Series: Progress in Mathematics, Vol. 245
2006, XX, 339 p., Hardcover
ISBN: 0-8176-4391-5
About this book
This monograph, divided into four parts, presents a comprehensive
treatment and systematic examination of cycle spaces of flag
domains. Assuming only a basic familiarity with the concepts of
Lie theory and geometry, this work presents a complete structure
theory for these cycle spaces, as well as their applications to
harmonic analysis and algebraic geometry.
Table of contents
* Dedication
* Acknowledgments
* Introduction
Part I: Introduction to Flag Domain Theory
Overview
* Structure of Complex Flag Manifolds
* Real Group Orbits
* Orbit Structure for Hermitian Symmetric Spaces
* Open Orbits
* The Cycle Space of a Flag Domain
Part II: Cycle Spaces as Universal Domains
Overview
* Universal Domains
* B-Invariant Hypersurfaces in Mz
* Orbit Duality via Momentum Geometry
* Schubert Slices in the Context of Duality
* Analysis of the Boundary of U
* Invariant Kobayashi?Hyperbolic Stein Domains
* Cycle Spaces of Lower-Dimensional Orbits
* Examples
Part III: Analytic and Geometric Concequences
Overview
* The Double Fibration Transform
* Variation of Hodge Structure
* Cycles in the K3 Period Domain
Part IV: The Full Cycle Space
Overview
* Combinatorics of Normal Bundles of Base Cycles
* Methods for Computing H1(C;O(E((q+0q)s)))
* Classification for Simple g0 with rank t < rank g
* Classification for rank t = rank g
* References
* Index
* Symbol Index
Series: Progress in Mathematics, Vol. 244
2006, XXII, 632 p. 41 illus., Hardcover
ISBN: 0-8176-4076-2
About this book
A tribute to the vision and legacy of Israel Moiseevich Gelfand,
the invited papers in this volume reflect the unity of
mathematics as a whole, with particular emphasis on the many
connections among the fields of geometry, physics, and
representation theory. Written by leading mathematicians, the
text is broadly divided into two sections: the first is devoted
to developments at the intersection of geometry and physics, and
the second to representation theory and algebraic geometry.
Topics include conformal field theory, K-theory, noncommutative
geometry, gauge theory, representations of infinite-dimensional
Lie algebras, and various aspects of the Langlands program.
Table of contents
P. Etingof, V. Retakh, and I.M. Singer (eds.): Preface
H. Cartan: A Tribute to I.M. Gelfand
Conference Program: An International Conference on "The
Unity of Mathematics"
I.M. Gelfand: Talk Given at the Dinner at Royal East Restaurant
on September 3, 2003
I.M. Gelfand: Mathematics as an Adequate Language
M. Atiyah: The Interaction between Geometry and Physics
A. Braverman, M. Finkelberg, and D. Gaitsgory: Uhlenbeck Spaces
via Affine Lie Algebras
H. Brezis: New Questions Related to the Topological Degree
T. Coates and A. Givental: Quantum Cobordisms and Formal Group
Laws
A. Connes: On the Foundations of Noncommutative Geometry
S. Debacker and D. Kazhdan: Stables Distributions Supported on
the Nilpotent Cone for the Group G2
V. Drinfeld: Infinite-Dimensional Vector Bundles in Algebraic
Geometry: An Introduction
L.D. Faddeev: Algebraic Lessons from the Theory of Quantum
Integrable Models
M. Kontsevich and Y. Soibelman: Affine Structures and Non-Archimedean
Analytic Spaces
B. Kostant and N. Wallach: Gelfand?Zeitlin Theory from the
Perspective of Classical Mechanics II
C-H. Liu, K. Liu, and S-T. Yau: Mirror Symmetry and Localizations
G. Lusztig: Character Sheaves and Generalizations
D. McDuff: Symplectomorphism Groups and Quantum Cohomology
M. Movshev and A. Schwarz: Algebraic Structure of Yang?Mills
Theory
N.A. Nekrasov and A. Okounkov: Seiberg?Witten Theory and Random
Partitions
A. Okounkov, N. Reshetikhin, and C. Vafa: Quantum Calabi?Yau and
Classical Crystals
A.M. Vershik: Gelfand?Tsetlin Algebras, Expectations, Inverse
Limits, Fourier Analysis
Series: Mathematics: Theory & Applications
2006, XIII, 479 p., Hardcover
ISBN: 0-8176-3247-6
About this book
Noncompact symmetric and locally symmetric spaces naturally
appear in many mathematical theories, including analysis (representation
theory, nonabelian harmonic analysis), number theory (automorphic
forms), algebraic geometry (modulae) and algebraic topology (cohomology
of discrete groups). In most applications it is necessary to form
an appropriate compactification of the space. The literature
dealing with such compactifications is vast. The main purpose of
this book is to introduce uniform constructions of most of the
known compactifications with emphasis on their geometric and
topological structures.
Table of contents
* Preface
* Introduction
Part I: Compactifications of Riemannian Symmetric Spaces
* Review of Classical Compactifications of Symmetric Spaces
* Uniform Construction of Compactifications of Symmetric Spaces
* Properties of Compactifications of Symmetric Spaces
Part II: Smooth Compactifications of Semisimple Symmetric Spaces
* Smooth Compactifications of Riemannian Symmetric Spaces G / K
* Semisimple Symmetric Spaces G / H
* The Real Points of Complex Symmetric Spaces Defined Over R
* The DeConcini?Procesi Compactification of a Complex Symmetric
Space and its Real Points
* The Oshima?Sekiguchi Compactification of G / K and Comparison
with G/Hw (R)
Part III: Compactifications of Locally Symmetric Spaces
* Classical Compactifications of Locally Symmetric Spaces
* Uniform Construction of Compactifications of Locally Symmetric
Spaces
* Properties of Compactifications of Locally Symmetric Spaces
* Subgroup Compactifications of o \ G
* Metric Properties of Compactifications of Locally Symmetric
Spaces o \ X
* References
* Index
Series: Lecture Notes in Mathematics, Vol. 1866
Volume package: Quantum Independent Increment Processes
2006, XV, 340 p., Softcover
ISBN: 3-540-24407-7
About this book
This is the second of two volumes containing the revised and
completed notes of lectures given at the school "Quantum
Independent Increment Processes: Structure and Applications to
Physics". This school was held at the Alfried-Krupp-Wissenschaftskolleg
in Greifswald in March, 2003, and supported by the Volkswagen
Foundation. The school gave an introduction to current research
on quantum independent increment processes aimed at graduate
students and non-specialists working in classical and quantum
probability, operator algebras, and mathematical physics.
The present second volume contains the following lectures: "Random
Walks on Finite Quantum Groups" by Uwe Franz and Rolf Gohm,
"Quantum Markov Processes and Applications in Physics"
by Burkhard Kummerer, Classical and Free Infinite Divisibility
and Levy Processes" by Ole E. Barndorff-Nielsen, Steen
Thorbjornsen, and "Levy Processes on Quantum Groups and Dual
Groups" by Uwe Franz.
Table of contents
Volume package: Modern Differential Geometry in Gauge Theories
2006, XVII, 293 p., Softcover
ISBN: 0-8176-4378-8
About this book
Differential geometry, in the classical sense, is developed
through the theory of smooth manifolds. Modern differential
geometry from the authorfs perspective is used in this work to
describe physical theories of a geometric character without using
any notion of calculus (smoothness). Instead, an axiomatic
treatment of differential geometry is presented via sheaf theory
(geometry) and sheaf cohomology (analysis). Using vector sheaves,
in place of bundles, based on arbitrary topological spaces, this
unique approach in general furthers new perspectives and
calculations that generate unexpected potential applications.
Modern Differential Geometry in Gauge Theories is a two-volume
research monograph that systematically applies a sheaf-theoretic
approach to such physical theories as gauge theory. Beginning
with Volume 1, the focus is on Maxwell fields. All the basic
concepts of this mathematical approach are formulated and used
thereafter to describe elementary particles, electromagnetism,
and geometric prequantization. Maxwell fields are fully examined
and classified in the language of sheaf theory and sheaf
cohomology. Continuing in Volume 2, this sheaf-theoretic approach
is applied to Yang?Mills fields in general.
Table of contents
General Preface.- Preface to Volume I.- Acknowledgements.-
Contents of Volume II.- Part I: Maxwell Fields: General Theory.-
The Rudiments of Abstract Differential Geometry.- Elementary
Particles: Sheaf-Theoretic Classification, by Spin-Structure,
According to Selesnick's Correspondence Principle.-
Electromagnetism.- Cohomological Classification of Maxwell and
Hermitian Maxwell Fields.- Geometric Prequantization.- References.-
Index of Notation.- Index.