2009, Approx. 805 p., Hardcover
ISBN: 978-3-540-89580-0
Due: March 19, 2009
Santalo (Spain 1911 - Argentina 2001) contributed to several branches of Geometry, perhaps his most outstanding achievement being his having laid the mathematical foundations of Stereology and its applications. Considerable power of abstraction, a brilliant geometric intuition and his outstanding gifts as a disseminator of science were among his virtues. The present volume contains a selection of his best papers.
Part I consists of a short biography and some photographs along with a complete list of his publications, classified into research papers, books, and articles on education and the popularization of mathematics, as well as a comprehensive analysis of his contribution to science.
Part II, the main part of the book, includes the selected papers, arranged in five sections, according to the nature of their contents: Differential Geometry, Integral Geometry, Convex Geometry, Affine Geometry, and Statistics and Stereology. Each section is preceded by a comment by a renowned specialist: Teufel, Langevin, Schneider, Leichtweiss, and Cruz-Orive respectively.
Finally, Part III emphasizes the influence of his work. It contains comments by several specialists about modern results based on, or closely related to, those of Santalo, some book reviews written by Santalo, as well as some reviews of his books. As a curious addendum, a ranking of his own articles, given by Santalo himself, is included.
From the Preface written by Simon Donaldson:
"The word geometry can cover many different things. The paths that can be traced from the most ancient concepts to sophisticated modern abstractions form one of the charms of the subject. We can all agree that the study of lines in three dimensional Euclidean space is a part of geometry. It is a wonderful idea that the set of all lines can itself be considered as a space, which has in turn its own geometry. Of course this leap into abstraction may seem commonplace now... Modern differential geometry provides the language and tools for doing calculus on such spaces and in particular for integration. Then we can talk about the volume of a set of lines, the mean value of a function on the space of lines and so on. This is the beginning of Integral Geometry, to which Luis Santalo contributed so much."
Preface by Simon K. Donaldson.- A Short Biography.- Scientific Work.- Some Photographs.- L. A. Santalo's Published Work.- List of Santalo's Papers Included in the Selection.- Selected Papers: Part I. Differential Geometry, with comments by E. Teufel.- Part II. Integral Geometry, with comments by R. Langevin.- Part III. Convex Geometry, with comments by R. Schneider.- Part IV. Affine Geometry, with comments by K. Leichtweiss.- Part V. Satistics and Stereology, with comments by L. M.. Cruz-Orive.- Coments on Some of Santalo's Papers.- Book Reviews.- On the Correspondence between Santalo and Vidal-Abascal.- Some Remarks on Two Classifications of Santalo's Papers.- Authorization.- Acknowledgments.
Series: Lecture Notes in Mathematics , Vol. 1972
2009, Approx. 410 p., Softcover
ISBN: 978-3-540-93912-2
Due: March 16, 2009
We are defining and studying an algebro-geometric analogue of Donaldson invariants by using moduli spaces of semistable sheaves with arbitrary ranks on a polarized projective surface.We are interested in relations among the invariants, which are natural generalizations of the "wall-crossing formula" and the "Witten conjecture" for classical Donaldson invariants.
Our goal is to obtain a weaker version of these relations, by systematically using the intrinsic smoothness of moduli spaces. According to the recent excellent work of L. Goettsche, H. Nakajima and K. Yoshioka, the wall-crossing formula for Donaldson invariants of projective surfaces can be deduced from such a weaker result in the rank two case!
1. Introduction.- 2. Preliminaries.- 3. Parabolic L-Bradlow pairs.- 4. Geometric Invariant Theory and Enhanced Master Space.- 5. Obstruction Theories of Moduli Stacks and Master Spaces.- 6. Virtual Fundamental Classes.- 7. Invariants.
Series: Frontiers in Mathematics
2009, Approx. 200 p., Softcover
ISBN: 978-3-0346-0003-3
Due: April 2009
Pseudoanalytic function theory generalizes and preserves many crucial features of complex analytic function theory. The Cauchy-Riemann system is replaced by a much more general first-order system with variable coefficients which turns out to be closely related to important equations of mathematical physics. This relation supplies powerful tools for studying and solving Schrodinger, Dirac, Maxwell, Klein-Gordon and other equations with the aid of complex-analytic methods. It has also given rise to a new method for solving Sturm-Liouville problems.
The book is dedicated to these recent developments in pseudoanalytic function theory and their applications as well as to multidimensional generalizations.
Introduction.- I. Pseudoanalytic function theory and second-order elliptic equations.- 1. Definitions and results from Bers’ theory.- 2. Second order equations.- 3. Formal powers.- 4. Cauchy’s integral formula.- 5. Complex Riccati equation.- II. Applications to Sturm-Liouville theory.- 6. Sturm-Liouville equation.- 7. Spectral problems and Darboux transformation.- III. Applications to real first-order systems.- 8. Beltrami fields.- 9. Static Maxwell system.- IV. Hyperbolic pseudoanalytic functions.- 10. Hyperbolic numbers and analytic functions.- 11. Hyperbolic pseudoanalytic functions.- 12. Klein-Gordon equation.- V. Bicomplex and biquaternionic pseudoanalytic functions and applications.- 13. The Dirac equation.- 14. Complex second order elliptic equations and bicomplex pseudoanalytic functions.- 15. Multidimensional second order equations.- Open problems.- Bibliography.- Index.
Series: Universitext
2009, Approx. 600 p. 10 illus., Softcover
ISBN: 978-0-387-89485-0
Due: June 2009
A modern introduction to the theory of numbers, emphasizing its connections with other branches of mathematics.
Suitable for first-year undergraduates through more advanced math students.
Presents a broad pictures of the nature of mathematics.
Undergraduate courses in mathematics are commonly of two types. On the one hand are courses in subjects?such as linear algebra or real analysis?with which it is considered that every student of mathematics should be acquainted. On the other hand are courses given by lecturers in their own areas of specialization, which are intended to serve as a preparation for research. But after taking courses of only these two types, students might not perceive the sometimes surprising interrelationships and analogies between different branches of mathematics, and students who do not go on to become professional mathematicians might never gain a clear understanding of the nature and extent of mathematics. Number Theory: An Introduction to Mathematics attempts to provide such an understanding of the nature and extent of mathematics. It is a modern introduction to the theory of numbers, emphasizing its connections with other branches of mathematics. Part A, which should be accessible to a first-year undergraduate, deals with elementary number theory. Part B is more advanced than the first and should give the reader some idea of the scope of mathematics today. The connecting theme is the theory of numbers. By exploring its many connections with other branches, we may obtain a broad picture.
Preface.- The Expanding Universe of Numbers.- Divisibility.- More on Divisibility.- Continued Fractions and their Uses.- Hadamard’s Determinant Problem.- Hensel’s P-Adic Numbers.- Notations.- Axioms.- The Arithmetic of Quadratic Forms.- The Geometry of Numbers.- The Number of Prime Numbers.- A Character Study.- Uniform Distribution and Ergodic Theory.- Elliptic Functions.- Connections with Number Theory.- Notations.- Axioms.- Index.