Lectures on Random Lozenge Tilings

Lectures on Random Lozenge Tilings
Title Lectures on Random Lozenge Tilings PDF eBook
Author Vadim Gorin
Publisher Cambridge University Press
Pages 261
Release 2021-09-09
Genre Language Arts & Disciplines
ISBN 1108843964

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This is the first book dedicated to reviewing the mathematics of random tilings of large domains on the plane.

The Tiling Book

The Tiling Book
Title The Tiling Book PDF eBook
Author Colin Adams
Publisher American Mathematical Society
Pages 310
Release 2023-08-28
Genre Mathematics
ISBN 1470474611

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Tiling theory provides a wonderful opportunity to illustrate both the beauty and utility of mathematics. It has all the relevant ingredients: there are stunning pictures; open problems can be stated without having to spend months providing the necessary background; and there are both deep mathematics and applications. Furthermore, tiling theory happens to be an area where many of the sub-fields of mathematics overlap. Tools can be applied from linear algebra, algebra, analysis, geometry, topology, and combinatorics. As such, it makes for an ideal capstone course for undergraduates or an introductory course for graduate students. This material can also be used for a lower-level course by skipping the more technical sections. In addition, readers from a variety of disciplines can read the book on their own to find out more about this intriguing subject. This book covers the necessary background on tilings and then delves into a variety of fascinating topics in the field, including symmetry groups, random tilings, aperiodic tilings, and quasicrystals. Although primarily focused on tilings of the Euclidean plane, the book also covers tilings of the sphere, hyperbolic plane, and Euclidean 3-space, including knotted tilings. Throughout, the book includes open problems and possible projects for students. Readers will come away with the background necessary to pursue further work in the subject.

Algebraic Varieties: Minimal Models and Finite Generation

Algebraic Varieties: Minimal Models and Finite Generation
Title Algebraic Varieties: Minimal Models and Finite Generation PDF eBook
Author Yujiro Kawamata
Publisher Cambridge University Press
Pages 263
Release 2024-06-30
Genre Mathematics
ISBN 1009344676

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The finite generation theorem is a major achievement of modern algebraic geometry. Based on the minimal model theory, it states that the canonical ring of an algebraic variety defined over a field of characteristic zero is a finitely generated graded ring. This graduate-level text is the first to explain this proof. It covers the progress on the minimal model theory over the last 30 years, culminating in the landmark paper on finite generation by Birkar-Cascini-Hacon-McKernan. Building up to this proof, the author presents important results and techniques that are now part of the standard toolbox of birational geometry, including Mori's bend and break method, vanishing theorems, positivity theorems and Siu's analysis on multiplier ideal sheaves. Assuming only the basics in algebraic geometry, the text keeps prerequisites to a minimum with self-contained explanations of terminology and theorems.

Symmetry in Graphs

Symmetry in Graphs
Title Symmetry in Graphs PDF eBook
Author Ted Dobson
Publisher Cambridge University Press
Pages 527
Release 2022-05-12
Genre Language Arts & Disciplines
ISBN 1108429068

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The first full-length book on the theme of symmetry in graphs, a fast-growing topic in algebraic graph theory.

Homological Theory of Representations

Homological Theory of Representations
Title Homological Theory of Representations PDF eBook
Author Henning Krause
Publisher Cambridge University Press
Pages 517
Release 2021-11-18
Genre Mathematics
ISBN 1108838898

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This book for advanced graduate students and researchers discusses representations of associative algebras and their homological theory.

Elements of ?-Category Theory

Elements of ?-Category Theory
Title Elements of ?-Category Theory PDF eBook
Author Emily Riehl
Publisher Cambridge University Press
Pages 781
Release 2022-02-10
Genre Mathematics
ISBN 1108837980

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This book develops the theory of infinite-dimensional categories by studying the universe, or ∞-cosmos, in which they live.

Dimension Groups and Dynamical Systems

Dimension Groups and Dynamical Systems
Title Dimension Groups and Dynamical Systems PDF eBook
Author Fabien Durand
Publisher Cambridge University Press
Pages 594
Release 2022-02-03
Genre Mathematics
ISBN 1108986099

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This book is the first self-contained exposition of the fascinating link between dynamical systems and dimension groups. The authors explore the rich interplay between topological properties of dynamical systems and the algebraic structures associated with them, with an emphasis on symbolic systems, particularly substitution systems. It is recommended for anybody with an interest in topological and symbolic dynamics, automata theory or combinatorics on words. Intended to serve as an introduction for graduate students and other newcomers to the field as well as a reference for established researchers, the book includes a thorough account of the background notions as well as detailed exposition – with full proofs – of the major results of the subject. A wealth of examples and exercises, with solutions, serve to build intuition, while the many open problems collected at the end provide jumping-off points for future research.