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Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Advanced Algebra includes chapters on modern algebra which treat various topics in commutative and noncommutative algebra and provide introductions to the theory of associative algebras, homological algebras, algebraic number theory, and algebraic geometry. Many examples and hundreds of problems are included, along with hints or complete solutions for most of the problems. Together the two books give the reader a global view of algebra and its role in mathematics as a whole.

An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects--elliptic curves and modular forms--come together in Eichler-Shimura theory, which constructs elliptic curves out of modular forms of a special kind. The converse, that all rational elliptic curves arise this way, is called the Taniyama-Weil Conjecture and is known to imply Fermat's Last Theorem. Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is, therefore, a window into a broad expanse of mathematics--including class field theory, arithmetic algebraic geometry, and group representations--in which the concidence of L functions relates analysis and algebra in the most fundamental ways. Developing, with many examples, the elementary theory of elliptic curves, the book goes on to the subject of modular forms and the first connections with elliptic curves. The last two chapters concern Eichler-Shimura theory, which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics, and no other explains Eichler-Shimura theory in such an accessible manner.

* Presents a comprehensive treatment with a global view of the subject * Rich in examples, problems with hints, and solutions, the book makes a welcome addition to the library of every mathematician

This book offers a systematic treatment--the first in book form--of the development and use of cohomological induction to construct unitary representations. George Mackey introduced induction in 1950 as a real analysis construction for passing from a unitary representation of a closed subgroup of a locally compact group to a unitary representation of the whole group. Later a parallel construction using complex analysis and its associated co-homology theories grew up as a result of work by Borel, Weil, Harish-Chandra, Bott, Langlands, Kostant, and Schmid. Cohomological induction, introduced by Zuckerman, is an algebraic analog that is technically more manageable than the complex-analysis construction and leads to a large repertory of irreducible unitary representations of reductive Lie groups. The book, which is accessible to students beyond the first year of graduate school, will interest mathematicians and physicists who want to learn about and take advantage of the algebraic side of the representation theory of Lie groups. Cohomological Induction and Unitary Representations develops the necessary background in representation theory and includes an introductory chapter of motivation, a thorough treatment of the "translation principle," and four appendices on algebra and analysis.

In this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes.

Systematically develop the concepts and tools that are vital to every mathematician, whether pure or applied, aspiring or established A comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics Included throughout are many examples and hundreds of problems, and a separate 55-page section gives hints or complete solutions for most.

Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole, presenting the subject matter in a forward-looking way that takes into account its historical development. Three prominent themes recur and blend together at times: the analogy between integers and polynomials in one variable over a field, the interplay between linear algebra and group theory, and the relationship between number theory and geometry. The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them. The presentation includes blocks of problems that introduce additional topics and applications to science and engineering to guide further study. Many examples and hundreds of problems are included, along with separate sections giving hints or complete solutions for most of the problems.

Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole. The presentation includes blocks of problems that introduce additional topics and applications to science and engineering to guide further study. Many examples and hundreds of problems are included, along with a separate 90-page section giving hints or complete solutions for most of the problems.

Basic Real Analysis: Along with a companion volume Advanced Real Analysis by Anthony W. KnappThis book and its companion volume Advanced Real Analysis systematicallydevelop concepts and tools in real analysis that are vital to every mathematician,whether pure or applied, aspiring or established. The two books together containwhat the young mathematician needs to know about real analysis in order tocommunicate well with colleagues in all branches of mathematics.The books are written as textbooks, and their primary audience is students whoare learning the material for the first time and who are planning a career in whichthey will use advanced mathematics professionally. Much of the material in thebooks corresponds to normal course work. Nevertheless, it is often the case thatcore mathematics curricula, time-limited as they are, do not include all the topicsthat one might like. Thus the book includes important topics that may be skippedin required courses but that the professional mathematician will ultimately wantto learn by self-study.The content of the required courses at each university reflects expectations ofwhat students need before beginning specialized study and work on a thesis. Theseexpectations vary from country to country and from university to university. Evenso, there seems to be a rough consensus about what mathematics a plenary lecturerat a broad international or national meeting may take as known by the audience.The tables of contents of the two books represent my own understanding of whatthat degree of knowledge is for real analysis today.Key topics and features of Basic Real Analysis are as follows:* Early chapters treat the fundamentals of real variables, sequences and seriesof functions, the theory of Fourier series for the Riemann integral, metricspaces, and the theoretical underpinnings of multivariable calculus and ordi-nary differential equations.* Subsequent chapters develop the Lebesgue theory in Euclidean and abstractspaces, Fourier series and the Fourier transform for the Lebesgue integral,point-set topology, measure theory in locally compact Hausdorff spaces, andthe basics of Hilbert and Banach spaces.* The subjects of Fourier series and harmonic functions are used as recurringmotivation for a number of theoretical developments.* The development proceeds from the particular to the general, often introducingexamples well before a theory that incorporates them.* More than 300 problems at the ends of chapters illuminate aspects of thetext, develop related topics, and point to additional applications. A separate55-page section "Hints for Solutions of Problems" at the end of the book givesdetailed hints for most of the problems, together with complete solutions formany.Beyond a standard calculus sequence in one and several variables, the mostimportant prerequisite for using Basic Real Analysis is that the reader alreadyknow what a proof is, how to read a proof, and how to write a proof. Thisknowledge typically is obtained from honors calculus courses, or from a coursein linear algebra, or from a first junior-senior course in real variables. In addition,it is assumed that the reader is comfortable with a modest amount of linear algebra,including row reduction of matrices, vector spaces and bases, and the associatedgeometry. A passing acquaintance with the notions of group, subgroup, andquotient is helpful as well.Chapters I-IV are appropriate for a single rigorous real-variables course andmay be used in either of two ways. For students who have learned about proofsfrom honors calculus or linear algebra, these chapters offer a full treatment of realvariables, leaving out only the more familiar parts near the beginning--such aselementary manipulations with limits, convergence tests for infinite series withpositive scalar terms, and routine facts about continuity and differentiability.

* Presents a comprehensive treatment with a global view of the subject * Rich in examples, problems with hints, and solutions, the book makes a welcome addition to the library of every mathematician