The Fourfold Way In Real Analysis
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Author | : André Unterberger |
Publisher | : Springer Science & Business Media |
Total Pages | : 228 |
Release | : 2006-06-15 |
Genre | : Mathematics |
ISBN | : 3764375450 |
The fourfold way starts with the consideration of entire functions of one variable satisfying specific estimates at infinity, both on the real line and the pure imaginary line. A major part of classical analysis, mainly that which deals with Fourier analysis and related concepts, can then be given a parameter-dependent analogue. The parameter is some real number modulo 2, the classical case being obtained when it is an integer. The space L2(R) has to give way to a pseudo-Hilbert space, on which a new translation-invariant integral still exists. All this extends to the n-dimensional case, and in the alternative to the metaplectic representation so obtained, it is the space of Lagrangian subspaces of R2n that plays the usual role of the complex Siegel domain. In fourfold analysis, the spectrum of the harmonic oscillator can be an arbitrary class modulo the integers. Even though the whole development touches upon notions of representation theory, pseudodifferential operator theory, and algebraic geometry, it remains completely elementary in all these aspects. The book should be of interest to researchers working in analysis in general, in harmonic analysis, or in mathematical physics.
Author | : André Unterberger |
Publisher | : Springer Science & Business Media |
Total Pages | : 133 |
Release | : 2008-09-03 |
Genre | : Mathematics |
ISBN | : 3540779108 |
This volume introduces an entirely new pseudodifferential analysis on the line, the opposition of which to the usual (Weyl-type) analysis can be said to reflect that, in representation theory, between the representations from the discrete and from the (full, non-unitary) series, or that between modular forms of the holomorphic and substitute for the usual Moyal-type brackets. This pseudodifferential analysis relies on the one-dimensional case of the recently introduced anaplectic representation and analysis, a competitor of the metaplectic representation and usual analysis. Besides researchers and graduate students interested in pseudodifferential analysis and in modular forms, the book may also appeal to analysts and physicists, for its concepts making possible the transformation of creation-annihilation operators into automorphisms, simultaneously changing the usual scalar product into an indefinite but still non-degenerate one.
Author | : André Unterberger |
Publisher | : Birkhäuser |
Total Pages | : 175 |
Release | : 2018-07-16 |
Genre | : Mathematics |
ISBN | : 3319927078 |
Classically developed as a tool for partial differential equations, the analysis of operators known as pseudodifferential analysis is here regarded as a possible help in questions of arithmetic. The operators which make up the main subject of the book can be characterized in terms of congruence arithmetic. They enjoy a Eulerian structure, and are applied to the search for new conditions equivalent to the Riemann hypothesis. These consist in the validity of certain parameter-dependent estimates for a class of Hermitian forms of finite rank. The Littlewood criterion, involving sums of Möbius coefficients, and the Weil so-called explicit formula, which leads to his positivity criterion, fit within this scheme, using in the first case Weyl's pseudodifferential calculus, in the second case Fuchs'. The book should be of interest to people looking for new possible approaches to the Riemann hypothesis, also to new perspectives on pseudodifferential analysis and on the way it combines with modular form theory. Analysts will have no difficulty with the arithmetic aspects, with which, save for very few exceptions, no previous acquaintance is necessary.
Author | : André Unterberger |
Publisher | : Birkhäuser |
Total Pages | : 222 |
Release | : 2009-09-03 |
Genre | : Mathematics |
ISBN | : 9783764391157 |
Author | : Ryoshi Hotta |
Publisher | : Springer Science & Business Media |
Total Pages | : 408 |
Release | : 2007-11-07 |
Genre | : Mathematics |
ISBN | : 081764363X |
D-modules continues to be an active area of stimulating research in such mathematical areas as algebraic, analysis, differential equations, and representation theory. Key to D-modules, Perverse Sheaves, and Representation Theory is the authors' essential algebraic-analytic approach to the theory, which connects D-modules to representation theory and other areas of mathematics. To further aid the reader, and to make the work as self-contained as possible, appendices are provided as background for the theory of derived categories and algebraic varieties. The book is intended to serve graduate students in a classroom setting and as self-study for researchers in algebraic geometry, representation theory.
Author | : Toshiyuki Kobayashi |
Publisher | : Springer Science & Business Media |
Total Pages | : 220 |
Release | : 2007-10-10 |
Genre | : Mathematics |
ISBN | : 0817646469 |
This volume uses a unified approach to representation theory and automorphic forms. It collects papers, written by leading mathematicians, that track recent progress in the expanding fields of representation theory and automorphic forms and their association with number theory and differential geometry. Topics include: Automorphic forms and distributions, modular forms, visible-actions, Dirac cohomology, holomorphic forms, harmonic analysis, self-dual representations, and Langlands Functoriality Conjecture, Both graduate students and researchers will find inspiration in this volume.
Author | : André Unterberger |
Publisher | : Birkhäuser |
Total Pages | : 208 |
Release | : 2015-06-22 |
Genre | : Mathematics |
ISBN | : 3319186574 |
The main results of this book combine pseudo differential analysis with modular form theory. The methods rely for the most part on explicit spectral theory and the extended use of special functions. The starting point is a notion of modular distribution in the plane, which will be new to most readers and relates under the Radon transformation to the classical one of modular form of the non-holomorphic type. Modular forms of the holomorphic type are addressed too in a more concise way, within a general scheme dealing with quantization theory and elementary, but novel, representation-theoretic concepts.
Author | : Detlef Gromoll |
Publisher | : Springer Science & Business Media |
Total Pages | : 185 |
Release | : 2009-03-28 |
Genre | : Mathematics |
ISBN | : 3764387157 |
Riemannian manifolds, particularly those with positive or nonnegative curvature, are constructed from only a handful by means of metric fibrations or deformations thereof. This text documents some of these constructions, many of which have only appeared in journal form. The emphasis is less on the fibration itself and more on how to use it to either construct or understand a metric with curvature of fixed sign on a given space.
Author | : Laurent Bartholdi |
Publisher | : Springer Science & Business Media |
Total Pages | : 432 |
Release | : 2005-12-09 |
Genre | : Mathematics |
ISBN | : 9783764374464 |
This book offers a panorama of recent advances in the theory of infinite groups. It contains survey papers contributed by leading specialists in group theory and other areas of mathematics. Topics include amenable groups, Kaehler groups, automorphism groups of rooted trees, rigidity, C*-algebras, random walks on groups, pro-p groups, Burnside groups, parafree groups, and Fuchsian groups. The accent is put on strong connections between group theory and other areas of mathematics.
Author | : Alan Richardson |
Publisher | : Westminster John Knox Press |
Total Pages | : 642 |
Release | : 1983-01-01 |
Genre | : Religion |
ISBN | : 9780664227487 |
The Westminter Dictionary of Christian Theology is an important reference for any pastor, scholar, or student of theology. The articles are clearly written, historically informative, and conceptually clarifying. The entries are arranged alphabetically for ease of use.