Introduces key topics on Geometric Invariant Theory through examples and
applications
Covers Hilbert classification of binary forms and Hitchin's theory on Higgs
bundles
Takes particular note of unstable objects in module problems
This book introduces key topics on Geometric Invariant Theory, a technique to obtaining
quotients in algebraic geometry with a good set of properties, through various examples. It
starts from the classical Hilbert classification of binary forms, advancing to the construction of
the moduli space of semistable holomorphic vector bundles, and to Hitchinfs theory on Higgs
bundles. The relationship between the notion of stability between algebraic, differential and
symplectic geometry settings is also covered. Unstable objects in moduli problems -- a result
of the construction of moduli spaces -- get specific attention in this work. The notion of the
Harder-Narasimhan filtration as a tool to handle them, and its relationship with GIT quotients,
provide instigating new calculations in several problems. Applications include a survey of
research results on correspondences between Harder-Narasimhan filtrations with the GIT
picture and stratifications of the moduli space of Higgs bundles.
Due 2021-03-24
1st ed. 2021, IV, 121 p. 16 illus., 12 illus. in color.
Softcover
ISBN 978-3-030-67828-9
Product category : Brief
Series : SpringerBriefs in Mathematics
Mathematics : Algebraic Geometry
The first unified exposition of Liouville and Riemann?Roch type theorems for
elliptic operators on abelian coverings
Gives a well-organized and self-contained exposition of the topic, including
new results
Intersects with geometric analysis, the spectral theory of periodic operators,
and their applications
This book is devoted to computing the index of elliptic PDEs on non-compact Riemannian
manifolds in the presence of local singularities and zeros, as well as polynomial growth at
infinity. The classical Riemann?Roch theorem and its generalizations to elliptic equations on
bounded domains and compact manifolds, due to Mazfya, Plameneskii, Nadirashvilli, Gromov
and Shubin, account for the contribution to the index due to a divisor of zeros and singularities.
On the other hand, the Liouville theorems of Avellaneda, Lin, Li, Moser, Struwe, Kuchment and
Pinchover provide the index of periodic elliptic equations on abelian coverings of compact
manifolds with polynomial growth at infinity, i.e. in the presence of a "divisor" at infinity. A
natural question is whether one can combine the Riemann?Roch and Liouville type results. This
monograph shows that this can indeed be done, however the answers are more intricate than
one might initially expect. Namely, the interaction between the finite divisor and the point at
infinity is non-trivial. The text is targeted towards researchers in PDEs, geometric analysis, and
mathematical physics.
Due 2021-03-20
1st ed. 2021, XII, 96 p. 2 illus., 1 illus. in color.
Softcover
ISBN 978-3-030-67427-4
Product category : Monograph
Series : Lecture Notes in Mathematics 2245
Mathematics : Global Analysis and Analysis on Manifolds
Presents four lectures by leading experts
Provides an in-depth exploration of recent advances
Includes extensive background and references
This book presents four lectures on Rees rings and blow-ups, Koszul modules with applications
to syzygies, Grobner bases and degenerations, and applications of Adams operations.
Commutative Algebra has witnessed a number of spectacular developments in recent years,
including the resolution of long-standing problems; the new techniques and perspectives are
leading to an extraordinary transformation in the field. The material contained in this volume,
based on lectures given at a workshop held in Levico Terme, Trento, in July 2019, highlights
some of these developments. The text will be a valuable asset to graduate students and
researchers in commutative algebra and related fields.
Due 2021-04-08
1st ed. 2021, XI, 118 p. 17 illus., 1 illus. in color.
Softcover
ISBN 978-3-030-65063-6
Product category : Contributed volume
Series : Lecture Notes in Mathematics, vol.2283
C.I.M.E. Foundation Subseries
Mathematics : Commutative Rings and Algebras
Presents three lectures by leading experts
Provides an in-depth exploration of recent advances
Includes extensive background and references
This volume covers contemporary aspects of geometric measure theory with a focus on
applications to partial differential equations, free boundary problems and water waves. It is
based on lectures given at the 2019 CIME summer school gGeometric Measure Theory and
Applications ? From Geometric Analysis to Free Boundary Problemsh which took place in
Cetraro, Italy, under the scientific direction of Matteo Focardi and Emanuele Spadaro. Providing
a description of the structure of measures satisfying certain differential constraints, and
covering regularity theory for Bernoulli type free boundary problems and water waves as well
as regularity theory for the obstacle problems and the developments leading to applications to
the Stefan problem, this volume will be of interest to students and researchers in
mathematical analysis and its applications.
Due 2021-03-12
1st ed. 2021, X, 110 p. 31illus.
Softcover
ISBN 978-3-030-65798-7
Product category : Contributed volume
Series : Lecture Notes in Mathematics, vol.2284
C.I.M.E. Foundation Subseries
Mathematics : Analysis
Die Theorie der regularen Graphen (The Theory of Regular Graphs), written by the Danish
Mathematician Julius Petersen in 1891,is often considered the first strictly theoretical paper
dealing with graphs. In the 130 years since then, regular graphs have been a common and
popular area of study. While regular graphs are typically considered to be graphs whose
vertices all have the same degree, a more general interpretation is that of graphs possessing
some common characteristic throughout their structure. During the past several decades,
however, there has been some increased interest in investigating graphs possessing a property
that is, in a sense, opposite to regularity. It is this topic with which this book deals, giving rise
to a study of what might be called irregularity in graphs. Here, various irregularity concepts
dealing with several topics in graph theory are described, such as degrees of vertices, graph
labelings, weightings, colorings, graph structures, Eulerian and Hamiltonian properties, graph
decompositions, and Ramsey-type problems
Due 2021-04-20
1st ed. 2021, XII, 126 p.
Softcover
ISBN 978-3-030-67992-7
Product category : Brief
Series : SpringerBriefs in Mathematics
Mathematics : Graph Theory
Develops a general sub-supersolution method for stationary and evolutionary
multi-valued variational inequalities
Provides a self-contained exposition of existence, comparison and enclosure
principles
Accessible to a wide audience of graduate students and researchers
This book focuses on a large class of multi-valued variational differential inequalities and
inclusions of stationary and evolutionary types with constraints reflected by subdifferentials of
convex functionals. Its main goal is to provide a systematic, unified, and relatively selfcontained
exposition of existence, comparison and enclosure principles, together with other
qualitative properties of multi-valued variational inequalities and inclusions. The problems
under consideration are studied in different function spaces such as Sobolev spaces, OrliczSobolev
spaces, Sobolev spaces with variable exponents, and Beppo-Levi spaces. A general and
comprehensive sub-supersolution method (lattice method) is developed for both stationary and
evolutionary multi-valued variational inequalities, which preserves the characteristic features of
the commonly known sub-supersolution method for single-valued, quasilinear elliptic and
parabolic problems. This method provides a powerful tool for studying existence and enclosure
properties of solutions when the coercivity of the problems under consideration fails. It can
also be used to investigate qualitative properties such as the multiplicity and location of
solutions or the existence of extremal solutions. This is the first in-depth treatise on the subsupersolution
(lattice) method for multi-valued variational inequalities without any variational
structures, together with related topics. The choice of the included materials and their
organization in the book also makes it useful and accessible to a large audience consisting of
graduate students and researchers in various areas of Mathematical Analysis and Theoretical
Physics.
Due 2021-04-11
1st ed. 2021, XVII, 584 p. 5 illus.
Hardcover
ISBN 978-3-030-65164-0
Product category : Monograph
Series : Springer Monographs in Mathematics
Mathematics : Analysis
Illustrates a variety of applications of spectral theory with a focus on
quantum physics
Provides potential research directions for students and numerous references
to more advanced treatments of many topics
Guides readers through topics using a progressive, concise approach
Allows instructors to easily adapt more advanced concepts to different
graduate level courses
This textbook provides a graduate-level introduction to the spectral theory of linear operators
on Banach and Hilbert spaces, guiding readers through key components of spectral theory and
its applications in quantum physics. Based on their extensive teaching experience, the authors
present topics in a progressive manner so that each chapter builds on the ones preceding.
Researchers and students alike will also appreciate the exploration of more advanced
applications and research perspectives presented near the end of the book. Beginning with a
brief introduction to the relationship between spectral theory and quantum physics, the authors
go on to explore unbounded operators, analyzing closed, adjoint, and self-adjoint operators.
Next, the spectrum of a closed operator is defined and the fundamental properties of Fredholm
operators are introduced. The authors then develop the Grushin method to execute the spectral
analysis of compact operators. The chapters that follow are devoted to examining Hille-Yoshida
and Stone theorems, the spectral analysis of self-adjoint operators, and trace-class and HilbertSchmidt
operators. The final chapter opens the discussion to several selected applications.
Throughout this textbook, detailed proofs are given, and the statements are illustrated by a
number of well-chosen examples. At the end, an appendix about foundational functional
analysis theorems is provided to help the uninitiated reader. A Guide to Spectral Theory:
Applications and Exercisesis intended for graduate students taking an introductory course in
spectral theory or operator theory. A background in linear functional analysis and partial
differential equations is assumed; basic knowledge of bounded linear operators is useful but
not required.
Due 2021-04-11
1st ed. 2021, XIV, 243 p. 1illus. in color.
Hardcover
ISBN 978-3-030-67461-8
Product category : Graduate/advanced undergraduate textbook
Series : Birkhauser Advanced Texts Basler Lehrbucher
Mathematics : Functional Analysis
Discusses topics in homological algebra and applications in topology,
commutative algebra, and algebraic geometry
Offers the third volume of a four-volume textbook series in Algebra
Is written by an expert with over 43 years of teaching experience
This book, the third book in the four-volume series in algebra, deals with important topics in
homological algebra, including abstract theory of derived functors, sheaf co-homology, and an
introduction to etale and l-adic co-homology. It contains four chapters which discuss homology
theory in an abelian category together with some important and fundamental applications in
geometry, topology, algebraic geometry (including basics in abstract algebraic geometry), and
group theory. The book will be of value to graduate and higher undergraduate students
specializing in any branch of mathematics. The author has tried to make the book selfcontained
by introducing relevant concepts and results required. Prerequisite knowledge of the
basics of algebra, linear algebra, topology, and calculus of several variables will be useful
Due 2021-04-10
1st ed. 2021, XV, 300 p. 32 illus.
Hardcover
ISBN 978-981-33-6325-0
Product category : Undergraduate textbook
Series : Infosys Science Foundation Series in Mathematical Sciences
Mathematics : Algebraic Topology