Provides a self-contained introduction to the Brauer group of schemes
Presents recent applications to rational points on varieties and to rationality
problems in algebraic geometry
Offers a detailed guide to the computation and finiteness of the Brauer group
for various classes of varieties
This monograph provides a systematic treatment of the Brauer group of schemes, from the
foundational work of Grothendieck to recent applications in arithmetic and algebraic geometry.
The importance of the cohomological Brauer group for applications to Diophantine equations
and algebraic geometry was discovered soon after this group was introduced by Grothendieck.
The Brauer?Manin obstruction plays a crucial role in the study of rational points on varieties
over global fields. The birational invariance of the Brauer group was recently used in a novel
way to establish the irrationality of many new classes of algebraic varieties. The book covers
the vast theory underpinning these and other applications. Intended as an introduction to
cohomological methods in algebraic geometry, most of the book is accessible to readers with a
knowledge of algebra, algebraic geometry and algebraic number theory at graduate level. Much
of the more advanced material is not readily available in book form elsewhere; notably, de
Jong’s proof of Gabber’s theorem, the specialisation method and applications of the Brauer
group to rationality questions, an in-depth study of the Brauer?Manin obstruction, and proof of
the finiteness theorem for the Brauer group of abelian varieties and K3 surfaces over finitely
generated fields. The book surveys recent work but also gives detailed proofs of basic
theorems, maintaining a balance between general theory and concrete examples. Over half a
century after Grothendieck's foundational seminars on the topic, The Brauer?Grothendieck
Group is a treatise that fills a longstanding gap in the literature, providing researchers,
including research students, with a valuable reference on a central object of algebraic and
arithmetic geometry.
Due 2021-08-25
1st ed. 2021, XIV, 421 p.
Hardcover
ISBN 978-3-030-74247-8
Product category : Monograph
Series : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of
Modern Surveys in Mathematics
Mathematics : Algebraic Geometry
The first book in the area of thermodynamic formalism, there are no obvious
comparable books
Provides insight into the current state of research
Accessible to a broad range of lecturers
This volume arose from a semester at CIRM-Luminy on “Thermodynamic Formalism:
Applications to Probability, Geometry and Fractals” which brought together leading experts in
the area to discuss topical problems and recent progress. It includes a number of surveys
intended to make the field more accessible to younger mathematicians and scientists wishing
to learn more about the area. Thermodynamic formalism has been a powerful tool in ergodic
theory and dynamical system and its applications to other topics, particularly Riemannian
geometry (especially in negative curvature), statistical properties of dynamical systems and
fractal geometry. This work will be of value both to graduate students and more senior
researchers interested in either learning about the main ideas and themes in thermodynamic
formalism, and research themes which are at forefront of research in this area.
Due 2021-08-10
1st ed. 2021, X, 390 p. 21 illus., 8 illus. in color.
Softcover
ISBN 978-3-030-74862-3
Product category : Contributed volume
Series : Lecture Notes in Mathematics
Mathematics : Dynamical Systems and Ergodic Theory
Gives a survey of current areas of active research
Written in an approachable manner suitable for graduate students
Provides updates, further clarifications and remarks
This textbook contains the lecture series originally delivered at the "Advanced Course on Limit
Cycles of Differential Equations" in the Centre de Recerca Matematica Barcelona in 2006.The
topics covered are the center-focus problem for polynomial vector fields, and the application of
Abelian integrals to limit cycle bifurcations. Both topics are related to Hilbert's sixteenth
problem. In particular, the book willbeof interest to students and researchers workingin the
qualitative theory of dynamical systems. This secondedition provides updates, further
clarifications and remarks, and includes an expanded list of references.
Due 2022-02-11
2nd ed. 2021, Approx. 120 p.
Softcover
ISBN 978-3-030-59655-2
Product category : Graduate/advanced undergraduate textbook
Series : Advanced Courses in Mathematics - CRM Barcelona
Mathematics : Analysis
https://doi.org/10.1142/12308 | May 2021
Pages: 336
This book is an introduction to fiber bundles and fibrations. But the ultimate goal is to make the reader feel comfortable with basic ideas in homotopy theory. The author found that the classification of principal fiber bundles is an ideal motivation for this purpose. The notion of homotopy appears naturally in the classification. Basic tools in homotopy theory such as homotopy groups and their long exact sequence need to be introduced. Furthermore, the notion of fibrations, which is one of three important classes of maps in homotopy theory, can be obtained by extracting the most essential properties of fiber bundles. The book begins with elementary examples and then gradually introduces abstract definitions when necessary. The reader is assumed to be familiar with point-set topology, but it is the only requirement for this book.
Preface
Acknowledgments
List of Figures
How to Bundle Fibers
Covering Spaces as a Toy Model
Basic Properties of Fiber Bundles
Classification of Fiber Bundles
Fibrations
Postscript
Appendices: Related Topics
The Meaning of Compact-Open Topology
Vector Bundles
Simplicial Techniques
Bibliography
Index
Graduate students and researchers in algebraic topology.
https://doi.org/10.1142/12295 | August 2021
Pages: 180
The second edition represents an ongoing effort to make probability accessible to students in a wide range of fields such as mathematics, statistics and data science, engineering, computer science, and business analytics. The book is written for those learning about probability for the first time. Revised and updated, the book is aimed specifically at statistics and data science students who need a solid introduction to the basics of probability.
While retaining its focus on basic probability, including Bayesian probability and the interface between probability and computer simulation, this edition's significant revisions are as follows:
Many extra motivational examples and problems
New material on Bayesian probability, including two famous court cases
New sections on real-world applications of the Poisson distribution
New sections on generating functions and the bivariate normal density
New chapter on Markov chains, including Markov chain Monte Carlo simulation
The approach followed in the book is to develop probabilistic intuition before diving into details. The best way to learn probability is by practising on a lot of problems. Many instructive problems together with problem-solving strategies are given. Answers to all problems and worked-out solutions to selected problems are also provided.
Combinatorics and Calculus for Probability
Basics of Probability
Useful Probability Distributions
Real-Life Examples of Poisson Probabilities
Monte Carlo Simulation and Probability
A Primer on Markov Chains
Solutions to Selected Problems
Undergraduate students in fields such as mathematics, statistics and data science, engineering, computer science and business analytics. Graduate students in natural and social sciences. Students taking a first course in probability, or a course on probability for statistics and data science. High school math teachers and STEM students. Students in natural and social sciences, economics and finance.
https://doi.org/10.1142/12100 | September 2021
Pages: 200
This book contains a complete detailed description of two classes of special numbers closely related to classical problems of the Theory of Primes. There is also extensive discussions of applied issues related to Cryptography.
In Mathematics, a Mersenne number (named after Marin Mersenne, who studied them in the early 17-th century) is a number of the form Mn = 2n ? 1 for positive integer n.
In Mathematics, a Fermat number (named after Pierre de Fermat who first studied them) is a positive integer of the form Fn = 2k+ 1, k=2n, where n is a non-negative integer.
Mersenne and Fermat numbers have many other interesting properties. Long and rich history, many arithmetic connections (with perfect numbers, with construction of regular polygons etc.), numerous modern applications, long list of open problems allow us to provide a broad perspective of the Theory of these two classes of special numbers, that can be useful and interesting for both professionals and the general audience.
Preliminaries
Prime Numbers
Mersenne Numbers
Fermat Numbers
Modern Applications
Zoo of Numbers
Mini Dictionary
Exercises
Advanced undergraduate, graduate students, researchers and the general public interested in Arithmetic, Number Theory, General Algebra, Cryptography and related fields.
September 2021
Pages: 380
This book provides a unique and highly accessible approach to singularity theory from the perspective of differential geometry of curves and surfaces. It is written by three leading experts on the interplay between two important fields ? singularity theory and differential geometry.
The book introduces singularities and their recognition theorems, and describes their applications to geometry and topology, restricting the objects of attention to singularities of plane curves and surfaces in the Euclidean 3-space. In particular, by presenting the singular curvature, which originated through research by the authors, the Gauss?Bonnet theorem for surfaces is generalized to those with singularities. The Gauss?Bonnet theorem is intrinsic in nature, that is, it is a theorem not only for surfaces but also for 2-dimensional Riemannian manifolds. The book also elucidates the notion of Riemannian manifolds with singularities.
These topics, as well as elementary descriptions of proofs of the recognition theorems, cannot be found in other books. Explicit examples and models are provided in abundance, along with insightful explanations of the underlying theory as well. Numerous figures and exercise problems are given, becoming strong aids in developing an understanding of the material.
Readers will gain from this text a unique introduction to the singularities of curves and surfaces from the viewpoint of differential geometry, and it will be a useful guide for students and researchers interested in this subject.
Planar Curves and Singular Points
Singularities of Surfaces
Proof of Criteria for Singularities
Application of Criteria for Singularities
Local Differential Geometry of Surfaces with Singularities
Gauss?Bonnet Type Formulas and Applications
Flat Surfaces
Proof of the Criterion for Swallowtails
Coherent Tangent Bundles
Contact Structures and Wave Fronts
Appendices:
The Division Lemma
Topics on Cusps
A Criterion for 4/3-cusps
Proof of the Criterion for Whitney Cusps
A Zakalyukin-Type Lemma
A Formula for Singular Curvature
Geometry of Cross Caps
Readership:
Advanced undergraduate and graduate students, and researchers interested in the singularity theory from the perspective of differential geometry of curves and surfaces.