Series: Universitext
* Starting from the basics, provides a coherent and up-to-date
account of various quantitative measures of symmetry arising in
convex geometry
* Contains over 90 challenging problems, with hints and references
to aid the reader
* Introduces a new sequence of mean Minkowski measures associated
to a convex body
This textbook treats two important and related matters in convex geometry: the
quantification of symmetry of a convex set*measures of symmetry*and the degree
to which convex sets that nearly minimize such measures of symmetry are themselves
nearly symmetric*the phenomenon of stability. By gathering the subject’s core ideas
and highlights around Grunbaum’s general notion of measure of symmetry, it paints a
coherent picture of the subject, and guides the reader from the basics to the state-of-theart.
The exposition takes various paths to results in order to develop the reader’s grasp
of the unity of ideas, while interspersed remarks enrich the material with a behind-thescenes
view of corollaries and logical connections, alternative proofs, and allied results
from the literature. Numerous illustrations elucidate definitions and key constructions,
and over 70 exercises*with hints and references for the more difficult ones*test and
sharpen the reader’s comprehension.
The presentation includes: a basic course covering foundational notions in convex
geometry, the three pillars of the combinatorial theory (the theorems of Caratheodory,
Radon, and Helly), critical sets and Minkowski measure, the Minkowski*Radon inequality,
and, to illustrate the general theory, a study of convex bodies of constant width; two
proofs of F. John’s ellipsoid theorem; a treatment of the stability of Minkowski measure,
the Banach*Mazur metric, and Groemer’s stability estimate for the Brunn*Minkowski
inequality; important specializations of Grunbaum’s abstract measure of symmetry,
such as Winternitz measure, the Rogers*Shepard volume ratio, and Guo’s Lp -Minkowski
measure; a construction by the author of a new sequence of measures of symmetry, the
kth mean Minkowski measure; and lastly, an intriguing application to the moduli space of
certain distinguished maps from a Riemannian homogeneous space to
spheres*illustrating the broad mathematical relevance of the book’s subject.
1st ed. 2015, XV, 305 p. 64 illus., 1 illus. in
Softcover
ISBN 978-3-319-23732-9
Series: Oberwolfach Seminars, Vol. 46
* First book dealing exclusively with this topic which has hitherto only
been treated in original research papers
* Develops relevant background and explains the ideas involved
* Short, concise text with topics ranging from classical results right up
to the most recent developments
* Suitable for graduate students with an interest in Riemannian
geometry
This book studies certain spaces of Riemannian metrics on both compact and noncompact
manifolds. These spaces are defined by various sign-based curvature conditions,
with special attention paid to positive scalar curvature and non-negative sectional
curvature, though we also consider positive Ricci and non-positive sectional curvature. If
we form the quotient of such a space of metrics under the action of the diffeomorphism
group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of
both the original space of metrics and the corresponding moduli space form the central
theme of this book. For example, what can be said about the connectedness or the various
homotopy groups of such spaces* We explore the major results in the area, but provide
sufficient background so that a non-expert with a grounding in Riemannian geometry can
access this growing area of research.
1st ed. 2015, X, 120 p. 3 illus.
Softcover
ISBN 978-3-0348-0947-4
Series: Springer Undergraduate Mathematics Series
* Introduces key ideas in real and complex analysis in the manner they
were discovered
* Motivates and explains many related developments in neighbouring
fields, including number theory, elliptic function theory, and potential
theory
* Displays many examples illustrating the merits of rigorous analysis
This book contains a history of real and complex analysis in the nineteenth century, from
the work of Lagrange and Fourier to the origins of set theory and the modern foundations
of analysis. It studies the works of many contributors including Gauss, Cauchy, Riemann,
and Weierstrass.
This book is unique owing to the treatment of real and complex analysis as overlapping,
inter-related subjects, in keeping with how they were seen at the time. It is suitable as
a course in the history of mathematics for students who have studied an introductory
course in analysis, and will enrich any course in undergraduate real or complex analysis.
1st ed. 2015, XVIII, 320 p. 71 illus.
Softcover
ISBN 978-3-319-23714-5
Series: Developments in Mathematics, Vol. 43
* A self-contained introduction to symmetric functions and their use in
counting problems
* First book to consider many of the methods and results presented
* Unifies a large number of results in the theory of permutation enumeration
* Numerous exercises with full solutions included throughout
This monograph provides a self-contained introduction to symmetric functions and their
use in enumerative combinatorics. It is the first book to explore many of the methods and
results that the authors present. Numerous exercises are included throughout, along with
full solutions, to illustrate concepts and also highlight many interesting mathematical
ideas.
The text begins by introducing fundamental combinatorial objects such as permutations
and integer partitions, as well as generating functions. Symmetric functions are
considered in the next chapter, with a unique emphasis on the combinatorics of
the transition matrices between bases of symmetric functions. Chapter 3 uses this
introductory material to describe how to find an assortment of generating functions
for permutation statistics, and then these techniques are extended to find generating
functions for a variety of objects in Chapter 4. The next two chapters present the
Robinson-Schensted-Knuth algorithm and a method for proving Polya’s enumeration
theorem using symmetric functions. Chapters 7 and 8 are more specialized than the
preceding ones, covering consecutive pattern matches in permutations, words, cycles,
and alternating permutations and introducing the reciprocity method as a way to define
ring homomorphisms with desirable properties.
Counting with Symmetric Functions will appeal to graduate students and researchers in
mathematics or related subjects who are interested in counting methods, generating
functions, or symmetric functions. The unique approach taken and results and exercises
explored by the authors make it an important contribution to the mathematical literature.
1st ed. 2015, X, 290 p. 223 illus.
Hardcover
ISBN 978-3-319-23617-9
Series: Grundlehren der mathematischen Wissenschaften, Vol. 351
* Presents a concise and authoritative potential-theoretic approach on
metastability
* Reduces questions of interest to the computation of capacities, that
in turn can be estimated by exploiting powerful variational principles
* Deduces detailed information on the spectral characteristics of the
generator of the dynamics
* Unveils the common universal features of metastable systems
through numerous examples
This monograph provides a concise presentation of a mathematical approach to
metastability, a wide-spread phenomenon in the dynamics of non-linear systems -
physical, chemical, biological or economic - subject to the action of temporal random
forces typically referred to as noise, based on potential theory of reversible Markov
processes.
The authors shed new light on the metastability phenomenon as a sequence of visits of
the path of the process to different metastable sets, and focuses on the precise analysis of
the respective hitting probabilities and hitting times of these sets.
The theory is illustrated with many examples, ranging from finite-state Markov chains,
finite-dimensional diffusions and stochastic partial differential equations, via mean-field
dynamics with and without disorder, to stochastic spin-flip and particle-hop dynamics and
probabilistic cellular automata, unveiling the common universal features of these systems
with respect to their metastable behaviour.
The monograph will serve both as comprehensive introduction and as reference for
graduate students and researchers interested in metastability.
1st ed. 2016, Approx. 600 p.
Hardcover
ISBN 978-3-319-24775-5