting interplay of algebra, geometry, and topology in K-theory provides a fascinating glimpse of the unity of mathematics.
This book is a comprehensive introduction to the subject of algebraic K-theory. It blends classical algebraic techniques for K0 and K1 with newer topological techniques for higher K-theory such as homotopy theory, spectra, and cohomological descent. The book takes the reader from the basics of the subject to the state of the art, including the calculation of the higher K-theory of number fields and the relation to the Riemann zeta function.
Graduate students and research mathematicians interested in number theory, homological algebra, and K-theory.
Projective modules and vector bundles
The Grothendieck group K0
K1 and K2 of a ring
Definitions of higher K-theory
The fundamental theorems of higher K-theory
The higher K-theory of fields
Nomenclature
Bibliography
Index
Mathematical Surveys and Monographs, Volume: 188
2013; approx. 328 pp; hardcover
ISBN-13: 978-0-8218-9470-5
Expected publication date is May 10, 2013.
This monograph provides a comprehensive and self-contained study on the theory of water waves equations, a research area that has been very active in recent years. The vast literature devoted to the study of water waves offers numerous asymptotic models. Which model provides the best description of waves such as tsunamis or tidal waves? How can water waves equations be transformed into simpler asymptotic models for applications in, for example, coastal oceanography? This book proposes a simple and robust framework for studying these questions.
The book should be of interest to graduate students and researchers looking for an introduction to water waves equations or for simple asymptotic models to describe the propagation of waves. Researchers working on the mathematical analysis of nonlinear dispersive equations may also find inspiration in the many (and sometimes new) models derived here, as well as precise information on their physical relevance.
Graduate students and research mathematicians interested in nonlinear PDEs and applications to oceanography.
The water waves equations and its asymptotic regimes
The Laplace equation
The Dirichlet-Neumann operator
Well-posedness of the water waves equations
Shallow water asymptotics: Systems. Part 1: Derivation
Shallow water asymptotics: Systems. Part 2: Justification
Shallow water asymptotics: Scalar equations
Deep water models and modulation equations
Water waves with surface tension
Appendix A. More on the Dirichlet-Neumann operator
Appendix B. Product and commutator estimates
Appendix C. Asymptotic models: A reader's digest
Bibliography