Series: Advanced Courses in Mathematics - CRM Barcelona
2010, Approx. 125 p., Softcover
ISBN: 978-3-0346-0212-9
Due: October 2009
Contains an expanded version of the lectures delivered by the authors at the CRM Barcelona in March 2008
The first text serves as an introduction to the work of Huisken-Sinestrari about the formation of singularities and surgery of the mean curvature
The second text presents recent developments on the field of isoperimetric inequalities, mostly based on the use of geometric flows, as well as applications of isoperimetric inequalities to hyperbolic geometry
Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.
Series: Universitext
2010, Approx. 290 p. 6 illus., Softcover
ISBN: 978-1-84882-888-9
Due: December 2009
Combines physical book form with material on the internet so that students can use the physical book as a textbook and lecturers can use the website as a source of exercises, as well as providing more advanced students with a higher degree of material
A Course on Finite Groups introduces the fundamentals of group theory to advanced undergraduate and beginning graduate students. Based on a series of lecture courses developed by the author over many years, the book starts with the basic definitions and examples and develops the theory to the point where a number of classic theorems can be proved. The topics covered include: homomorphisms and isomorphisms; actions; Sylow theory; products and Abelian groups; series, and nilpotent and soluble groups; and an introduction to the classification of the finite simple groups.
A number of groups are described in detail and the reader is encouraged to work with one of the many computer algebra packages available to construct and experience "actual" groups for themselves in order to develop a deeper understanding of the theory and the significance of the theorems. Numerous problems, of varying levels of difficulty, help to test understanding.
A brief resume of the basic set theory and number theory required for the text is provided in an appendix, and a wealth of extra resources is available online at www.springer.com, including: hints and/or full solutions to all of the exercises; extension material for many of the chapters, covering more challenging topics and results for further study; and two additional chapters providing an introduction to group representation theory.
The Group Concept.- Elementary Group Properties.- Group Construction and Representation.- Homomorphisms.- Action and the Orbit-Stabiliser Theorem.- p-Groups and Sylow Theory.- Products and Abelian Groups.- Groups of Order 24, Three Examples.- Series, Jordan Holder Theorem and the Extension Problem.- Nilpotency.- Solubility.- Simple Groups of Order Less Than 10000.- Representation and Character Theory.- Character Tables and Theorems of Burnside and Frobenius.- Appendices
Series: Sources and Studies in the History of Mathematics and Physical Sciences
2010, XIV, 246 p. 54 illus., Hardcover
ISBN: 978-1-4419-1313-5
Due: December 2009
In the 18th century purely scientific interests as well as the practical necessities of navigation motivated the development of new theories and techniques to accurately describe celestial and lunar motion. Tobias Mayer, a German mathematician and astronomer, was among the most notable scientists of the time in the area of lunar theory.
"Between Theory and Observations" presents a detailed and rigorous account of Tobias Mayer’s work; his famous contribution is his extensive set of lunar tables, which were the most accurate of their time. This book gives a complete and accurate account, not to be found elsewhere in the literature, of Tobias Mayer's important contributions to the study of lunar motion.
The book highlights and examines three of Mayer's major achievements:
- The computational scheme embodied in Mayer's lunar tables is examined and traced back to the scheme of Newton's 1702 lunar theory with its decidedly non-dynamical characteristics.
- Mayer's dynamical lunar theory is compared to Euler's work in celestial mechanics of the same period. Evidence is presented refuting the commonly held opinion that Mayer's lunar theory was simply a modification of Euler's theory.
- Mayer's technique of adjusting the coefficients of his lunar tables to fit an extensive collection of observational data is examined in detail. The scale of Mayer's effort was unprecedented and preceded the invention of the least squares method by half a century.
This volume is intended for historians of mathematics and/or astronomy as well as anyone interested in the historical development of the theory of lunar motion.
List of Figures. List of Displays.- 1. Introduction.- 2. The quest for lunar theory.- 3. The poineer's work.- 4. A manual to the tables.- 5. Theoria Lunae.- 6. The Horrocks legacy.- 7. Multisteps in the Theoria Lunae.- 8. Hausbackene Combinationen.- 9. Some aspects of model fitting.- 10. Concluding observations.- Appendix A. Lunar equations: versions and aliases.- Appendix B. Spreadsheet contents.- Appendix C. Manuscript sources.- References.- Index.
Series: Graduate Texts in Mathematics , Vol. 257
2010, Approx. 240 p. 19 illus., Hardcover
ISBN: 978-1-4419-1595-5
Due: December 2009
The first textbook on deformation theory
Bestselling Springer author, Robin Hartshorne
Text contains plenty of motivation, enhanced with numerous exercises and examples
The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Hartshorne studies first what happens over small infinitesimal deformations, and then gradually builds up to more global situations, using methods pioneered by Kodaira and Spencer in the complex analytic case, and adapted and expanded in algebraic geometry by Grothendieck.
Topics include:
* deformations over the dual numbers;
* smoothness and the infinitesimal lifting property;
* Zariski tangent space and obstructions to deformation problems;
* pro-representable functors of Schlessinger;
* infinitesimal study of moduli spaces such as the Hilbert scheme, Picard scheme, moduli of curves, and moduli of stable vector bundles.
The author includes numerous exercises, as well as important examples illustrating various aspects of the theory. This text is based on a graduate course taught by the author at the University of California, Berkeley.
Preface.- Getting Started.- Higher Order Deformations.- Formal Moduli.- Global Questions.- References.
Series: Progress in Mathematics , Vol. 278
2010, XX, 208 p. 1 illus., Hardcover
ISBN: 978-0-8176-4874-9
Due: December 2009
Includes papers written by leading experts in the field of algebraic geometry
Offers articles exploring various topics in algebraic geometry
Focuses on characteristic 0 and modular invariant theory
A group of Gerry Schwarz’s colleagues and collaborators gathered at the Fields Institute in Toronto for a mathematical festschrift in honor of his 60th birthday. This volume is an outgrowth of that event, covering the wide range of mathematics to which Gerry Schwarz has either made fundamental contributions or stimulated others to pursue. The articles are a sampling of modern day algebraic geometry with associated group actions from its leading experts, with a particular focus on characteristic 0 and modular invariant theory.
Preface.- List of Contributors.- Michel Brion: "Some basis results on actions of non-affine algebraic groups".- Abraham Broer: "On Chevalley-Shephard-Todd's theorem in positive characteristic".-Jonathan Elmer and Peter Fleischmann: "On the depth of modular invariant rings for the groups $C_p×C_p$".- Daniel Daigle and Gene Freudenberg: "Families of affine fibrations".- Daniel Greb and Peter Heinzner: "Kahlerian reduction in steps".- Aloysius Helminck: "On orbit decompositions for symmetric k-varieties".- Bertram Kostant: "Root systems for Levi factors and Borel-de Siebenthal theory".- Hanspeter Kraft and Nolan R Wallach: "Polarizations and nullcone of representations of reductive groups".- R. James Shank and David Wehlau: "Decomposing symmetric powers of certain modular representations of cyclic groups".- Will Traves: "Differential operators on Grassmann varieties".
Series: Progress in Mathematics , Vol. 281
2010, XIV, 266 p., Hardcover
ISBN: 978-0-8176-4931-9
Due: November 2009
This book is concerned with the interplay between the theory of operator semigroups and spectral theory
The basics on operator semigroups are concisely covered in this self-contained text
Part I deals with the Hille-Yosida and Lumer-Phillips characterizations of semigroup generators, the Trotter-Kato approximation theorem, Kato’s unified treatment of the exponential formula and the Trotter product formula, the Hille-
Phillips perturbation theorem, and Stone’s representation of unitary semigroups
Part II explores generalizations of spectral theory’s connection to operator semigroups
Is suitable for a graduate seminar on operator semigroups or for self study
The theory of operator semigroups was essentially discovered in the early 1930s. Since then, the theory has developed into a rich and exciting area of functional analysis and has been applied to various mathematical topics such as Markov processes, the abstract Cauchy problem, evolution equations, and mathematical physics.
This self-contained monograph focuses primarily on the theoretical connection between the theory of operator semigroups and spectral theory. Divided into three parts with a total of twelve distinct chapters, this book gives an in-depth account of the subject with numerous examples, detailed proofs, and a brief look at a few applications.
* The Hille-Yosida and Lumer?Phillips characterizations of semigroup generators
* The Trotter?Kato approximation theorem
* Kato’s unified treatment of the exponential formula and the Trotter product formula
* The Hille?Phillips perturbation theorem, and Stone’s representation of unitary semigroups
* Generalizations of spectral theory’s connection to operator semigroups
* A natural generalization of Stone’s spectral integral representation to a Banach space setting
With a collection of miscellaneous exercises at the end of the book and an introductory chapter examining the basic theory involved, this monograph is suitable for second-year graduate students interested in operator semigroups.
Introduction.- Part I. General Theory. Basic theory. The semi-simplicity space. Analyticity. The semigroup as a function of its generator. Large parameter. Boundary values.- Part II. Generalizations. Pre-semigroups. The semi-simplicity space. Families of unbounded symmetric operators. Dependence on parameters.- Notes and References. Bibliography