Format: Hardback, 548 pages, height x width: 235x155 mm,
4 Illustrations, black and white; XII, 548 p. 4 illus.
Series: Springer Monographs in Mathematics
Pub. Date: 06-Sep-2022
ISBN-13: 9789811911842
This book furnishes a comprehensive treatment of differential graded Lie algebras, L-infinity algebras, and their use in deformation theory. We believe it is the first textbook devoted to this subject, although the first chapters are also covered in other sources with a different perspective.
Deformation theory is an important subject in algebra and algebraic geometry, with an origin that dates back to Kodaira, Spencer, Kuranishi, Gerstenhaber, and Grothendieck. In the last 30 years, a new approach, based on ideas from rational homotopy theory, has made it possible not only to solve long-standing open problems, but also to clarify the general theory and to relate apparently different features. This approach works over a field of characteristic 0, and the central role is played by the notions of differential graded Lie algebra, L-infinity algebra, and Maurer?Cartan equations.
The book is written keeping in mind graduate students with a basic knowledge of homological algebra and complex algebraic geometry as utilized, for instance, in the book by K. Kodaira, Complex Manifolds and Deformation of Complex Structures. Although the main applications in this book concern deformation theory of complex manifolds, vector bundles, and holomorphic maps, the underlying algebraic theory also applies to a wider class of deformation problems, and it is a prerequisite for anyone interested in derived deformation theory.
Researchers in algebra, algebraic geometry, algebraic topology, deformation theory, and noncommutative geometry are the major targets for the book.
1. An Overview of Deformation Theory of Complex Manifolds.-
2. Lie Algebras.-
3. Functors of Artin Rings.-
4. Infinitesimal Deformations of Complex Manifolds and Vector Bundles.-
5. Differential Graded Lie Algebras.-
6. Maurer-Cartan Equation and Deligne Groupoids.-
7. Totalization and Descent of Deligne Groupoids.-
8. Deformations of Complex Manifolds and Holomorphic Maps.-
9. Poisson, Gerstenhaber and Batalin-Vilkovisky Algebras.-
10. L1-algebras.-
11. Coalgebras and Coderivations.-
12. L1-morphisms.-
13. Formal Kuranishi Families and Period Maps.- References.
Format: Hardback, 323 pages, height x width: 235x155 mm,
14 Tables, color; 15 Illustrations, color; 8 Illustrations, black and white;
XII, 323 p. 23 illus., 15 illus. in color.
Series: Universitext
Pub. Date: 15-Jun-2022
ISBN-13: 9783030979010
This text is intended for a one-semester course in cryptography at the advanced undergraduate/Master's degree level. It is suitable for students from various STEM backgrounds, including engineering, mathematics, and computer science, and may also be attractive for researchers and professionals who want to learn the basics of cryptography. Advanced knowledge of computer science or mathematics (other than elementary programming skills) is not assumed. The book includes more material than can be covered in a single semester. The Preface provides a suggested outline for a single semester course, though instructors are encouraged to select their own topics to reflect their specific requirements and interests. Each chapter contains a set of carefully written exercises which prompts review of the material in the chapter and expands on the concepts. Throughout the book, problems are stated mathematically, then algorithms are devised to solve the problems. Students are tasked to write computer programs (in C++ or GAP) to implement the algorithms. The use of programming skills to solve practical problems adds extra value to the use of this text.
This book combines mathematical theory with practical applications to computer information systems. The fundamental concepts of classical and modern cryptography are discussed in relation to probability theory, complexity theory, modern algebra, and number theory. An overarching theme is cyber security: security of the cryptosystems and the key generation and distribution protocols, and methods of cryptanalysis (i.e., code breaking). It contains chapters on probability theory, information theory and entropy, complexity theory, and the algebraic and number theoretic foundations of cryptography. The book then reviews symmetric key cryptosystems, and discusses one-way trap door functions and public key cryptosystems including RSA and ElGamal. It contains a chapter on digital signature schemes, including material on message authentication and forgeries, and chapters on key generation and distribution. It contains a chapter on elliptic curve cryptography, including new material on the relationship between singular curves, algebraic groups and Hopf algebras.
1. Introduction to Cryptography.-
2. Introduction to Probability.-
3. Information Theory and Entropy.-
4. Introduction to Complexity Theory.-
5. Algebraic Foundations: Groups-
6. Algebraic Foundations: Rings and Fields.-
7. Advanced Topics in Algebra.- 8.Symmetric Key Cryptography.-
9. Public Key Cryptography.-
10. Digital Signature Schemes.-
11. Key Generation.-
12. Key Distribution.-
13. Elliptic Curves in Cryptography.-
14. Singular Curves.-
15. Bibliography.- Index.
Format: Paperback / softback, 87 pages,
height x width: 235x155 mm, 13 Tables, color;
15 Illustrations, color; 32 Illustrations, black and white;
XVIII, 87 p. 47 illus., 15 illus. in color
Series: Lecture Notes in Mathematics 2303
Pub. Date: 16-Aug-2022
ISBN-13: 9783031044274
This volume provides a unified and accessible account of recent developments regarding the real homotopy type of configuration spaces of manifolds. Configuration spaces consist of collections of pairwise distinct points in a given manifold, the study of which is a classical topic in algebraic topology. One of this theory’s most important questions concerns homotopy invariance: if a manifold can be continuously deformed into another one, then can the configuration spaces of the first manifold be continuously deformed into the configuration spaces of the second? This conjecture remains open for simply connected closed manifolds. Here, it is proved in characteristic zero (i.e. restricted to algebrotopological invariants with real coefficients), using ideas from the theory of operads. A generalization to manifolds with boundary is then considered. Based on the work of Campos, Ducoulombier, Lambrechts, Willwacher, and the author, the book covers a vast array of topics, including rational homotopy theory, compactifications, PA forms, propagators, Kontsevich integrals, and graph complexes, and will be of interest to a wide audience.
Overview of the volume.- Configuration spaces of manifolds.- Configuration spaces of closed manifolds.- Configuration spaces of manifolds with boundary.- Configuration spaces and operads.
Bibliog. data: 1st ed. 2022. 2023.
xiii, 578 S. XIII, 578 p. 468 illus. 235 mm
Series: Problem Books in Mathematics
ISBN-13: 9783030978488
This textbook teaches the transformations of plane Euclidean geometry through problems, offering a transformation-based perspective on problems that have appeared in recent years at mathematics competitions around the globe, as well as on some classical examples and theorems. It is based on the combined teaching experience of the authors (coaches of several Mathematical Olympiad teams in Brazil, Romania and the USA) and presents comprehensive theoretical discussions of isometries, homotheties and spiral similarities, and inversions, all illustrated by examples and followed by myriad problems left for the reader to solve. These problems were carefully selected and arranged to introduce students to the topics by gradually moving from basic to expert level. Most of them have appeared in competitions such as Mathematical Olympiads or in mathematical journals aimed at an audience interested in mathematics competitions, while some are fundamental facts of mathematics discussed in the framework of geometric transformations. The book offers a global view of the geometric content of today"s mathematics competitions, bringing many new methods and ideas to the attention of the public.Talented high school and middle school students seeking to improve their problem-solving skills can benefit from this book, as well as high school and college instructors who want to add nonstandard questions to their courses. People who enjoy solving elementary math problems as a hobby will also enjoy this work.
Introduction.- Part I: Problems - 1. Isometries.- 2. Homotheties and Spiral Similarities.- 3. Inversions.- 4. A Synthesis.- Part II: Hints - 5. Isometries.- 6. Homotheties and Spiral Similarities.- 7. Inversions.- 8. A Synthesis.- Part III: Solutions - 9. Isometries.- 10. Homotheties and Spiral Similarities.- 11. Inversions.- 12. A Synthesis.
Razvan Gelca is a Professor at Texas Tech University, USA, and holds a PhD in Mathematics from the University of Iowa, USA (1997). He coached the US International Mathematical Olympiad Team and co-authored, with Titu Andreescu, the books "Putnam and Beyond," published by Springer, and "Mathematical Olympiad Challenges" (Birkhauser), both now in their 2nd edition. Ionut Onisor is a Professor at the Colegiul National de Informatica Tudor Vianu, Romania. Carlos Yuzo Shine teaches at the Colegio Etapa, Brazil. He studied Production Engineering at the University of S?o Paulo, Brazil, and holds a Master"s degree (2018) from Texas A&M University, USA. He has been coordinating and coaching Mathematical Olympiad teams in Brazil since 2002.