Translations of Mathematical Monographs
2018; 200 pp; Hardcover
MSC: Primary 16; 14; Secondary 20; 12
Print ISBN: 978-1-4704-4072-5
This book is devoted to arithmetic geometry with special attention given to the unramified Brauer group of algebraic varieties and its most striking applications in birational and Diophantine geometry. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, Weil restriction of scalars, algebraic tori, the Hasse principle, Brauer-Manin obstruction, and etale cohomology. The book contains a detailed presentation of an example of a stably rational but not rational variety, which is presented as series of exercises with detailed hints. This approach is aimed to help the reader understand crucial ideas without being lost in technical details. The reader will end up with a good working knowledge of the Brauer group and its important geometric applications, including the construction of unirational but not stably rational algebraic varieties, a subject which has become fashionable again in connection with the recent breakthroughs by a number of mathematicians.
Graduate students and researchers interested in algebraic geometry and algebraic number theory.
Contemporary Mathematics, Volume: 711
2018; 168 pp; Softcover
MSC: Primary 81; 17;
Print ISBN: 978-1-4704-3717-6
This book contains the proceedings of the AMS Special Session on Vertex Algebras and Geometry, held from October 8?9, 2016, and the mini-conference on Vertex Algebras, held from October 10?11, 2016, in Denver, Colorado.
The papers cover vertex algebras in connection with geometry and tensor categories, with topics in vertex rings, chiral algebroids, the Higgs branch conjecture, and applicability and use of vertex tensor categories.
Graduate students and research mathematicians interested in vertex algebras and applications.
Contemporary Mathematics, Volume: 712
2018; 355 pp; Softcover
MSC: Primary 13; 14; 32; 53;
Print ISBN: 978-1-4704-3488-5
This volume contains the proceedings of the conference Local and Global Methods in Algebraic Geometry, held from May 12?15, 2016, at the University of Illinois at Chicago, in honor of Lawrence Ein's 60th birthday.
The articles cover a broad range of topics in algebraic geometry and related fields, including birational geometry and moduli theory, analytic and positive characteristic methods, geometry of surfaces, singularity theory, hyper-Kahler geometry, rational points, and rational curves.
Graduate students and research mathematicians interested in algebraic geometry and related fields.
University Lecture Series, Volume: 71
2018; 168 pp; Softcover
MSC: Primary 05; 60;
Print ISBN: 978-1-4704-4397-9
A central object of this book is the discrete Laplace operator on finite and infinite graphs. The eigenvalues of the discrete Laplace operator have long been used in graph theory as a convenient tool for understanding the structure of complex graphs. They can also be used in order to estimate the rate of convergence to equilibrium of a random walk (Markov chain) on finite graphs. For infinite graphs, a study of the heat kernel allows to solve the type problem?a problem of deciding whether the random walk is recurrent or transient.
This book starts with elementary properties of the eigenvalues on finite graphs, continues with their estimates and applications, and concludes with heat kernel estimates on infinite graphs and their application to the type problem.
The book is suitable for beginners in the subject and accessible to undergraduate and graduate students with a background in linear algebra I and analysis I. It is based on a lecture course taught by the author and includes a wide variety of exercises. The book will help the reader to reach a level of understanding sufficient to start pursuing research in this exciting area.
Undergraduate and graduate students and researchers interested in random walks on graphs and groups.
AMS Chelsea Publishing, Volume: 384
2018; 212 pp; Hardcover
MSC: Primary 11;
Print ISBN: 978-1-4704-4694-9
Growing out of a course designed to teach Gauss's Disquisitiones Arithmeticae to honors-level undergraduates, Flath's Introduction to Number Theory focuses on Gauss's theory of binary quadratic forms. It is suitable for use as a textbook in a course or self-study by advanced undergraduates or graduate students who possess a basic familiarity with abstract algebra. The text treats a variety of topics from elementary number theory including the distribution of primes, sums of squares, continued factions, the Legendre, Jacobi and Kronecker symbols, the class group and genera. But the focus is on quadratic reciprocity (several proofs are given including one that highlights
The clarity of the author's vision is matched by the clarity of his exposition. This is a book that reveals the discovery of the quadratic core of algebraic number theory. It should be on the desk of every instructor of introductory number theory as a source of inspiration, motivation, examples, and historical insight.
Undergraduate and graduate students and researchers interested in the history of number theory.