Series:De Gruyter Expositions in Mathematics
Hardcover
To be published: July 2017
ISBN 978-3-11-035986-2
Geometric Analysis is one of the most active research fields nowadays.
The interplay between geometric and analytic techniques is at the core
of recent remarkable advances in Differential Geometry and Topology. However,
the majority of the monographs and books on the subject focus on intrinsic
Riemannian Geometry techniques and applications. A systematic treatment
of problems involving the extrinsic curvature of submanifolds is still
missing in the literature. In particular, up to our knowledge, there are
no references joining the geometric and analytic aspects of the theory
of hypersurfaces which are solutions of variational problems as it is the
case of prescribed mean curvature hypersurfaces. These problems arise naturally
in the context of geometric functionals in General Relativity and elasticity
theory. This book is aimed to be a comprehensive introduction of the basic
geometric facts and PDEs tools as well as to some current research topics
on hypersurfaces with prescribed mean curvature in Riemannian manifolds.
We also intend to present a unified and simple presentation of classical
results on stability, existence of minimal foliations and geometric and
analytic inequalities in submanifolds.
The structure of the book can be roughly described as follows: the first
chapter presents basic facts on tensor and exterior calculus what fixes
the notation and makes the book self-contained in what refers to the geometric
language. The main tools of elliptic and parabolic PDEs theory are presented
in Chapter 2. Although this is a classical topic our presentation is going
to be adapted to the context of Riemannian manifolds. The variational problem
we are interested in are discussed in Chapter 3 which also provides a collection
of several important theorems on characterization and stability of hypersurfaces
with constant (higher order) mean curvature. The mean curvature PDE is
studied in depth in Chapter 4 and its parabolic counterpart, the mean curvature
flow of graphs, is the subject of Chapter 5. The last chapter concerns
higher order mean curvature equations as examples of fully nonlinear problems
arising from the geometry of hypersurfaces in Riemannian spaces.
Researchers and Ph.D. students in Differential Geometry and Analysis of
PDE may find in this book a compendium of techniques and applications of
elliptic and parabolic PDEs to geometric problems. It also provides a structured
presentation of some of the most important results in the field of constant
mean curvature hypersurfaces. Since it is intended to be self-contained,
it might be also used as a reference textbook for advanced topics courses
in Differential Geometry and Analysis.
Details
24.0 x 17.0 cm
Approx. x, 290 pages
5 Fig.
Language:
English
Type of Publication:
Monograph
Keyword(s):
Geometric Analysis; Geometric variational problems; Hypersurfaces; Mean curvature; Elliptic and parabolic PDEs
Subjects
Mathematics > Analysis
Mathematics > Geometry and Topology
Munster Lectures in Mathematics
ISBN print 978-3-03719-165-1,
July 2016, 148 pages, softcover, 17 x 24 cm.
Free probability is a probability theory dealing with variables having
the highest degree of noncommutativity, an aspect found in many areas (quantum
mechanics, free group algebras, random matrices etc). Thirty years after
its foundation, it is a well-established and very active field of mathematics.
Originating from Voiculescu’s attempt to solve the free group factor problem
in operator algebras, free probability has important connections with random
matrix theory, combinatorics, harmonic analysis, representation theory
of large groups, and wireless communication.
These lecture notes arose from a masterclass in Munster, Germany and present
the state of free probability from an operator algebraic perspective. This
volume includes introductory lectures on random matrices and combinatorics
of free probability (Speicher), free monotone transport (Shlyakhtenko),
free group factors (Dykema), free convolution (Bercovici), easy quantum
groups (Weber), and a historical review with an outlook (Voiculescu). In
order to make it more accessible, the exposition features a chapter on
basics in free probability, and exercises for each part.
This book is aimed at master students to early career researchers familiar with basic notions and concepts from operator algebras.
Keywords: Free probability, operator algebras, random matrices, free monotone
transport, free group factors, free convolution, compact quantum groups,
easy quantum groups, noncrossing partitions, free independence, entropy,
max-stable laws, exchangeability
ISBN: 978-1-119-10766-8
400 pages
November 2016
This book is a new edition of a title originally published in1992. No other
book has been published that treats inverse spectral and inverse scattering
results by using the so called Poisson summation formula and the related
study of singularities. This book presents these in a closed and comprehensive
form, and the exposition is based on a combination of different tools and
results from dynamical systems, microlocal analysis, spectral and scattering
theory.
The content of the first edition is still relevant, however the new edition
will include several new results established after 1992; new text will
comprise about a third of the content of the new edition. The main chapters
in the first edition in combination with the new chapters will provide
a better and more comprehensive presentation of importance for the applications
inverse results. These results are obtained by modern mathematical techniques
which will be presented together in order to give the readers the opportunity
to completely understand them. Moreover, some basic generic properties
established by the authors after the publication of the first edition establishing
the wide range of applicability of the Poison relation will be presented
for first time in the new edition of the book.
Advanced studies in pure mathematics, Volume 68
Nicolas Dutertre ; Topology and geometry of real singularities 1
David Mond ; Singularities of mappings and the vanishing homology of images and discriminants 41
Le Dung Tr´ang ; Notes on non-isolated singularities 143
Hans Sch¨onemann ; Algorithms for primary decomposition in Singular 171
Toru Ohmoto ; Singularities of maps and characteristic classes 191
Valery G. Romanovski and Douglas S. Shafer ? Centers and
limit cycles in polynomial systems of ordinary differential equations 267
Advanced studies in pure mathematics, Volume 69
Igor Dolgachev ; A brief introduction to Enriques surfaces 1
Eduard Looijenga ; Moduli spaces and locally symmetric varieties 33
Nicole Mestrano and Carlos Simpson ; Moduli of sheaves 77
Hiraku Nakajima ? More lectures on Hilbert schemes of points on surfaces 173
Iku Nakamura ; Compactification by GIT-stability of the moduli space of abelian varieties 207
Research Articles
Valery Alexeev ; Divisors on Burniat surfaces 287
Marian Aprodu, Gavril Farkas, and Angela Ortega ; Restricted
Lazarsfeld ; Mukai bundles and canonical curves 303
Ionut， Ciocan-Fontanine and Bumsig Kim ; Big I-functions 323
Brendan Hassett, Andrew Kresch, and Yuri Tschinkel ;
On the moduli of degree 4 Del Pezzo surfaces 349
Daniel Huybrechts ; On derived categories of K3 surfaces,
symplectic automorphisms and the Conway group 387
JongHae Keum ; Order 40 automorphisms of K3 surfaces 407
Viacheslav V. Nikulin ; K¨ahlerian K3 surfaces and Niemeier lattices, II 421
K￣ota Yoshioka ; Bridgeland’s stability and the positive cone of
the moduli spaces of stable objects on an abelian surface 473