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Lectures on Random Lozenge Tilings

Lectures on Random Lozenge Tilings
Author: Vadim Gorin
Publisher: Cambridge University Press
Total 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
Author: Colin Adams
Publisher: American Mathematical Society
Total 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
Author: Yujiro Kawamata
Publisher: Cambridge University Press
Total 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
Author: Ted Dobson
Publisher: Cambridge University Press
Total 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
Author: Henning Krause
Publisher: Cambridge University Press
Total 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
Author: Emily Riehl
Publisher: Cambridge University Press
Total 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
Author: Fabien Durand
Publisher: Cambridge University Press
Total 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.


Statistical Mechanics

Statistical Mechanics
Author: Scott Sheffield
Publisher: American Mathematical Soc.
Total Pages: 358
Release:
Genre: Science
ISBN: 0821886975

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Thirty-three Miniatures

Thirty-three Miniatures
Author: Jiří Matoušek
Publisher: American Mathematical Soc.
Total Pages: 196
Release: 2010
Genre: Mathematics
ISBN: 0821849778

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This volume contains a collection of clever mathematical applications of linear algebra, mainly in combinatorics, geometry, and algorithms. Each chapter covers a single main result with motivation and full proof in at most ten pages and can be read independently of all other chapters (with minor exceptions), assuming only a modest background in linear algebra. The topics include a number of well-known mathematical gems, such as Hamming codes, the matrix-tree theorem, the Lovasz bound on the Shannon capacity, and a counterexample to Borsuk's conjecture, as well as other, perhaps less popular but similarly beautiful results, e.g., fast associativity testing, a lemma of Steinitz on ordering vectors, a monotonicity result for integer partitions, or a bound for set pairs via exterior products. The simpler results in the first part of the book provide ample material to liven up an undergraduate course of linear algebra. The more advanced parts can be used for a graduate course of linear-algebraic methods or for seminar presentations. Table of Contents: Fibonacci numbers, quickly; Fibonacci numbers, the formula; The clubs of Oddtown; Same-size intersections; Error-correcting codes; Odd distances; Are these distances Euclidean?; Packing complete bipartite graphs; Equiangular lines; Where is the triangle?; Checking matrix multiplication; Tiling a rectangle by squares; Three Petersens are not enough; Petersen, Hoffman-Singleton, and maybe 57; Only two distances; Covering a cube minus one vertex; Medium-size intersection is hard to avoid; On the difficulty of reducing the diameter; The end of the small coins; Walking in the yard; Counting spanning trees; In how many ways can a man tile a board?; More bricks--more walls?; Perfect matchings and determinants; Turning a ladder over a finite field; Counting compositions; Is it associative?; The secret agent and umbrella; Shannon capacity of the union: a tale of two fields; Equilateral sets; Cutting cheaply using eigenvectors; Rotating the cube; Set pairs and exterior products; Index. (STML/53)


The Mathematical Gardner

The Mathematical Gardner
Author: David A. Klarner
Publisher: Springer Science & Business Media
Total Pages: 393
Release: 2012-12-06
Genre: Science
ISBN: 1468466860

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-~- T he articles in this book are dedicated to Martin Gardner, the world's greatest expositor and popularizer of mathematics. While our papers are confined to this single subject, Gardner's interests and accomplishments have a wide range of subjects. Hence, we have entitled the book the Mathematical Gardner, and would like to see other volumes such as the Magical, the Literary, the Philosophical, or the Scientific Gardner accompany it. Of course, our title is also an appropriate pun, for Martin Gardner's relationship to the mathematical community is similar to a gardener's relationship to a beautiful flower garden. The contributors to this volume comprise only a small part of a large body of mathematicians whose work has been nurtured by its exposition in "Mathematical Games"; Martin's column which appears every month in Scientific American. More than just a mathematical journalist, Martin connects his readers by passing along problems and information and stimulating creative activity. Thus, he is a force behind the scenes as well as a public figure. Two people were particularly helpful in putting this book together.