Using techniques from abstract algebraic geometry that have been developed over recent decades, Professor Fujita develops classification theories of such pairs using invariants that are polarized higher-dimensional versions of the genus of algebraic curves. The heart of the book is the theory of D-genus and sectional genus developed by the author, but numerous related topics are discussed or surveyed. Proofs are given in full in the central part of the development, but background and technical results are sometimes sketched in when the details are not essential for understanding the key ideas.

PMThis volume is devoted to the use of helices as a method for studying exceptional vector bundles, an important and natural concept in algebraic geometry. The work arises out of a series of seminars organized in Moscow by A.N. Rudakov. The first article sets up the general machinery, and later ones explore its use in various contexts. As to be expected, the approach is concrete; the theory is considered for quadrics, ruled surfaces, K3 surfaces and PP^T3(C

This is the most comprehensive survey of the mathematical life of the legendary Paul Erdos (1913-1996), one of the most versatile and prolific mathematicians of our time. For the first time, all the main areas of Erdos' research are covered in a single project. Because of overwhelming response from the mathematical community, the project now occupies over 1000 pages, arranged into two volumes. These volumes contain both high level research articles as well as key articles that survey some of the cornerstones of Erdos' work, each written by a leading world specialist in the field. A special chapter "Early Days", rare photographs, and art related to Erdos complement this striking collection. A unique contribution is the bibliography on Erdos' publications: the most comprehensive ever published. This new edition, dedicated to the 100th anniversary of Paul Erdos' birth, contains updates on many of the articles from the two volumes of the first edition, several new articles from prominent mathematicians, a new introduction, and more biographical information about Paul Erdos with an updated list of publications. The second volume contains chapters on graph theory and combinatorics, extremal and Ramsey theory, and a section on infinity that covers Erdos' research on set theory. All of these chapters are essentially updated, particularly the extremal theory chapter that contains a survey of flag algebras, a new technique for solving extremal problems.

This is the fifth conference in a bi-annual series, following conferences in Besancon, Limoges, Irsee and Toronto. The meeting aims to bring together different strands of research in and closely related to the area of Iwasawa theory. During the week before the conference in a kind of summer school a series of preparatory lectures for young mathematicians was provided as an introduction to Iwasawa theory.

Iwasawa theory is a modern and powerful branch of number theory and can be traced back to the Japanese mathematician Kenkichi Iwasawa, who introduced the systematic study of Z_p-extensions and p-adic L-functions, concentrating on the case of ideal class groups. Later this would be generalized to elliptic curves. Over the last few decades considerable progress has been made in automorphic Iwasawa theory, e.g. the proof of the Main Conjecture for GL(2) by Kato and Skinner & Urban. Techniques such as Hida s theory of p-adic modular forms and big Galois representations play a crucial part. Also a non-commutative Iwasawa theory of arbitrary p-adic Lie extensions has been developed. This volume aims to present a snapshot of the state of art of Iwasawa theory as of 2012. In particular it offers an introduction to Iwasawa theory (based on a preparatory course by Chris Wuthrich) and a survey of the proof of Skinner & Urban (based on a lecture course by Xin Wan)."

This textbook provides an essential introduction to Lie groups, presenting the theory from its fundamental principles. Lie groups are a special class of groups that are studied using differential and integral calculus methods. As a mathematical structure, a Lie group combines the algebraic group structure and the differentiable variety structure. Studies of such groups began around 1870 as groups of symmetries of differential equations and the various geometries that had emerged. Since that time, there have been major advances in Lie theory, with ramifications for diverse areas of mathematics and its applications. Each chapter of the book begins with a general, straightforward introduction to the concepts covered; then the formal definitions are presented; and end-of-chapter exercises help to check and reinforce comprehension. Graduate and advanced undergraduate students alike will find in this book a solid yet approachable guide that will help them continue their studies with confidence.

This book includes selected papers presented at the MIMS (Mediterranean Institute for the Mathematical Sciences) - GGTM (Geometry and Topology Grouping for the Maghreb) conference, held in memory of Mohammed Salah Baouendi, a most renowned figure in the field of several complex variables, who passed away in 2011. All research articles were written by leading experts, some of whom are prize winners in the fields of complex geometry, algebraic geometry and analysis. The book offers a valuable resource for all researchers interested in recent developments in analysis and geometry.

This book provides an overview of the latest progress on rationality questions in algebraic geometry. It discusses new developments such as universal triviality of the Chow group of zero cycles, various aspects of stable birationality, cubic and Fano fourfolds, rationality of moduli spaces and birational invariants of group actions on varieties, contributed by the foremost experts in their fields. The question of whether an algebraic variety can be parametrized by rational functions of as many variables as its dimension has a long history and played an important role in the history of algebraic geometry. Recent developments in algebraic geometry have made this question again a focal point of research and formed the impetus to organize a conference in the series of conferences on the island of Schiermonnikoog. The book follows in the tradition of earlier volumes, which originated from conferences on the islands Texel and Schiermonnikoog.

This book includes 58 selected articles that highlight the major contributions of Professor Radha Charan Gupta-a doyen of history of mathematics-written on a variety of important topics pertaining to mathematics and astronomy in India. It is divided into ten parts. Part I presents three articles offering an overview of Professor Gupta's oeuvre. The four articles in Part II convey the importance of studies in the history of mathematics. Parts III-VII constituting 33 articles, feature a number of articles on a variety of topics, such as geometry, trigonometry, algebra, combinatorics and spherical trigonometry, which not only reveal the breadth and depth of Professor Gupta's work, but also highlight his deep commitment to the promotion of studies in the history of mathematics. The ten articles of part VIII, present interesting bibliographical sketches of a few veteran historians of mathematics and astronomy in India. Part IX examines the dissemination of mathematical knowledge across different civilisations. The last part presents an up-to-date bibliography of Gupta's work. It also includes a tribute to him in Sanskrit composed in eight verses.

This volume collects papers based on talks given at the conference "Geometrias'19: Polyhedra and Beyond", held in the Faculty of Sciences of the University of Porto between September 5-7, 2019 in Portugal. These papers explore the conference's theme from an interdisciplinary standpoint, all the while emphasizing the relevance of polyhedral geometry in contemporary academic research and professional practice. They also investigate how this topic connects to mathematics, art, architecture, computer science, and the science of representation. Polyhedra and Beyond will help inspire scholars, researchers, professionals, and students of any of these disciplines to develop a more thorough understanding of polyhedra.

The aim of this monograph is to give an overview of various classes of in?ni- dimensional Lie groups and their applications, mostly in Hamiltonian - chanics, ?uid dynamics, integrable systems, and complex geometry. We have chosen to present the unifying ideas of the theory by concentrating on speci?c typesandexamplesofin?nite-dimensionalLiegroups. Ofcourse, theselection of the topics is largely in?uenced by the taste of the authors, but we hope thatthisselectioniswideenoughtodescribevariousphenomenaarisinginthe geometry of in?nite-dimensional Lie groups and to convince the reader that they are appealing objects to study from both purely mathematical and more applied points of view. This book can be thought of as complementary to the existing more algebraic treatments, in particular, those covering the str- ture and representation theory of in?nite-dimensional Lie algebras, as well as to more analytic ones developing calculus on in?nite-dimensional manifolds. This monograph originated from advanced graduate courses and mi- courses on in?nite-dimensional groups and gauge theory given by the ?rst author at the University of Toronto, at the CIRM in Marseille, and at the Ecole Polytechnique in Paris in 2001-2004. It is based on various classical and recentresultsthathaveshapedthisnewlyemergedpartofin?nite-dimensional geometry and group theory. Our intention was to make the book concise, relatively self-contained, and useful in a graduate course. For this reason, throughout the text, we have included a large number of problems, ranging from simple exercises to open questions

This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A key theme of the Conference and Summer School was the interplay between theory, algorithms and experiment. The 14 papers offer readers both, instructional courses on the latest algorithms for computing modular and automorphic forms, as well as original research articles reporting on the latest developments in the field. The three Summer School lectures provide an introduction to modern algorithms together with some theoretical background for computations of and with modular forms, including computing cohomology of arithmetic groups, algebraic automorphic forms, and overconvergent modular symbols. The 11 Conference papers cover a wide range of themes related to computations with modular forms, including lattice methods for algebraic modular forms on classical groups, a generalization of the Maeda conjecture, an efficient algorithm for special values of p-adic Rankin triple product L-functions, arithmetic aspects and experimental data of Bianchi groups, a theoretical study of the real Jacobian of modular curves, results on computing weight one modular forms, and more.

The International Mathematical Olympiad (IMO) is the World Championship Competition for High School students, and is held annually in a different country. More than eighty countries are involved.
Containing numerous exercises, illustrations, hints and solutions, presented in a lucid and thought- provoking style, this text provides a wide range of skills required in competitions such as the Mathematical Olympiad.
More than fifty problems in Euclidean geometry invo9lving integers and rational numbers are presented. Early chapters cover elementary problems while later sections break new ground in certain areas and area greater challenge for the more adventurous reader. The text is ideal for Mathematical
Olympiad training and also serves as a supplementary text for student in pure mathematics, particularly number theory and geometry.
Dr. Christopher Bradley was formerly a Fellow and Tutor in Mathematics at Jesus College, Oxford, Deputy Leader of the British Mathematical Olympiad Team and for several years Secretary of the British Mathematical Olympiad Committee.

This seminal text on Fourier-Mukai Transforms in Algebraic Geometry by a leading researcher and expositor is based on a course given at the Institut de Mathematiques de Jussieu in 2004 and 2005. Aimed at postgraduate students with a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. Including notions from other areas, e.g. singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs are given and exercises aid the reader throughout.

This book investigates the geometry of the quaternion and octonion algebras. Following a comprehensive historical introduction, the special properties of 3- and 4-dimensional Euclidean spaces are illuminated using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less familiar octonion algebra, concentrating on its remarkable "triality symmetry" after an appropriate study of Moufang loops. The arithmetics of the quaternions and octonions are also described, and the book concludes with a new theory of octonion factorization. Topics covered include: - history - the geometry of complex numbers - quaternions and 3-dimensional groups - quaternions and 4-dimensional groups - the Hurwitz integral quaternions - the composition algebras - Moufang loops - octonions and 8-dimensional geometry - integral octonions - the octonion projective plane

This volume presents modern trends in the area of symmetries and their applications based on contributions from the workshop "Lie Theory and Its Applications in Physics", held near Varna, Bulgaria, in June 2015. Traditionally, Lie theory is a tool to build mathematical models for physical systems.Recently, the trend has been towards geometrization of the mathematical description of physical systems and objects. A geometric approach to a system yields in general some notion of symmetry, which is very helpful in understanding its structure. Geometrization and symmetries are employed in their widest sense, embracing representation theory, algebraic geometry, number theory, infinite-dimensional Lie algebras and groups, superalgebras and supergroups, groups and quantum groups, noncommutative geometry, symmetries of linear and nonlinear partial differential operators (PDO), special functions, and others. Furthermore, the necessary tools from functional analysis are included.

The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sharing, frameproof codes, and broadcast encryption. Suitable for researchers and graduate students in mathematics and computer science, this self-contained book is one of the first to focus on many topics in cryptography involving algebraic curves. After supplying the necessary background on algebraic curves, the authors discuss error-correcting codes, including algebraic geometry codes, and provide an introduction to elliptic curves. Each chapter in the remainder of the book deals with a selected topic in cryptography (other than elliptic curve cryptography). The topics covered include secret sharing schemes, authentication codes, frameproof codes, key distribution schemes, broadcast encryption, and sequences. Chapters begin with introductory material before featuring the application of algebraic curves.

Covering topics in algebraic geometry, coding theory, and cryptography, this volume presents interdisciplinary group research completed for the February 2016 conference at the Institute for Pure and Applied Mathematics (IPAM) in cooperation with the Association for Women in Mathematics (AWM). The conference gathered research communities across disciplines to share ideas and problems in their fields and formed small research groups made up of graduate students, postdoctoral researchers, junior faculty, and group leaders who designed and led the projects. Peer reviewed and revised, each of this volume's five papers achieves the conference's goal of using algebraic geometry to address a problem in either coding theory or cryptography. Proposed variants of the McEliece cryptosystem based on different constructions of codes, constructions of locally recoverable codes from algebraic curves and surfaces, and algebraic approaches to the multicast network coding problem are only some of the topics covered in this volume. Researchers and graduate-level students interested in the interactions between algebraic geometry and both coding theory and cryptography will find this volume valuable.

The first part of this book provides an elementary and self-contained exposition of classical Galois theory and its applications to the questions of solvability of algebraic equations in explicit form. The second part describes a surprising analogy between the fundamental theorem of Galois theory and classification of coverings over a topological space. The third part contains a geometric description of finite algebraic extensions of the field of meromorphic functions on a Riemann surface and provides an introduction to topological Galois theory developed by the author. All results are presented in the same elementary and self-contained manner as classical Galois theory. Due to this feature, the book will be useful and interesting to readers with very different background in mathematics, from undergraduate students to researchers.

Based on the author's course for first-year graduate students this well-written text explains how the tools of algebraic geometry and of number theory can be applied to a study of curves. The book starts by introducing the essential background material and includes 600 exercises.

"...A nice feature of the book [is] that at various points the authors provide examples, or rather counterexamples, that clearly show what can go wrong...This is a nicely-written book [that] studies algebraic differential modules in several variables." --Mathematical Reviews

This book is a collection of the lectures and talks presented in the Tohoku Forum for Creativity in the thematic year 2015 'Fundamental Problems in Quantum Physics: Strings, Black Holes and Quantum Information', and related events in the period 2014-2016.This volume especially contains an overview of recent developments in the theory of strings and membranes, as well as topological field theory.

Recent developments in various algebraic structures and the applications of those in different areas play an important role in Science and Technology. One of the best tools to study the non-linear algebraic systems is the theory of Near-rings.The forward note by G

Based on lectures held at the 8th edition of the series of summer schools in Villa de Leyva since 1999, this book presents an introduction to topics of current interest at the interface of geometry, algebra, analysis, topology and theoretical physics. It is aimed at graduate students and researchers in physics or mathematics, and offers an introduction to the topics discussed in the two weeks of the summer school: operator algebras, conformal field theory, black holes, relativistic fluids, Lie groupoids and Lie algebroids, renormalization methods, spectral geometry and index theory for pseudo-differential operators.

This book provides comprehensive analysis of dynamical systems in tropical geometry, which include the author's significant discoveries and pioneering contributions. Tropical geometry is a kind of dynamical scale transform which connects real rational dynamics with piecewise linear one presented by max and plus algebras. A comparison method is given which estimates orbits corresponding to different rational dynamics by reduction to the piecewise linear dynamics.Both rational and piecewise linear dynamics appear in many important branches of mathematics. Tropical geometry can play a role or function to bridge between different subjects in mathematics. This book contains detailed accounts of basic strategy on how to apply tropical geometry to analysis in various mathematical subjects by presenting several applications which include: a rough classification of partial differential equations from the point of view of global behavior of solutions; construction of the infinite quasi-recursive rational dynamics, based on the automaton of the Burnside group by Aleshin-Grigorchuk; study on nearly periodicity of the pentagram map on the moduli space of the twisted polygons; spectral coincidence between lamplighter group in theory of automata groups and Box and ball systems corresponding to KdV equation in soliton theory.This book is self-contained, and detailed accounts of theory of automata groups, BBS and the pentagram map are also included.