In recent years, the old idea that gauge theories and string theories are equivalent has been implemented and developed in various ways, and there are by now various models where the string theory / gauge theory correspondence is at work. One of the most important examples of this correspondence relates Chern-Simons theory, a topological gauge theory in three dimensions which describes knot and three-manifold invariants, to topological string theory, which is deeply related to Gromov-Witten invariants. This has led to some surprising relations between three-manifold geometry and enumerative geometry. This book gives the first coherent presentation of this and other related topics. After an introduction to matrix models and Chern-Simons theory, the book describes in detail the topological string theories that correspond to these gauge theories and develops the mathematical implications of this duality for the enumerative geometry of Calabi-Yau manifolds and knot theory. It is written in a pedagogical style and will be useful reading for graduate students and researchers in both mathematics and physics willing to learn about these developments.

Boundaries and Hulls of Euclidean Graphs: From Theory to Practice presents concepts and algorithms for finding convex, concave and polygon hulls of Euclidean graphs. It also includes some implementations, determining and comparing their complexities. Since the implementation is application-dependent, either centralized or distributed, some basic concepts of the centralized and distributed versions are reviewed. Theoreticians will find a presentation of different algorithms together with an evaluation of their complexity and their utilities, as well as their field of application. Practitioners will find some practical and real-world situations in which the presented algorithms can be used.

This book is aimed at graduate students and researchers in physics and mathematics who seek to understand the basics of supersymmetry from a mathematical point of view. It provides a bridge between the physical and mathematical approaches to the superworld. The physicist who is devoted to learning the basics of supergeometry can find a friendly approach here, since only the concepts that are strictly necessary are introduced. On the other hand, the mathematician who wants to learn from physics will find that all the mathematical assumptions are firmly rooted in physical concepts. This may open up a channel of communication between the two communities working on different aspects of supersymmetry.Starting from special relativity and Minkowski space, the idea of conformal space and superspace is built step by step in a mathematically rigorous way, and always connecting with the ideas and notation used in physics. While the book is mainly devoted to these important physical examples of superspaces, it can also be used as an introduction to the field of supergeometry, where a reader can ease into the subject without being overwhelmed with the technical difficulties.

This new edition of Six Simple Twists: The Pleat Pattern Approach to Origami Tessellation Design introduces an innovative pleat pattern technique for origami designs that is easily accessible to anyone who enjoys the geometry of paper. The book begins with six basic forms meant to ease the reader into the style, and then systematically scaffolds the instructions to build a strong understanding of the techniques, leading to instructions on a limitless number of patterns. It then describes a process of designing additional building blocks. At the end, what emerges is a fascinating artform that will enrich folders for many years. Unlike standard, project-based origami books, Six Simple Twists focuses on how to design, rather than construct. In this thoroughly updated second edition, the book explores new techniques and example tessellations, with full-page images, and mathematical analysis of the patterns. A reader will, through practice, gain the ability to create still more complex and fascinating designs. Key Features Introduces the reader to origami tessellations and demonstrates their place in the origami community New layout and instructional approach restructure the book from the ground up Addresses common tessellation questions, such as what types of paper are best to use, and how this artform rose in popularity Teaches the reader how to grid a sheet of paper and the importance of the pre-creases Gives the reader the ability to create and understand tessellations through scaffolded instruction Includes exercises to test understanding Introduces a new notation system for precisely describing pleat intersections Analyzes pleat intersections mathematically using geometrically-focused models, including information about Brocard points

This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-theorists, such as longitudes, the homological complexity of many-variable Laurent polynomial rings, the fact that links are not usually boundary links, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completion and algebraic closure of the link group, and disc links. Invariants of the types considered here play an essential role in many applications of knot theory to other areas of topology.This second edition introduces two new chapters - twisted polynomial invariants and singularities of plane curves. Each replaces brief sketches in the first edition. Chapter 2 has been reorganized, and new material has been added to four other chapters.

Since the 1950s control theory has established itself as a major mathematical discipline, particularly suitable for application in a number of research fields, including advanced engineering design, economics and the medical sciences. However, since its emergence, there has been a need to rethink and extend fields such as calculus of variations, differential geometry and nonsmooth analysis, which are closely tied to research on applications. Today control theory is a rich source of basic abstract problems arising from applications, and provides an important frame of reference for investigating purely mathematical issues. In many fields of mathematics, the huge and growing scope of activity has been accompanied by fragmentation into a multitude of narrow specialties. However, outstanding advances are often the result of the quest for unifying themes and a synthesis of different approaches. Control theory and its applications are no exception. Here, the interaction between analysis and geometry has played a crucial role in the evolution of the field. This book collects some recent results, highlighting geometrical and analytical aspects and the possible connections between them. Applications provide the background, in the classical spirit of mutual interplay between abstract theory and problem-solving practice.

Let G be the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristic p.LetI be a pro-p Iwahori subgroup of G and let R be a commutative quasi-Frobenius ring. If H = R[I\G/I] denotes the pro-p Iwahori- Hecke algebra of G over R we clarify the relation between the category of H-modules and the category of G-equivariant coefficient systems on the semisimple Bruhat-Tits building of G.IfR is a field of characteristic zero this yields alternative proofs of the exactness of the Schneider-Stuhler resolution and of the Zelevinski conjecture for smooth G-representations generated by their I-invariants. In general, it gives a description of the derived category of H-modules in terms of smooth G-representations and yields a functor to generalized (?, ?)-modules extending the constructions of Colmez, Schneider and Vigneras.

This volume discusses the classical subjects of Euclidean, affine and projective geometry in two and three dimensions, including the classification of conics and quadrics, and geometric transformations. These subjects are important both for the mathematical grounding of the student and for applications to various other subjects. They may be studied in the first year or as a second course in geometry. The material is presented in a geometric way, and it aims to develop the geometric intuition and thinking of the student, as well as his ability to understand and give mathematical proofs. Linear algebra is not a prerequisite, and is kept to a bare minimum. The book includes a few methodological novelties, and a large number of exercises and problems with solutions. It also has an appendix about the use of the computer programme MAPLEV in solving problems of analytical and projective geometry, with examples.

This is a collection of surveys on important mathematical ideas, their origin, their evolution and their impact in current research. The authors are mathematicians who are leading experts in their fields. The book is addressed to all mathematicians, from undergraduate students to senior researchers, regardless of the specialty.

This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles' resolution of Fermat's Last Theorem.

This book is a systematic presentation of the solution of one of the fundamental problems of the theory of random dynamical systems - the problem of topological classification and structural stability of linear hyperbolic random dynamical systems. As a relatively new and fast expanding field of research, this theory attracts the attention of researchers from various fields of science. It unites and develops the classical deterministic theory of dynamical systems and probability theory, hence finds many applications in a very wide range of disciplines from physics to biology to engineering, finance and economics. Recent developments call for a systematic presentation of the theory. Mathematicians working in the theory of dynamical systems, stochastic dynamics as well as those interested in applications of mathematical systems with random noise will find this timely book a valuable reference and rich source of modern mathematical methods and results.

This volume discusses the classical subjects of Euclidean, affine and projective geometry in two and three dimensions, including the classification of conics and quadrics, and geometric transformations. These subjects are important both for the mathematical grounding of the student and for applications to various other subjects. They may be studied in the first year or as a second course in geometry.The material is presented in a geometric way, and it aims to develop the geometric intuition and thinking of the student, as well as his ability to understand and give mathematical proofs. Linear algebra is not a prerequisite, and is kept to a bare minimum.The book includes a few methodological novelties, and a large number of exercises and problems with solutions. It also has an appendix about the use of the computer program MAPLEV in solving problems of analytical and projective geometry, with examples.

A polynomial identity for an algebra (or a ring) $A$ is a polynomial in noncommutative variables that vanishes under any evaluation in $A$. An algebra satisfying a nontrivial polynomial identity is called a PI algebra, and this is the main object of study in this book, which can be used by graduate students and researchers alike. The book is divided into four parts. Part 1 contains foundational material on representation theory and noncommutative algebra. In addition to setting the stage for the rest of the book, this part can be used for an introductory course in noncommutative algebra. An expert reader may use Part 1 as reference and start with the main topics in the remaining parts. Part 2 discusses the combinatorial aspects of the theory, the growth theorem, and Shirshov's bases. Here methods of representation theory of the symmetric group play a major role. Part 3 contains the main body of structure theorems for PI algebras, theorems of Kaplansky and Posner, the theory of central polynomials, M. Artin's theorem on Azumaya algebras, and the geometric part on the variety of semisimple representations, including the foundations of the theory of Cayley-Hamilton algebras. Part 4 is devoted first to the proof of the theorem of Razmyslov, Kemer, and Braun on the nilpotency of the nil radical for finitely generated PI algebras over Noetherian rings, then to the theory of Kemer and the Specht problem. Finally, the authors discuss PI exponent and codimension growth. This part uses some nontrivial analytic tools coming from probability theory. The appendix presents the counterexamples of Golod and Shafarevich to the Burnside problem.

This unique book overturns our ideas about non-Euclidean geometry and the fine-structure constant, and attempts to solve long-standing mathematical problems. It describes a general theory of 'recursive' hyperbolic functions based on the 'Mathematics of Harmony,' and the 'golden,' 'silver,' and other 'metallic' proportions. Then, these theories are used to derive an original solution to Hilbert's Fourth Problem for hyperbolic and spherical geometries. On this journey, the book describes the 'golden' qualitative theory of dynamical systems based on 'metallic' proportions. Finally, it presents a solution to a Millennium Problem by developing the Fibonacci special theory of relativity as an original physical-mathematical solution for the fine-structure constant. It is intended for a wide audience who are interested in the history of mathematics, non-Euclidean geometry, Hilbert's mathematical problems, dynamical systems, and Millennium Problems.See Press Release: Application of the mathematics of harmony - Golden non-Euclidean geometry in modern math

Cartan geometries were the first examples of connections on a principal bundle. They seem to be almost unknown these days, in spite of the great beauty and conceptual power they confer on geometry. The aim of the present book is to fill the gap in the literature on differential geometry by the missing notion of Cartan connections. Although the author had in mind a book accessible to graduate students, potential readers would also include working differential geometers who would like to know more about what Cartan did, which was to give a notion of "espaces généralisés" (= Cartan geometries) generalizing homogeneous spaces (= Klein geometries) in the same way that Riemannian geometry generalizes Euclidean geometry. In addition, physicists will be interested to see the fully satisfying way in which their gauge theory can be truly regarded as geometry.

The book covers all the fundamental aspects of generating fractals through L-system. Also it provides insight to various researches in this area for generating fractals through L-system approach & estimating dimensions. Also it discusses various applications of L-system fractals.

Key Features:
- Fractals generated from L-System including hybrid fractals
- Dimension calculation for L-system fractals
- Images & codes for L-system fractals
- Research directions in the area of L-system fractals
- Usage of various freely downloadable tools in this area

- Fractals generated from L-System including hybrid fractals
- Dimension calculation for L-system fractals
- Images & codes for L-system fractals
- Research directions in the area of L-system fractals
- Usage of various freely downloadable tools in this area

This book discusses how to design "good" geometric puzzles: two-dimensional dissection puzzles, polyhedral dissections, and burrs. It outlines major categories of geometric puzzles and provides examples, sometimes going into the history and philosophy of those examples. The author presents challenges and thoughtful questions, as well as practical design and woodworking tips to encourage the reader to build his own puzzles and experiment with his own designs. Aesthetics, phychology, and mathematical considerations all factor into the definition of the quality of a puzzle.

A one-stop reference to fractional factorials and related orthogonal arrays.
Presenting one of the most dynamic areas of statistical research, this book offers a systematic, rigorous, and up-to-date treatment of fractional factorial designs and related combinatorial mathematics. Leading statisticians Aloke Dey and Rahul Mukerjee consolidate vast amounts of material from the professional literature--expertly weaving fractional replication, orthogonal arrays, and optimality aspects. They develop the basic theory of fractional factorials using the calculus of factorial arrangements, thereby providing a unified approach to the study of fractional factorial plans. An indispensable guide for statisticians in research and industry as well as for graduate students, Fractional Factorial Plans features:
* Construction procedures of symmetric and asymmetric orthogonal arrays.
* Many up-to-date research results on nonexistence.
* A chapter on optimal fractional factorials not based on orthogonal arrays.
* Trend-free plans, minimum aberration plans, and search and supersaturated designs.
* Numerous examples and extensive references.

A First Course in Chaotic Dynamical Systems: Theory and Experiment, Second Edition The long-anticipated revision of this well-liked textbook offers many new additions. In the twenty-five years since the original version of this book was published, much has happened in dynamical systems. Mandelbrot and Julia sets were barely ten years old when the first edition appeared, and most of the research involving these objects then centered around iterations of quadratic functions. This research has expanded to include all sorts of different types of functions, including higher-degree polynomials, rational maps, exponential and trigonometric functions, and many others. Several new sections in this edition are devoted to these topics. The area of dynamical systems covered in A First Course in Chaotic Dynamical Systems: Theory and Experiment, Second Edition is quite accessible to students and also offers a wide variety of interesting open questions for students at the undergraduate level to pursue. The only prerequisite for students is a one-year calculus course (no differential equations required); students will easily be exposed to many interesting areas of current research. This course can also serve as a bridge between the low-level, often non-rigorous calculus courses, and the more demanding higher-level mathematics courses. Features More extensive coverage of fractals, including objects like the Sierpinski carpet and others that appear as Julia sets in the later sections on complex dynamics, as well as an actual chaos "game." More detailed coverage of complex dynamical systems like the quadratic family and the exponential maps. New sections on other complex dynamical systems like rational maps. A number of new and expanded computer experiments for students to perform. About the Author Robert L. Devaney is currently professor of mathematics at Boston University. He received his PhD from the University of California at Berkeley under the direction of Stephen Smale. He taught at Northwestern University and Tufts University before coming to Boston University in 1980. His main area of research is dynamical systems, primarily complex analytic dynamics, but also including more general ideas about chaotic dynamical systems. Lately, he has become intrigued with the incredibly rich topological aspects of dynamics, including such things as indecomposable continua, Sierpinski curves, and Cantor bouquets.

This volume contains the proceedings of the ICM 2018 satellite school and workshop $K$-theory conference in Argentina. The school was held from July 16-20, 2018, in La Plata, Argentina, and the workshop was held from July 23-27, 2018, in Buenos Aires, Argentina. The volume showcases current developments in $K$-theory and related areas, including motives, homological algebra, index theory, operator algebras, and their applications and connections. Papers cover topics such as $K$-theory of group rings, Witt groups of real algebraic varieties, coarse homology theories, topological cyclic homology, negative $K$-groups of monoid algebras, Milnor $K$-theory and regulators, noncommutative motives, the classification of $C^*$-algebras via Kasparov's $K$-theory, the comparison between full and reduced $C^*$-crossed products, and a proof of Bott periodicity using almost commuting matrices.

From the reviews: "The 2nd (slightly enlarged) edition of the van Lint's book is a short, concise, mathematically rigorous introduction to the subject. Basic notions and ideas are clearly presented from the mathematician's point of view and illustrated on various special classes of codes...This nice book is a must for every mathematician wishing to introduce himself to the algebraic theory of coding." European Mathematical Society Newsletter, 1993 "Despite the existence of so many other books on coding theory, this present volume will continue to hold its place as one of the standard texts...." The Mathematical Gazette, 1993

This volume is based on lectures given at the highly successful three-week Summer School on Geometry, Topology and Dynamics of Character Varieties held at the National University of Singapore's Institute for Mathematical Sciences in July 2010.Aimed at graduate students in the early stages of research, the edited and refereed articles comprise an excellent introduction to the subject of the program, much of which is otherwise available only in specialized texts. Topics include hyperbolic structures on surfaces and their degenerations, applications of ping-pong lemmas in various contexts, introductions to Lorenzian and complex hyperbolic geometry, and representation varieties of surface groups into PSL(2, ) and other semi-simple Lie groups. This volume will serve as a useful portal to students and researchers in a vibrant and multi-faceted area of mathematics.

The aim of Summable Spaces and Their Duals, Matrix Transformations and Geometric Properties is to discuss primarily about different kinds of summable spaces, compute their duals and then characterize several matrix classes transforming one summable space into other. The book also discusses several geometric properties of summable spaces, as well as dealing with the construction of summable spaces using Orlicz functions, and explores several structural properties of such spaces. Each chapter contains a conclusion section highlighting the importance of results, and points the reader in the direction of possible new ideas for further study. Features Suitable for graduate schools, graduate students, researchers and faculty, and could be used as a key text for special Analysis seminars Investigates different types of summable spaces and computes their duals Characterizes several matrix classes transforming one summable space into other Discusses several geometric properties of summable spaces Examines several possible generalizations of Orlicz sequence spaces

Subanalytic and semialgebraic sets were introduced for topological and systematic investigations of real analytic and algebraic sets. One of the author's purposes is to show that almost all (known and unknown) properties of subanalytic and semialgebraic sets follow abstractly from some fundamental axioms. Another is to develop methods of proof that use finite processes instead of integration of vector fields. The proofs are elementary, but the results obtained are new and significant - for example, for singularity theorists and topologists. Further, the new methods and tools developed provide solid foundations for further research by model theorists (logicians) who are interested in applications of model theory to geometry. A knowledge of basic topology is required.

The importance of mathematics competitions has been widely recognized for three reasons: they help to develop imaginative capacity and thinking skills whose value far transcends mathematics; they constitute the most effective way of discovering and nurturing mathematical talent; and they provide a means to combat the prevalent false image of mathematics held by high school students, as either a fearsomely difficult or a dull and uncreative subject. This book provides a comprehensive training resource for competitions from local and provincial to national Olympiad level, containing hundreds of diagrams, and graced by many light-hearted cartoons. It features a large collection of what mathematicians call "beautiful" problems - non-routine, provocative, fascinating, and challenging problems, often with elegant solutions. It features careful, systematic exposition of a selection of the most important topics encountered in mathematics competitions, assuming little prior knowledge. Geometry, trigonometry, mathematical induction, inequalities, Diophantine equations, number theory, sequences and series, the binomial theorem, and combinatorics - are all developed in a gentle but lively manner, liberally illustrated with examples, and consistently motivated by attractive "appetiser" problems, whose solution appears after the relevant theory has been expounded.
Each chapter is presented as a "toolchest" of instruments designed for cracking the problems collected at the end of the chapter. Other topics, such as algebra, co-ordinate geometry, functional equations and probability, are introduced and elucidated in the posing and solving of the large collection of miscellaneous problems in thefinal toolchest.
An unusual feature of this book is the attention paid throughout to the history of mathematics - the origins of the ideas, the terminology and some of the problems, and the celebration of mathematics as a multicultural, cooperative human achievement.
As a bonus the aspiring "mathlete" may encounter, in the most enjoyable way possible, many of the topics that form the core of the standard school curriculum.