ISBN: 978-3-030-89299-9
Provides the first complete and thorough treatment of GIT from a differential geometric viewpoint
Treats Hamiltonian group actions on general, not necessarily projective, compact Kahler manifolds
Presents another route to many of the main results of Mumford's theory which will have long lasting significance
This book provides an introduction to geometric invariant theory from a differential geometric viewpoint.
It is inspired by certain infinite-dimensional analogues of geometric invariant theory that arise naturally in several
different areas of geometry. The central ingredients are the moment-weight inequality relating the Mumford
numerical invariants to the norm of the moment map, the negative gradient flow of the moment map squared,
and the Kempf--Ness function. The exposition is essentially self-contained, except for an appeal to the Lojasiewicz g
radient inequality. A broad variety of examples illustrate the theory, and five appendices cover essential topics that
go beyond the basic concepts of differential geometry. The comprehensive bibliography will be a valuable resource
for researchers.
The book is addressed to graduate students and researchers interested in geometric invariant theory and related
subjects. It will be easily accessible to readers with a basic understanding of differential geometry and does
not require any knowledge of algebraic geometry.
ISBN: 978-1-4614-5292-8
Provides an accessible and broad overview to the field of commutative algebra
Includes expository contributions written by leaders in the field
Accessible to graduate students and researchers who are new to the area
Introduction
This contributed volume brings together the highest quality expository papers written by leaders and talented junior
mathematicians in the field of Commutative Algebra. Contributions cover a very wide range of topics, including core
areas in Commutative Algebra and also relations to Algebraic Geometry, Algebraic Combinatorics, Hyperplane
Arrangements, Homological Algebra, and String Theory. The book aims to showcase the area, especially for the
benefit of junior mathematicians and researchers who are new to the field; it will aid them in broadening their
background and to gain a deeper understanding of the current research in this area. Exciting developments are
surveyed and many open problems are discussed with the aspiration to inspire the readers and foster further research.
Castelnuovo-Mumford regularityCharacteristic pCohomological DegreesCommutative AlgebraHilbert-Kunz multiplicity
Provides self-contained and comprehensive round up of the theory of evolutionary equations
The matter is confined to elementary Hilbert space theory and complex analysis
Easy access to challenging theory of time-dependent parties differential equations
Many different equations accessible with a single approach
This open access book provides a solution theory for time-dependent partial differential equations,
which classically have not been accessible by a unified method. Instead of using sophisticated
techniques and methods, the approach is elementary in the sense that only Hilbert space methods
and some basic theory of complex analysis are required. Nevertheless, key properties of solutions
can be recovered in an elegant manner. Moreover, the strength of this method is demonstrated
by a large variety of examples, showing the applicability of the approach of evolutionary equations in
various fields. Additionally, a quantitative theory for evolutionary equations is developed.
The text is self-contained, providing an excellent source for a first study on evolutionary equations and a
decent guide to the available literature on this subject, thus bridging the gap to state-of-the-art mathematical
research.
Explores applications of Carleman estimates to study stabilization and controllability properties of PDEs
Covers necessary background material in detail
Complemented by a second volume that considers Carleman estimates in Euclidean space
This monograph explores applications of Carleman estimates in the study of stabilization and controllability
properties of partial differential equations, including the stabilization property of the damped wave equation
and the null-controllability of the heat equation. All analysis is performed in the case of open sets in the
Euclidean space; a second volume will extend this treatment to Riemannian manifolds.
The first three chapters illustrate the derivation of Carleman estimates using pseudo-differential calculus with
a large parameter. Continuation issues are then addressed, followed by a proof of the logarithmic stabilization
of the damped wave equation by means of two alternative proofs of the resolvent estimate for the generator
of a damped wave semigroup. The authors then discuss null-controllability of the heat equation, its equivalence
with observability, and how the spectral inequality allows one to either construct a control function or prove
the observability inequality. The final part of the book is devoted to the exposition of some necessary background
material: the theory of distributions, invariance under change of variables, elliptic operators with Dirichlet data and
associated semigroup, and some elements from functional analysis and semigroup theory.
Explores applications of Carleman estimates to study stabilization and controllability properties of PDEs
Covers all necessary background material in detail
Complemented by a second volume that extends treatment to Riemannian manifolds
This monograph explores applications of Carleman estimates in the study of stabilization and controllability properties
of partial differential equations, including quantified unique continuation, logarithmic stabilization of the wave equation,
and null-controllability of the heat equation. Where the first volume derived these estimates in regular open sets in
Euclidean space and Dirichlet boundary conditions, here they are extended to Riemannian manifolds and more general
boundary conditions.
The book begins with the study of Lopatinskii-Sapiro boundary conditions for the Laplace-Beltrami operator, followed by
derivation of Carleman estimates for this operator on Riemannian manifolds. Applications of Carleman estimates are
explored next: quantified unique continuation issues, a proof of the logarithmic stabilization of the boundary-damped
wave equation, and a spectral inequality with general boundary conditions to derive the null-controllability result for the
heat equation. Two additional chapters consider some more advanced results on Carleman estimates.
The final part of the book is devoted to exposition of some necessary background material: elements of differential and
Riemannian geometry, and Sobolev spaces and Laplace problems on Riemannian manifolds.