Hawkins, T., University of Boston, MA, USA
The Emergence of the Theory of Lie Groups
An Essay in the History of Mathematics 1869-1926
2000. Approx. 575 pp. 50 figs.
0-387-98963-3
Lie groups arose in the study of the mathematical properties of rotations; like physical rotations, they depend on a parameter (such as the angle of rotation) that can be varied in a continuous manner, and like rotations in more than 2 dimensions (whose result depends on the order in which they are performed), they form a non-commutative group; the derivative of the elements with respect to the parameter gives rise to a set of operators whose algebraic properties are themselves of interest: the Lie algebra. These ideas can of course be generalized to other physical transformations, and they play an important role in the development of 20th century mathematics and mathematical physics. The great Swedish mathematician Sophus Lie (1849-1899) developed the general theory of transformations in the 1870s, and the first part of the book properly focuses on his work. In the second part the central figure is Wilhelm Killing (1847-1923), who developed structure and classification of semisimple Lie algebras. The third part focuses on the developments of the representation of Lie algebras, particularly in the work of Elie Cartan (1869-1951). The book concludes with the work of Hermann Weyl and his contemporaries on the structure and representation of Lie groups which serve to bring together much of the earlier work into a coherent theory while at the same time opening up significant avenues for further work.
Contents: Preface.- The Geometrical Origins of Lie's theory.- Jacobi & The Analytical Origins of Lie's
Theory.- Lie's Theory of Transformation Groups 1874-1893.- Non-euclidean Geometry & Weierstrassian
Mathematics.- Killing & the Structure of Lie Algebras.- The Doctoral Thesis of Elie Cartan.- Lie's School &
Linear Representations.- Cartan's Trilogy: 1913-14.- The G?ttingen School of Hilbert.- The Berlin Algebraists:
Frobenius & Schur.- From Relativity to Representations.- Weyl's Great Papers of 1925 & 1926.- References.-
Index.
Series: Sources and Studies in the History of Mathematics and Physical Series.
Fields: Group Theory
Written for: Mathematicians, historians of science
Book category: Monograph
Publication language: English
Publication date: May 2000
Herman, J., Brno, Czech Republic
Kucera, R., Masaryk University, Brno, Czech Republic
Simsa, J., Academy of Sciences of the Czech Republic, Brno, Czech Republic
Dilcher, K., Dalhousie University of Halifax, NS, CanadaEquations and Inequalities
Elementary Problems and Theorems in Algebra and Number Theory
2000. Approx. 360 pp.
0-387-98942-0
A look at solving problems in three areas of classical elementary mathematics: equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here. With approximately 330 examples and 760 exercises.
Contents: Algebraic Identities and Equations.- Algebraic Inequalities.- Number Theory.- Hints and Answers.
Fields: Number Theory; Algebra
Written for: Graduate math students, undergraduate math students, mathematicians
Book category: Undergraduate Textbook
Publication language: English
Publication date: May 2000
Arnold, D., Baylor University, Waco, TX, USA
Abelian Groups and Representations of Finite Partially Ordered Sets
2000. Approx. 330 pp.
0-387-98982-X
The theme of this book is an exposition of connections between representations of finite partially ordered sets and abelian groups. Emphasis is placed throughout on classification, a description of the objects up to isomorphism, and computation of representation type, a measure of when classification is feasible.
David M. Arnold is the Ralph and Jean Storm Professor of Mathematics at Baylor University. He is the author of "Finite Rank Torsion Free Abelian Groups and Rings" published in the Springer-Verlag Lecture Notes in Mathematics series, a co-editor for two volumes of conference proceedings, and the author of numerous articles in mathematical research journals.
Contents: * Representations of posets over a field * Torsion-free Abelian groups * Butler groups *
Representations over a discrete valuation ring * Almost completely decomposable groups * Representations over fields and exact sequences * Finite rank Butler groups * Applications of representations and Butler groups
Fields: Algebra; Group Theory
Written for: Mathematicians, graduate math students
Book category: Monograph
Publication language: English
Publication date: July 2000
Eisenbud, D., Mathematical Sciences Research Institute, Berkeley, CA, USA
Harris, J., Harvard University, Cambridge, MA, USAThe Geometry of Schemes
2000. Approx. 305 pp. 40 figs.
0-387-98638-3
The theory of schemes is the foundation for algebraic geometry proposed and elaborated by Alexander
Grothendieck and his coworkers. It has allowed major progress in classical areas of algebraic geometry such as
invariant theory and the moduli of curves. It integrates algebraic number theory with algebraic geometry, fulfilling the dreams of earlier generations of number theorists. This integration has led to proofs of some of the major conjectures in number theory (Deligne's proof of the Weil Conjectures, Faltings proof of the Mordell Conjecture).
This book is intended to bridge the chasm between a first course in classical algebraic geometry and a technical treatise on schemes. It focuses on examples, and strives to show "what is going on" behind the definitions. There are many exercises to test and extend the reader's understanding. The prerequisites are modest: a little commutative algebra and an acquaintance with algebraic varieties, roughly at the level of a one-semester course. The book aims to show schemes in relation to other geometric ideas, such as the theory of manifolds. Some familiarity with these ideas is helpful, though not required.
Contents: Basic Definitions.- Examples.- Projective Schemes.- Classical Constructions.- Local Constructions.- Schemes and Functors.
Series: Graduate Texts in Mathematics.VOL. 197
Harris, J.M., Appalachian State University, Boone, NC, USA
Hirst, J., Appalachian State University, Boone, NC, USA
Mossinghoff, M.J., Appalachian State University, Boone, NC, USAA Course in Combinatorics and Graph Theory
2000. Approx. 200 pp. 70 figs
0-387-98736-3
These notes were first used in an introductory course team taught by the authors at Appalachian State University to advanced undergraduates and beginning graduates. The text was written with four pedagogical goals in mind: offer a variety of topics in one course, get to the main themes and tools as efficiently as possible, show the relationships between the different topics, and include recent results to convince students that mathematics is a living discipline.
Contents: Graph Theory: Introductory Concepts. Trees. Planarity. Colorings. Matchings. Ramsey Theory.
References.- Combinatorics: Three Basic Problems. Binomial Coefficients. The Principle of Inclusion and
Exclusion. Generating Functions. Polya's Theory of Counting. More Numbers. Stable Marriage. References.-
Infinite Combinatorics and Graph Theory: Pigeons and Trees. Ramsey Revisited. ZFC. The Return of der
Koenig. Ordinals, Cardinals, and Many Pigeons. Incompleteness and Coardinals.- Weakly Compact Cardinals.
Finite Combinatorics with Infinite Consequences. Points of Departure. References.
Series: Undergraduate Texts in Mathematics.
Fields: Combinatorial Mathematics/Graph Theory and Discrete Mathematics
Written for: Undergraduate math students, graduate math students, mathematicians, computer scientists
Book category: Undergraduate Textbook
Publication language: English
Publication date: May 2000
Carter, M., Massey University, Palmerston North, New Zealand
Brunt, B.van, Massey University, Palmerston North, New ZealandThe Lebesgue-Stieltjes Integral
A Practical Introduction
2000. Approx. 230 pp. 45 figs.
0-387-95012-5
While mathematics students generally meet the Riemann integral early in their undergraduate studies, those whose interests lie more in the direction of applied mathematics will probably find themselves needing to use the Lebesgue or Lebesgue-Stieltjes Integral before they have acquired the necessary theoretical background. This book is aimed at exactly this group of readers. The authors introduce the Lebesgue-Stieltjes integral on the real line as a natural extension of the Riemann integral, making the treatment as practical as possible. They discuss the evaluation of Lebesgue-Stieltjes integrals in detail, as well as the standard convergence theorems, and conclude with a brief discussion of multivariate integrals and surveys of L spaces plus some applications. The whole is rounded off with exercises that extend and illustrate the theory, as well as providing practice in the techniques.
Contents: Real Numbers * Some Analytic Preliminaries * The Riemann Integral * The Lebesgue-Stieltjes Integral * Properties of the Integral * Integral Calculus * Double and Repeated Integrals * The Lebesgue Spaces L^p * Hilbert Spaces and L^2 * Epilogue
Series: Undergraduate Texts in Mathematics.
Fields: Real Functions,Measure and Integration
Written for: Undergraduate math students, applied mathematicians, pure mathematicians,
graduate math students
Book category: Undergraduate Textbook
Publication language: English
Publication date: June 2000