Series: Graduate Texts in Mathematics, Vol. 275
1st ed. 2017, VIII, 362 p. 40 illus., 10 illus. in color.
Hardcover
ISBN 978-3-319-55082-4
* Narrative provides a panorma of some of the high points in the
history of differential geometry
* Problems are presented in each chapter with selected solutions and
hints given at the end of the book
* Accessible to graduate students of mathematics and physics
This text presents a graduate-level introduction to differential geometry for mathematics
and physics students. The exposition follows the historical development of the concepts
of connection and curvature with the goal of explaining the Chern*Weil theory of
characteristic classes on a principal bundle. Along the way we encounter some of
the high points in the history of differential geometry, for example, Gauss' Theorema
Egregium and the Gauss*Bonnet theorem. Exercises throughout the book test the
readerfs understanding of the material and sometimes illustrate extensions of the theory.
Initially, the prerequisites for the reader include a passing familiarity with manifolds. After
the first chapter, it becomes necessary to understand and manipulate differential forms. A
knowledge of de Rham cohomology is required for the last third of the text.
Prerequisite material is contained in author's text An Introduction to Manifolds, and
can be learned in one semester. For the benefit of the reader and to establish common
notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to
make the exposition more self-contained, sections on algebraic constructions such as the
tensor product and the exterior power are included.
Differential geometry, as its name implies, is the study of geometry using differential
calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not
until the nineteenth century, with the work of Gauss on surfaces and Riemann on the
curvature tensor, that differential geometry flourished and its modern foundation was
laid. Over the past one hundred years, differential geometry has proven indispensable
to an understanding of the physical world, in Einstein's general theory of relativity,
in the theory of gravitation, in gauge theory, and now in string theory. Differential
geometry is also useful in topology, several complex variables, algebraic geometry,
complex manifolds, and dynamical systems, among other fields. The field has even found
applications to group theory as in Gromov's work and to probability theory as in Diaconis's
work. It is not too far-fetched to argue that differential geometry should be in every
mathematician's arsenal.
Series: C.I.M.E. Foundation Subseries, Vol. 2179
1st ed. 2017, IX, 215 p. 21 illus., 1 illus. in color.
Softcover
ISBN 978-3-319-54513-4
* High level of lectures
* Written by experts
* With numerous figures
Collating different aspects of Vector-valued Partial Differential Equations and Applications,
this volume is based on the 2013 CIME Course with the same name which took place at
Cetraro, Italy, under the scientific direction of John Ball and Paolo Marcellini. It contains
the following contributions: The pullback equation (Bernard Dacorogna), The stability of the
isoperimetric inequality (Nicola Fusco), Mathematical problems in thin elastic sheets: scaling
limits, packing, crumpling and singularities (Stefan Muller), and Aspects of PDEs related
to fluid flows (Vladimir Sverak). These lectures are addressed to graduate students and
researchers in the field.
Series: Lecture Notes in Mathematics, Vol. 2182
1st ed. 2017, X, 441 p.
Softcover
ISBN 978-3-319-54360-4
* Detailed and complete real-variable theory of Musielak-Orlicz Hardy
type function spaces
* Detailed and self-contained arguments for the main results
* Presents some applications to endpoint or sharp problems of
analysis
* Detailed references and more known related results of each chapter
The main purpose of this book is to give a detailed and complete survey of recent progress
related to the real-variable theory of Musielak*Orlicz Hardy-type function spaces, and to lay
the foundations for further applications.
The real-variable theory of function spaces has always been at the core of harmonic analysis.
Recently, motivated by certain questions in analysis, some more general Musielak*Orlicz
Hardy-type function spaces were introduced. These spaces are defined via growth functions
which may vary in both the spatial variable and the growth variable. By selecting special
growth functions, the resulting spaces may have subtler and finer structures, which are
necessary in order to solve various endpoint or sharp problems.
This book is written for graduate students and researchers interested in function spaces and, in
particular, Hardy-type spaces.
Series: Frontiers in Mathematics
1st ed. 2017, Approx. 290 p.
Softcover
ISBN 978-3-319-50354-7
* Focuses on the mathematical interplay between Eulerian and
Lagrangian formulations and its significance for numerical methods
* Explains the structure of modern corner-based Lagrangian solvers
* Details the linear-quadratic structure of Lagrangian fluxes
originating from continuum mechanics
This book focuses on the interplay between Eulerian and Lagrangian conservation laws
for systems that admit physical motivation and originate from continuum mechanics.
Ultimately, it highlights what is specific to and beneficial in the Lagrangian approach
and its numerical methods. The two first chapters present a selection of well-known
features of conservation laws and prepare readers for the subsequent chapters, which are
dedicated to the analysis and discretization of Lagrangian systems.
The text is at the frontier of applied mathematics and scientific computing and appeals to
students and researchers interested in Lagrangian-based computational fluid dynamics.
It also serves as an introduction to the recent corner-based Lagrangian finite volume
techniques.
Series: Universitext
1st ed. 2017, VII, 273 p.
Softcover
ISBN 978-3-319-54373-4
* Written in the authorfs inimitable style
* Includes full background on topological groups and group quotients
* Contains many exercises and concrete examples
* Requires only basic knowledge of topology and algebra
This textbook covers the general theory of Lie groups. By first considering the case of
linear groups (following von Neumann's method) before proceeding to the general case,
the reader is naturally introduced to Lie theory.
Written by a master of the subject and influential member of the Bourbaki group, the
French edition of this textbook has been used by several generations of students. This
translation preserves the distinctive style and lively exposition of the original. Requiring
only basics of topology and algebra, this book offers an engaging introduction to Lie
groups for graduate students and a valuable resource for researchers.