Series: Encyclopaedia of Mathematical Sciences, Vol. 139
Volume package: Invariant Theory and Algebraic Transformation Groups
1st Edition., 2011, 233 p., Hardcover
ISBN: 978-3-642-17403-2
Due: February 2011
This book covers the modular invariant theory of finite groups, the case when the characteristic of the field divides the order of the group. It explains a theory that is more complicated than the study of the classical non-modular case, and it describes many open questions.
Largely self-contained, the book develops the theory from its origins up to modern results. It explores many examples, illustrating the theory and its contrast with the better understood non-modular setting. It details techniques for the computation of invariants for many modular representations of finite groups, especially the case of the cyclic group of prime order. It includes detailed examples of many topics as well as a quick survey of the elements of algebraic geometry and commutative algebra as they apply to invariant theory. The book is aimed at both graduate students and researchers?an introduction to many important topics in modern algebra within a concrete setting for the former, an exploration of a fascinating subfield of algebraic geometry for the latter.
1 First Steps.- 2 Elements of Algebraic Geometry and Commutative Algebra.- 3 Applications of Commutative Algebra to Invariant Theory.- 4 Examples.- 5 Monomial Orderings and SAGBI Bases.- 6 Block Bases.- 7 The Cyclic Group Cp.- 8 Polynomial Invariant Rings.- 9 The Transfer.- 10 Invariant Rings via Localization.- 11 Rings of Invariants which are Hypersurfaces.- 12 Separating Invariants.- 13 Using SAGBI Bases to Compute Rings of Invariants.- 14 Ladders.- References.- Index.
Series: Lecture Notes in Mathematics, Vol. 2013
1st Edition., 2011, X, 260 p., Softcover
ISBN: 978-3-642-17412-4
Due: February 2011
This volume offers a well-structured overview of existent computational approaches to Riemann surfaces and those currently in development. The authors of the contributions represent the groups providing publically available numerical codes in this field. Thus this volume illustrates which software tools are available and how they can be used in practice. In addition examples for solutions to partial differential equations and in surface theory are presented. The intended audience of this book is twofold. It can be used as a textbook for a graduate course in numerics of Riemann surfaces, in which case the standard undergraduate background, i.e., calculus and linear algebra, is required. In particular, no knowledge of the theory of Riemann surfaces is expected; the necessary background in this theory is contained in the Introduction chapter. At the same time, this book is also intended for specialists in geometry and mathematical physics applying the theory of Riemann surfaces in their research. It is the first book on numerics of Riemann surfaces that reflects the progress made in this field during the last decade, and it contains original results. There are a growing number of applications that involve the evaluation of concrete characteristics of models analytically described in terms of Riemann surfaces. Many problem settings and computations in this volume are motivated by such concrete applications in geometry and mathematical physics.
Introduction to Compact Riemann Surfaces.- Computing with plane algebraic curves and Riemann surfaces: the algorithms of the Maple package “algcurves”.- Algebraic curves and Riemann surfaces in Matlab.- Computing Poincare Theta Series for Schottky Groups.- Uniformizing real hyperelliptic M-curves using the Schottky-Klein prime function.- Numerical Schottky Uniformizations: Myrberg’s Opening Process.- Period Matrices of Polyhedral Surfaces.- On the spectral theory of the Laplacian on compact polyhedral surfaces of arbitrary genus.
Series: Lecture Notes in Mathematics, Vol. 2014
Subseries: History of Mathematics Subseries
1st Edition. Translation of the French Original Version, Springer 2009., 2011, VI, 332 p., Softcover
ISBN: 978-3-642-17853-5
Due: January 2011
How did Pierre Fatou and Gaston Julia create what we now call Complex Dynamics, in the context of the early twentieth century and especially of the First World War? The book is based partly on new, unpublished sources.Who were Pierre Fatou, Gaston Julia, Paul Montel? New biographical information is given on the little known mathematician that was Pierre Fatou. How did the serious injury of Julia during WWI influence mathematical life in France?
"Audin’s book is indeed filled with marvelous biographical information and analysis, dealing not just with the men mentioned in the book’s title but a large number of other players, too … . the book under review addresses itself to scholars for whom the history of mathematics has a particular resonance and especially to mathematicians active, or even with merely an interest in, complex dynamics. … presents it all to the reader in a very appealing form." (Michael Berg, The Mathematical Association of America, October, 2009)
I The Great Prize, the framework.- I.1 The iteration problem in 1915.- I.2 The protagonists around 1917?1918.- I.3 The war.- I.4 Iteration, a few definitions and notation.- I.5 Normal families.- I.6 Relation to functional equations.- II The Great Prize of Mathematical.- II.1 Year 1917.- II.2 Year 1918.- III The memoirs.- III.1 Julia’s memoir.- III.2 The (three) memoir(s) of Fatou.- III.3 Comments (in the first person).- III.4 To summarise.- IV After Fatou and Julia.- IV.1 Stop.- IV.2 Hausdorff distance (1914) and dimension (1919).- IV.3 Irregular points, J-points, O-points (1925?1927).- IV.4 The centre problem (1927?1942).-IV.5 Holomorphic dynamics.-V On Pierre Fatou.-V.1 Childhood and youth of Fatou.- V.2 What do we know of Pierre Fatou?.- V.3 Continuation of Fatou’s career.- V.4 Fatou’s thesis.- V.5 Fatou as a mathematician.-V.6 Fatou as an astronomer.- V.7 Teaching and candidatures of Fatou.- V.8 Fatou and other mathematicians.- V.9 Death of Fatou.- VI A controversy in 1965.- VI.1 The protagonists, from 1918 to 1965.- VI.2 Relations between Julia and Montel, in the 1930’s.- VI.3 The third centenary of the Institut de France VI.4.- As a conclusion: O for a biography of Gaston Julia.- References.-Index.
Series: Theoretical and Mathematical Physics
330 p., 5 images, 1st edition
1st Edition., 2011, 330 p. 5 illus., Hardcover
ISBN: 978-94-007-0204-2
Due: March 20, 2011
This book gives a detailed and self-contained introduction into the theory of spectral functions, with an emphasis on their applications to quantum field theory. All methods are illustrated with applications to specific physical problems from the forefront of current research, such as finite-temperature field theory, D-branes, quantum solitons and noncommutativity. In the first part of the book, necessary background information on differential geometry and quantization, including less standard material, is collected. The second part of the book contains a detailed description of main spectral functions and methods of their calculation. In the third part, the theory is applied to several examples (D-branes, quantum solitons, anomalies, noncommutativity). This book addresses advanced graduate students and researchers in mathematical physics with basic knowledge of quantum field theory and differential geometry. The aim is to prepare readers to use spectral functions in their own research, in particular in relation to heat kernels and zeta functions.
1 Preface.- 2 Notation Index I The Basics: 3 Geometrical Background.- 4 Quantum fields II Spectral geometry: 5 Operators and their spectra.- 6 Spectral functions.- 7 Non-linear spectral problems.- 8 Anomalies and Index Theorem III Applications: 9 Effective action.- 10 Anomalies in quantum field theories.- 11 Vacuum energy.- 12 Open strings and Born-Infeld action.- 13 Noncommutative geometry and field theory IV Problem solving: 14 Solutions to exercises.