In this book we give a unified interpretation of confluences, contiguity relations and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The integral representations and series expansions of their solutions are also within our interpretation. As an application to Fuchsian differential equations on the Riemann sphere, we construct a universal model of Fuchsian differential equations with a given spectral type, in particular, we construct a single ordinary differential equation without apparent singularities corresponding to any rigid local system on the Riemann sphere, whose existence was an open problem presented by N. Katz. Furthermore we obtain fundamental properties of the solutions of the rigid Fuchsian differential equations such as their connection coefficients and the necessary and sufficient condition for the irreducibility of their monodromy groups. We give many examples calculated by our fractional calculus.
2012, 203p, ISBN: 978-4-86497-016-7
9780230365483 | Hardcover
9780230365490 | Paperback
324 pages |
11 Feb 2013
What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Godel's platonism, up to the the current debate on Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem.
Preface
Terminological Conventions
Introduction
The Origins
From Frege to Godel (through Hilbert)
Benacerraf's Arguments
Non-conservative Responses to Benacerraf's Dilemma
Conservative Responses to Benacerraf's Dilemma
The Indispensability Argument: Structure and Basic Notions
The Indispensability Argument: The Debate
Conclusion
Notes
References
Index
Cloth | March 2013 | ISBN: 9780691151755
288 pp. | 7 x 10 | 66 line illus. 1 table.
This graduate-level textbook is the first pedagogical synthesis of the field of topological insulators and superconductors, one of the most exciting areas of research in condensed matter physics. Presenting the latest developments, while providing all the calculations necessary for a self-contained and complete description of the discipline, it is ideal for graduate students and researchers preparing to work in this area, and it will be an essential reference both within and outside the classroom.
The book begins with simple concepts such as Berry phases, Dirac fermions, Hall conductance and its link to topology, and the Hofstadter problem of lattice electrons in a magnetic field. It moves on to explain topological phases of matter such as Chern insulators, two- and three-dimensional topological insulators, and Majorana p-wave wires. Additionally, the book covers zero modes on vortices in topological superconductors, time-reversal topological superconductors, and topological responses/field theory and topological indices. The book also analyzes recent topics in condensed matter theory and concludes by surveying active subfields of research such as insulators with point-group symmetries and the stability of topological semimetals. Problems at the end of each chapter offer opportunities to test knowledge and engage with frontier research issues. Topological Insulators and Topological Superconductors will provide graduate students and researchers with the physical understanding and mathematical tools needed to embark on research in this rapidly evolving field.
B. Andrei Bernevig is the Eugene and Mary Wigner Assistant Professor of Theoretical Physics at Princeton University. Taylor L. Hughes is an assistant professor in the condensed matter theory group at the University of Illinois, Urbana-Champaign.
"This book presents an array of increasingly complicated problems centered around the idea of the topology of a band in k-space and the theorem that the Chern number determines the Hall effect. It will be invaluable to those who are in tune with the conceptual structure of modern band theory as reconfigured by Haldane and his fellow travelers."--Philip W. Anderson, Nobel Laureate in Physics
"One of the most exciting developments in condensed matter physics over the last seven or eight years has been the topic of topological insulators and superconductors. The present book, by one of the original pioneers in this area, is a very up-to-date and comprehensive introduction to the theory of these systems. It will be extremely useful to both graduate students and more senior researchers."--Anthony J. Leggett, University of Illinois, Urbana-Champaign
"An authoritative and sophisticated introduction to the mathematics of topological insulation."--Robert B. Laughlin, Stanford University
"This book gives the first comprehensive introduction to the theory of topological insulators and superconductors--an exciting new field in condensed matter physics. The authors contributed to the discovery of the first topological insulator HgTe, and now present a readable account accessible to most graduate students."--Shoucheng Zhang, Stanford University
"This excellent book introduces a relatively new topic in condensed matter physics. The material is well developed and sufficient detail is given for students to follow arguments and derivations. With a hands-on, no-nonsense approach, Topological Insulators and Topological Superconductors will be a mainstay in the field for years to come."--Marcel Franz, University of British Columbia
Series:London Mathematical Society Monographs
Cloth | March 2013 | ISBN: 9780691156545
588 pp. | 7 x 10
By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties.
PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary. This book explains in detail the following topics about PEL-type Shimura varieties and their compactifications:
A construction of smooth integral models of PEL-type Shimura varieties by defining and representing moduli problems of abelian schemes with PEL structures
An analysis of the degeneration of abelian varieties with PEL structures into semiabelian schemes, over noetherian normal complete adic base rings
A construction of toroidal and minimal compactifications of smooth integral models of PEL-type Shimura varieties, with detailed descriptions of their structure near the boundary
Through these topics, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai).
Kai-Wen Lan is assistant professor of mathematics at the University of Minnesota.
Paper | April 2013 | ISBN: 9780691157153
Cloth | April 2013 | ISBN: 9780691157122
320 pp. | 6 x 9 | 3 line illus.
This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process.
Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Holder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right half-plane. Epstein and Mazzeo also demonstrate precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.
Preface xi
1 Introduction 1
1.1 Generalized Kimura Diffusions 3
1.2 Model Problems 5
1.3 Perturbation Theory 9
1.4 Main Results 10
1.5 Applications in Probability Theory 13
1.6 Alternate Approaches 14
1.7 Outline of Text 16
1.8 Notational Conventions 20
I Wright-Fisher Geometry and the Maximum Principle 23
2 Wright-Fisher Geometry 25
2.1 Polyhedra and Manifolds with Corners 25
2.2 Normal Forms and Wright-Fisher Geometry 29
3 Maximum Principles and Uniqueness Theorems 34
3.1 Model Problems 34
3.2 Kimura Diffusion Operators on Manifolds with Corners 35
3.3 Maximum Principles for theHeat Equation 45
II Analysis of Model Problems 49
4 The Model Solution Operators 51
4.1 The Model Problemin 1-dimension 51
4.2 The Model Problem in Higher Dimensions 54
4.3 Holomorphic Extension 59
4.4 First Steps Toward Perturbation Theory 62
5 Degenerate Holder Spaces 64
5.1 Standard Holder Spaces 65
5.2 WF-Holder Spaces in 1-dimension 66
6 Holder Estimates for the 1-dimensional Model Problems 78
6.1 Kernel Estimates for Degenerate Model Problems 80
6.2 Holder Estimates for the 1-dimensional Model Problems 89
6.3 Propertiesof the Resolvent Operator 103
7 Holder Estimates for Higher Dimensional CornerModels 107
7.1 The Cauchy Problem 109
7.2 The Inhomogeneous Case 122
7.3 The Resolvent Operator 135
8 Holder Estimates for Euclidean Models 137
8.1 Holder Estimates for Solutions in the Euclidean Case 137
8.2 1-dimensional Kernel Estimates 139
9 Holder Estimates for General Models 143
9.1 The Cauchy Problem 145
9.2 The Inhomogeneous Problem 149
9.3 Off-diagonal and Long-time Behavior 166
9.4 The Resolvent Operator 169
III Analysis of Generalized Kimura Diffusions 179
10 Existence of Solutions 181
10.1 WF-Holder Spaces on a Manifold with Corners 182
10.2 Overview of the Proof 187
10.3 The Induction Argument 191
10.4 The Boundary Parametrix Construction 194
10.5 Solution of the Homogeneous Problem 205
10.6 Proof of the Doubling Theorem 208
10.7 The Resolvent Operator and C0-Semi-group 209
10.8 Higher Order Regularity 211
11 The Resolvent Operator 218
11.1 Construction of the Resolvent 220
11.2 Holomorphic Semi-groups 229
11.3 DiffusionsWhere All Coefficients Have the Same Leading Homogeneity 230
12 The Semi-group on C0(P) 235
12.1 The Domain of the Adjoint 237
12.2 The Null-space of L 240
12.3 Long Time Asymptotics 243
12.4 Irregular Solutions of the Inhomogeneous Equation 247
A Proofs of Estimates for the Degenerate 1-d Model 251
A.1 Basic Kernel Estimates 252
A.2 First Derivative Estimates 272
A.3 Second Derivative Estimates 278
A.4 Off-diagonal and Large-t Behavior 291
Bibliography 301
Index 305
Paper | April 2013 |ISBN: 9780691157764
Cloth | April 2013 |ISBN: 9780691157757
184 pp. | 6 x 9 | 5 line illus.
Since its introduction by Friedhelm Waldhausen in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing Waldhausen's program from more than thirty years ago.
The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory.
The proof has two main parts. The essence of the first part is a "desingularization," improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.
Friedhelm Waldhausen is professor emeritus of mathematics at Bielefeld University. Bjorn Jahren is professor of mathematics at the University of Oslo. John Rognes is professor of mathematics at the University of Oslo.