This unique book explores the world of q, known technically as basic hypergeometric series, and represents the author's personal and life-long study-inspired by Ramanujan-of aspects of this broad topic. While the level of mathematical sophistication is graduated, the book is designed to appeal to advanced undergraduates as well as researchers in the field. The principal aims are to demonstrate the power of the methods and the beauty of the results. The book contains novel proofs of many results in the theory of partitions and the theory of representations, as well as associated identities. Though not specifically designed as a textbook, parts of it may be presented in course work; it has many suitable exercises. After an introductory chapter, the power of q-series is demonstrated with proofs of Lagrange's four-squares theorem and Gauss's two-squares theorem. Attention then turns to partitions and Ramanujan's partition congruences. Several proofs of these are given throughout the book. Many chapters are devoted to related and other associated topics. One highlight is a simple proof of an identity of Jacobi with application to string theory. On the way, we come across the Rogers-Ramanujan identities and the Rogers-Ramanujan continued fraction, the famous "forty identities" of Ramanujan, and the representation results of Jacobi, Dirichlet and Lorenz, not to mention many other interesting and beautiful results. We also meet a challenge of D.H. Lehmer to give a formula for the number of partitions of a number into four squares, prove a "mysterious" partition theorem of H. Farkas and prove a conjecture of R.Wm. Gosper "which even Erdos couldn't do." The book concludes with a look at Ramanujan's remarkable tau function.

The main purpose of these lectures is first to briefly survey the fundamental con nection between the representation theory of the symmetric group Sn and the theory of symmetric functions and second to show how combinatorial methods that arise naturally in the theory of symmetric functions lead to efficient algorithms to express various prod ucts of representations of Sn in terms of sums of irreducible representations. That is, there is a basic isometry which maps the center of the group algebra of Sn, Z(Sn), to the space of homogeneous symmetric functions of degree n, An. This basic isometry is known as the Frobenius map, F. The Frobenius map allows us to reduce calculations involving characters of the symmetric group to calculations involving Schur functions. Now there is a very rich and beautiful theory of the combinatorics of symmetric functions that has been developed in recent years. The combinatorics of symmetric functions, then leads to a number of very efficient algorithms for expanding various products of Schur functions into a sum of Schur functions. Such expansions of products of Schur functions correspond via the Frobenius map to decomposing various products of irreducible representations of Sn into their irreducible components. In addition, the Schur functions are also the characters of the irreducible polynomial representations of the general linear group over the complex numbers GLn(C)."

Number theory has a wealth of long-standing problems, the study of which over the years has led to major developments in many areas of mathematics. This volume consists of seven significant chapters on number theory and related topics. Written by distinguished mathematicians, key topics focus on multipartitions, congruences and identities (G. Andrews), the formulas of Koshliakov and Guinand in Ramanujan's Lost Notebook (B.C. Berndt, Y. Lee, and J. Sohn), alternating sign matrices and the Weyl character formulas (D.M. Bressoud), theta functions in complex analysis (H.M. Farkas), representation functions in additive number theory (M.B. Nathanson), and mock theta functions, ranks, and Maass forms (K. Ono), and elliptic functions (M. Waldschmidt).

All of the surveys were outgrowths of featured talks given during the Special Year in Number Theory and Combinatorics at the University of Florida, Gainesville, 2004-2005, and describe major progress on a broad range of topics.

This volume is intended for mathematicians and graduate students interested in number theory and related areas.

This book is devoted to the study of rational and integral points on higher- dimensional algebraic varieties. It contains research papers addressing the arithmetic geometry of varieties which are not of general type, with an em- phasis on how rational points are distributed with respect to the classical, Zariski and adelic topologies. The book gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. The techniques involve explicit geometric con- structions, ideas from the minimal model program in algebraic geometry as well as analytic number theory and harmonic analysis on adelic groups. In recent years there has been substantial progress in our understanding of the arithmetic of algebraic surfaces. Five papers are devoted to cubic surfaces: Basile and Fisher study the existence of rational points on certain diagonal cubics, Swinnerton-Dyer considers weak approximation and Broberg proves upper bounds on the number of rational points on the complement to lines on cubic surfaces. Peyre and Tschinkel compare numerical data with conjectures concerning asymptotics of rational points of bounded height on diagonal cubics of rank ~ 2. Kanevsky and Manin investigate the composition of points on cubic surfaces. Satge constructs rational curves on certain Kummer surfaces. Colliot-Thelene studies the Hasse principle for pencils of curves of genus 1. In an appendix to this paper Skorobogatov produces explicit examples of Enriques surfaces with a Zariski dense set of rational points.

This book covers the following three topics in a manner accessible to graduate students who have an understanding of algebraic number theory and scheme theoretic algebraic geometry:

1. An elementary construction of Shimura varieties as moduli of abelian schemes

2. p-adic deformation theory of automorphic forms on Shimura varieties

3. A simple proof of irreducibility of the generalized Igusa tower over the Shimura variety

The book starts with a detailed study of elliptic and Hilbert modular forms and reaches to the forefront of research of Shimura varieties associated with general classical groups. The method of constructing p-adic analytic families and the proof of irreducibility was recently discovered by the author. The area covered in this book is now a focal point of research worldwide with many far-reaching applications that have led to solutions of longstanding problems and conjectures. Specifically, the use of p-adic elliptic and Hilbert modular forms have proven essential in recent breakthroughs in number theory (for example, the proof of Fermat's Last Theorem and the Shimura-Taniyama conjecture by A. Wiles and others).

Haruzo Hida is Professor of Mathematics at University of California, Los Angeles. His previous books include Modular Forms and Galois Cohomology (Cambridge University Press 2000) and Geometric Modular Forms and Elliptic Curves (World Scientific Publishing Company 2000).

Finslerian Laplacians have arisen from the demands of modelling the modern world. However, the roots of the Laplacian concept can be traced back to the sixteenth century. Its phylogeny and history are presented in the Prologue of this volume. The text proper begins with a brief introduction to stochastically derived Finslerian Laplacians, facilitated by applications in ecology, epidemiology and evolutionary biology. The mathematical ideas are then fully presented in section II, with generalizations to Lagrange geometry following in section III. With section IV, the focus abruptly shifts to the local mean-value approach to Finslerian Laplacians and a Hodge-de Rham theory is developed for the representation on real cohomology classes by harmonic forms on the base manifold. Similar results are proved in sections II and IV, each from different perspectives. Modern topics treated include nonlinear Laplacians, Bochner and Lichnerowicz vanishing theorems, WeitzenbAck formulas, and Finslerian spinors and Dirac operators. The tools developed in this book will find uses in several areas of physics and engineering, but especially in the mechanics of inhomogeneous media, e.g. Cofferat continua. Audience: This text will be of use to workers in stochastic processes, differential geometry, nonlinear analysis, epidemiology, ecology and evolution, as well as physics of the solid state and continua.

This contributed volume provides readers with an overview of the most recent developments in the mathematical fields related to fractals, including both original research contributions, as well as surveys from many of the leading experts on modern fractal theory and applications. It is an outgrowth of the Conference of Fractals and Related Fields III, that was held on September 19-25, 2015 in ile de Porquerolles, France. Chapters cover fields related to fractals such as harmonic analysis, multifractal analysis, geometric measure theory, ergodic theory and dynamical systems, probability theory, number theory, wavelets, potential theory, partial differential equations, fractal tilings, combinatorics, and signal and image processing. The book is aimed at pure and applied mathematicians in these areas, as well as other researchers interested in discovering the fractal domain.

This interdisciplinary book covers a wide range of subjects, from pure mathematics (knots, braids, homotopy theory, number theory) to more applied mathematics (cryptography, algebraic specification of algorithms, dynamical systems) and concrete applications (modeling of polymers and ionic liquids, video, music and medical imaging). The main mathematical focus throughout the book is on algebraic modeling with particular emphasis on braid groups. The research methods include algebraic modeling using topological structures, such as knots, 3-manifolds, classical homotopy groups, and braid groups. The applications address the simulation of polymer chains and ionic liquids, as well as the modeling of natural phenomena via topological surgery. The treatment of computational structures, including finite fields and cryptography, focuses on the development of novel techniques. These techniques can be applied to the design of algebraic specifications for systems modeling and verification. This book is the outcome of a workshop in connection with the research project Thales on Algebraic Modeling of Topological and Computational Structures and Applications, held at the National Technical University of Athens, Greece in July 2015. The reader will benefit from the innovative approaches to tackling difficult questions in topology, applications and interrelated research areas, which largely employ algebraic tools.

The book describes developments on some well-known problems regarding the relationship between orders of finite groups and that of their automorphism groups. It is broadly divided into three parts: the first part offers an exposition of the fundamental exact sequence of Wells that relates automorphisms, derivations and cohomology of groups, along with some interesting applications of the sequence. The second part offers an account of important developments on a conjecture that a finite group has at least a prescribed number of automorphisms if the order of the group is sufficiently large. A non-abelian group of prime-power order is said to have divisibility property if its order divides that of its automorphism group. The final part of the book discusses the literature on divisibility property of groups culminating in the existence of groups without this property. Unifying various ideas developed over the years, this largely self-contained book includes results that are either proved or with complete references provided. It is aimed at researchers working in group theory, in particular, graduate students in algebra.

A new and complete treatment of semi-abelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of Siegel moduli space, with most of the results being published for the first time. Highlights of the book include a classification of semi-abelian schemes, construction of the toroidal and the minimal compactification over the integers, heights for abelian varieties over number fields, and Eichler integrals in several variables, together with a new approach to Siegel modular forms. A valuable source of reference for researchers and graduate students interested in algebraic geometry, Shimura varieties or diophantine geometry.

This book includes a self-contained approach of the general theory of quadratic forms and integral Euclidean lattices, as well as a presentation of the theory of automorphic forms and Langlands' conjectures, ranging from the first definitions to the recent and deep classification results due to James Arthur. Its connecting thread is a question about lattices of rank 24: the problem of p-neighborhoods between Niemeier lattices. This question, whose expression is quite elementary, is in fact very natural from the automorphic point of view, and turns out to be surprisingly intriguing. We explain how the new advances in the Langlands program mentioned above pave the way for a solution. This study proves to be very rich, leading us to classical themes such as theta series, Siegel modular forms, the triality principle, L-functions and congruences between Galois representations. This monograph is intended for any mathematician with an interest in Euclidean lattices, automorphic forms or number theory. A large part of it is meant to be accessible to non-specialists.

The aim of this book is to familiarize the reader with fundamental topics in number theory: theory of divisibility, arithmetrical functions, prime numbers, geometry of numbers, additive number theory, probabilistic number theory, theory of Diophantine approximations and algebraic number theory. The author tries to show the connection between number theory and other branches of mathematics with the resultant tools adopted in the book ranging from algebra to probability theory, but without exceeding the undergraduate students who wish to be acquainted with number theory, graduate students intending to specialize in this field and researchers requiring the present state of knowledge.

A conference on Harmonic Analysis on Reductive Groups was held at Bowdoin College in Brunswick, Maine from July 31 to August 11, 1989. The stated goal of the conference was to explore recent advances in harmonic analysis on both real and p-adic groups. It was the first conference since the AMS Summer Sym posium on Harmonic Analysis on Homogeneous Spaces, held at Williamstown, Massachusetts in 1972, to cover local harmonic analysis on reductive groups in such detail and to such an extent. While the Williamstown conference was longer (three weeks) and somewhat broader (nilpotent groups, solvable groups, as well as semisimple and reductive groups), the structure and timeliness of the two meetings was remarkably similar. The program of the Bowdoin Conference consisted of two parts. First, there were six major lecture series, each consisting of several talks addressing those topics in harmonic analysis on real and p-adic groups which were the focus of intensive research during the previous decade. These lectures began at an introductory level and advanced to the current state of research. Sec ond, there was a series of single lectures in which the speakers presented an overview of their latest research."

New statements of problems arose recently demanding thorough ana lysis. Notice, first of all, the statements of problems using adjoint equations which gradually became part of our life. Adjoint equations are capable to bring fresh ideas to various problems of new technology based on linear and nonlinear processes. They became part of golden fund of science through quantum mechanics, theory of nuclear reactors, optimal control, and finally helped in solving many problems on the basis of perturbation method and sensitivity theory. To emphasize the important role of adjoint problems in science one should mention four-dimensional analysis problem and solution of inverse problems. This range of problems includes first of all problems of global climate changes on our planet, state of environment and protection of environ ment against pollution, preservation of the biosphere in conditions of vigorous growth of population, intensive development of industry, and many others. All this required complex study of large systems: interac tion between the atmosphere and oceans and continents in the theory of climate, cenoses in the biosphere affected by pollution of natural and anthropogenic origin. Problems of local and global perturbations and models sensitivity to input data join into common complex system."

This textbook introduces exciting new developments and cutting-edge results on the theme of hyperbolicity. Written by leading experts in their respective fields, the chapters stem from mini-courses given alongside three workshops that took place in Montreal between 2018 and 2019. Each chapter is self-contained, including an overview of preliminaries for each respective topic. This approach captures the spirit of the original lectures, which prepared graduate students and those new to the field for the technical talks in the program. The four chapters turn the spotlight on the following pivotal themes: The basic notions of o-minimal geometry, which build to the proof of the Ax-Schanuel conjecture for variations of Hodge structures; A broad introduction to the theory of orbifold pairs and Campana's conjectures, with a special emphasis on the arithmetic perspective; A systematic presentation and comparison between different notions of hyperbolicity, as an introduction to the Lang-Vojta conjectures in the projective case; An exploration of hyperbolicity and the Lang-Vojta conjectures in the general case of quasi-projective varieties. Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces is an ideal resource for graduate students and researchers in number theory, complex algebraic geometry, and arithmetic geometry. A basic course in algebraic geometry is assumed, along with some familiarity with the vocabulary of algebraic number theory.

Combinatorics and finite fields are of great importance in modern applications such as in the analysis of algorithms, in information and communication theory, and in signal processing and coding theory. This book contains survey articles on topics such as difference sets, polynomials, and pseudorandomness.

Linear Algebra: Algorithms, Applications, and Techniques, Fourth Edition offers a modern and algorithmic approach to computation while providing clear and straightforward theoretical background information. The book guides readers through the major applications, with chapters on properties of real numbers, proof techniques, matrices, vector spaces, linear transformations, eigen values, and Euclidean inner products. Appendices on Jordan canonical forms and Markov chains are included for further study. This useful textbook presents broad and balanced views of theory, with key material highlighted and summarized in each chapter. To further support student practice, the book also includes ample exercises with answers and hints.

This book is devoted to mean-square and weak approximations of solutions of stochastic differential equations (SDE). These approximations represent two fundamental aspects in the contemporary theory of SDE. Firstly, the construction of numerical methods for such systems is important as the solutions provided serve as characteristics for a number of mathematical physics problems. Secondly, the employment of probability representations together with a Monte Carlo method allows us to reduce the solution of complex multidimensional problems of mathematical physics to the integration of stochastic equations. Along with a general theory of numerical integrations of such systems, both in the mean-square and the weak sense, a number of concrete and sufficiently constructive numerical schemes are considered. Various applications and particularly the approximate calculation of Wiener integrals are also dealt with. This book is of interest to graduate students in the mathematical, physical and engineering sciences, and to specialists whose work involves differential equations, mathematical physics, numerical mathematics, the theory of random processes, estimation and control theory.

A collection of expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held at Boston University. The purpose of the conference, and indeed of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to show, at long last, that Fermats Last Theorem is true. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. In recognition of the historical significance of Fermats Last Theorem, the volume concludes by reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Indispensable for students and professional mathematicians alike.

These proceedings collect several number theory articles, most of which were written in connection to the workshop WIN4: Women in Numbers, held in August 2017, at the Banff International Research Station (BIRS) in Banff, Alberta, Canada. It collects papers disseminating research outcomes from collaborations initiated during the workshop as well as other original research contributions involving participants of the WIN workshops. The workshop and this volume are part of the WIN network, aimed at highlighting the research of women and gender minorities in number theory as well as increasing their participation and boosting their potential collaborations in number theory and related fields.

Model theory is the meta-mathematical study of the concept of mathematical truth. After Afred Tarski coined the term Theory of Models in the early 1950's, it rapidly became one of the central most active branches of mathematical logic. In the last few decades, ideas that originated within model theory have provided powerful tools to solve problems in a variety of areas of classical mathematics, including algebra, combinatorics, geometry, number theory, and Banach space theory and operator theory. The two volumes of Beyond First Order Model Theory present the reader with a fairly comprehensive vista, rich in width and depth, of some of the most active areas of contemporary research in model theory beyond the realm of the classical first-order viewpoint. Each chapter is intended to serve both as an introduction to a current direction in model theory and as a presentation of results that are not available elsewhere. All the articles are written so that they can be studied independently of one another. This second volume contains introductions to real-valued logic and applications, abstract elementary classes and applications, interconnections between model theory and function spaces, nonstucture theory, and model theory of second-order logic. Features A coherent introduction to current trends in model theory. Contains articles by some of the most influential logicians of the last hundred years. No other publication brings these distinguished authors together. Suitable as a reference for advanced undergraduate, postgraduates, and researchers. Material presented in the book (e.g, abstract elementary classes, first-order logics with dependent sorts, and applications of infinitary logics in set theory) is not easily accessible in the current literature. The various chapters in the book can be studied independently.

Even three decades ago, the words 'combinatorial algebra' contrasting, for in stance, the words 'combinatorial topology,' were not a common designation for some branch of mathematics. The collocation 'combinatorial group theory' seems to ap pear first as the title of the book by A. Karras, W. Magnus, and D. Solitar [182] and, later on, it served as the title of the book by R. C. Lyndon and P. Schupp [247]. Nowadays, specialists do not question the existence of 'combinatorial algebra' as a special algebraic activity. The activity is distinguished not only by its objects of research (that are effectively given to some extent) but also by its methods (ef fective to some extent). To be more exact, we could approximately define the term 'combinatorial algebra' for the purposes of this book, as follows: So we call a part of algebra dealing with groups, semi groups , associative algebras, Lie algebras, and other algebraic systems which are given by generators and defining relations {in the first and particular place, free groups, semigroups, algebras, etc. )j a part in which we study universal constructions, viz. free products, lINN-extensions, etc. j and, finally, a part where specific methods such as the Composition Method (in other words, the Diamond Lemma, see [49]) are applied. Surely, the above explanation is far from covering the full scope of the term (compare the prefaces to the books mentioned above).

In the present book, we have put together the basic theory of the units and cuspidal divisor class group in the modular function fields, developed over the past few years. Let i) be the upper half plane, and N a positive integer. Let r(N) be the subgroup of SL (Z) consisting of those matrices == 1 mod N. Then r(N)\i) 2 is complex analytic isomorphic to an affine curve YeN), whose compactifi cation is called the modular curve X(N). The affine ring of regular functions on yeN) over C is the integral closure of C j] in the function field of X(N) over C. Here j is the classical modular function. However, for arithmetic applications, one considers the curve as defined over the cyclotomic field Q(JlN) of N-th roots of unity, and one takes the integral closure either of Q j] or Z j], depending on how much arithmetic one wants to throw in. The units in these rings consist of those modular functions which have no zeros or poles in the upper half plane. The points of X(N) which lie at infinity, that is which do not correspond to points on the above affine set, are called the cusps, because of the way they look in a fundamental domain in the upper half plane. They generate a subgroup of the divisor class group, which turns out to be finite, and is called the cuspidal divisor class group."

This book collects more than thirty contributions in memory of Wolfgang Schwarz, most of which were presented at the seventh International Conference on Elementary and Analytic Number Theory (ELAZ), held July 2014 in Hildesheim, Germany. Ranging from the theory of arithmetical functions to diophantine problems, to analytic aspects of zeta-functions, the various research and survey articles cover the broad interests of the well-known number theorist and cherished colleague Wolfgang Schwarz (1934-2013), who contributed over one hundred articles on number theory, its history and related fields. Readers interested in elementary or analytic number theory and related fields will certainly find many fascinating topical results among the contributions from both respected mathematicians and up-and-coming young researchers. In addition, some biographical articles highlight the life and mathematical works of Wolfgang Schwarz.

Introduction to Number Theory covers the essential content of an introductory number theory course including divisibility and prime factorization, congruences, and quadratic reciprocity. The instructor may also choose from a collection of additional topics. Aligning with the trend toward smaller, essential texts in mathematics, the author strives for clarity of exposition. Proof techniques and proofs are presented slowly and clearly. The book employs a versatile approach to the use of algebraic ideas. Instructors who wish to put this material into a broader context may do so, though the author introduces these concepts in a non-essential way. A final chapter discusses algebraic systems (like the Gaussian integers) presuming no previous exposure to abstract algebra. Studying general systems urges students realize unique factorization into primes is a more subtle idea than may at first appear; students will find this chapter interesting, fun and quite accessible. Applications of number theory include several sections on cryptography and other applications to further interest instructors and students alike.