FORMAT: Hardback
ISBN: 9781009170321
LENGTH: 268 pages DIMENSIONS: 229 x 152 mm
AVAILABILITY: Not yet published - available from June 2022
Part of Cambridge Tracts in Mathematics
Point-counting results for sets in real Euclidean space have found remarkable applications
to diophantine geometry, enabling significant progress on the Andre?Oort and Zilber?Pink
conjectures. The results combine ideas close to transcendence theory with the strong tameness
properties of sets that are definable in an o-minimal structure, and thus the material treated
connects ideas in model theory, transcendence theory, and arithmetic. This book describes
the counting results and their applications along with their model-theoretic and transcendence
connections. Core results are presented in detail to demonstrate the flexibility of the method,
while wider developments are described in order to illustrate the breadth of the diophantine
conjectures and to highlight key arithmetical ingredients. The underlying ideas are elementary
and most of the book can be read with only a basic familiarity with number theory and complex
algebraic geometry. It serves as an introduction for postgraduate students and researchers
to the main ideas, results, problems, and themes of current research.
The only integrated book treatment of this material
Sets out the various different ways in which point-counting is applied, beyond the basic case of special-point problems
Gives model-theoretic, transcendence-theoretic and arithmetic context, demonstrating how key arithmetic
results and conjectures fit in
1. Introduction
Part I. Point-Counting and Diophantine Applications:
2. Point-counting
3. Multiplicative Manin?Mumford
4. Powers of the Modular Curve as Shimura Varieties
5. Modular Andre?Oort
6. Point-Counting and the Andre?Oort Conjecture
Part II. O-Minimality and Point-Counting:
7. Model theory and definable sets
8. O-minimal structures
9. Parameterization and point-counting
10. Better bounds
11. Point-counting and Galois orbit bounds
12. Complex analysis in O-minimal structures
Part III. Ax?Schanuel Properties:
13. Schanuel's conjecture and Ax?Schanuel
14. A formal setting
15. Modular Ax?Schanuel
16. Ax?Schanuel for Shimura varieties
17. Quasi-periods of elliptic curves
Part IV. The Zilber?Pink Conjecture:
18. Sources
19. Formulations
20. Some results
21. Curves in a power of the modular curve
22. Conditional modular Zilber?Pink
23. O-minimal uniformity
24. Uniform Zilber?Pink
References
List of notation
Index.
Part of New Mathematical Monographs
FORMAT: Hardback
ISBN: 9781108838672
LENGTH: 500 pages DIMENSIONS: 229 x 152 mm
AVAILABILITY: Not yet published - available from July 2022
This book presents the probabilistic methods around Hardy martingales for an audience interested
in applications to complex, harmonic, and functional analysis. Building on work of Bourgain, Garling,
Maurey, Pisier, and Varopoulos, it discusses in detail those martingale spaces that reflect characteristic
qualities of complex analytic functions. Its particular themes are holomorphic random variables on
Wiener space, and Hardy martingales on the infinite torus product, and their numerous deep applications
to the geometry and classification of complex Banach spaces, e.g. the embedding of L1 into L1/H1,
the isomorphic classification theorem for the class of poly-disk algebras, or the real variables characterization
of Banach spaces with the analytic Radon Nikodym property. Including key background material on stochastic
analysis and Banach space theory, it is suitable for a wide spectrum of researchers and graduate students
working in classical and functional analysis.
Presents the theory of Hardy martingales for an audience interested in applications to complex, harmonic,
and functional analysis
Includes important core material on stochastic analysis and Banach space theory
Suitable for a wide spectrum of researchers and graduate students working in classical and functional analysis
Preface
1. Stochastic Holomorphy
2. Hardy Martingales
3. Embedding L1 in L1=H01
4. Embedding L1 in X or L1/X 5. Isomorphic Invariants
6. Operators on Lp(L1)
7. Formative Examples
Bibliography
Notation index
Subject index.
Part of London Mathematical Society Lecture Note Series
FORMAT: PaperbackI
SBN: 9781108969031
LENGTH: 200 pages DIMENSIONS: 228 x 152 mm
AVAILABILITY: Not yet published - available from August 2022
This modern introduction to operator theory on spaces with indefinite inner product discusses
the geometry and the spectral theory of linear operators on these spaces, the deep interplay with
complex analysis, and applications to interpolation problems. The text covers the key results from
the last four decades in a readable way with full proofs provided throughout. Step by step, the reader
is guided through the intricate geometry and topology of spaces with indefinite inner product, before
progressing to a presentation of the geometry and spectral theory on these spaces. The author
carefully highlights where difficulties arise and what tools are available to overcome them.
With generous background material included in the appendices, this text is an excellent resource
for graduate students as well as researchers in operator theory, functional analysis, and related areas.
Introduces the concepts step by step and provides full proofs of all the results to make the theory accessible
to readers without specialised training
Explains the deep connections between operator theory on spaces with indefinite inner product and complex analysis
Discusses applications in areas such as interpolation theory, dilation theory, and realisations of operator-valued functions
1. Inner product spaces
2. Angular operators
3. Subspaces of Kre?n spaces
4. Linear operators on Kre?n spaces
5. Selfadjoint projections and unitary operators
6. Techniques of induced Kre?n spaces
7. Plus/minus-operators
8. Geometry of contractive operators
9. Invariant maximal semidefinite subspaces
10. Hankel operators and interpolation problems
11. Spectral theory for selfadjoint operators
12. Quasi-contractions
13. More on definitisable operators
Appendix
References
Symbol index
Subject index
FORMAT: Paperback
ISBN: 9781108708791
LENGTH: 254 pages DIMENSIONS: 244 x 170 x 14 mm WEIGHT: 0.414kg
CONTAINS: 11 b/w illus. 4 tables
AVAILABILITY: Not yet published - available from September 2022
Sets are central to mathematics and its foundations, but what are they? I
n this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses
their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with
which these conceptions are associated. He shows that the conceptual landscape includes not only
the naive and iterative conceptions but also the limitation of size conception, the definite conception,
the stratified conception and the graph conception. In addition, he presents a novel, minimalist account
of the iterative conception which does not require the existence of a relation of metaphysical dependence
between a set and its members. His book will be of interest to researchers and advanced students in logic
and the philosophy of mathematics.
The first book-length treatment of conceptions of set
Provides historical background on a variety of competing approaches and explores the mathematical theories
with which they are associated
Sets out a novel account and defence of the iterative conception of set
1. Concepts and conceptions
2. The iterative conception
3. Challenges to the iterative conception
4. The naive conception
5. The limitation of size conception
6. The stratified conception
7. The graph conception.
ISBN 9781032173191
Published September 30, 2021 by CRC Press
312 Pages 3 B/W Illustrations
This book is aimed at both experts and non-experts with an interest in getting acquainted with sequence spaces,
matrix transformations and their applications. It consists of several new results which are part of the recent
research on these topics. It provides different points of view in one volume, e.g. their topological properties,
geometry and summability, fuzzy valued study and more.
This book presents the important role sequences and series play in everyday life, it covers geometry of
Banach Sequence Spaces, it discusses the importance of generalized limit, it offers spectrum and fine
spectrum of several linear operators and includes fuzzy valued sequences which exhibits the study of
sequence spaces in fuzzy settings.
This book is the main attraction for those who work in Sequence Spaces, Summability Theory and would also
serve as a good source of reference for those involved with any topic of Real or Functional Analysis.
1. Basic Functional Analysis. 2. Geometric Properties of Some Sequence Spaces. 3. Infinite Matrices.
4. Difference Sequence Spaces. 5. Almost Convergence and Classes of Related Matrix Transformations.
6. Spectrum of Some Triangle Matrices on Some Sequence Spaces. 7. Sets of Fuzzy Valued Sequences and Series.
ISBN 9780367535230
December 29, 2021 Forthcoming by CRC Press
482 Pages 43 B/W Illustrations
This book provides advanced undergraduate physics and mathematics students with an accessible
yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics.
Readers, working through the book, will obtain a thorough understanding of symmetry principles and their
application in mechanics, field theory, and general relativity, and in addition acquire the necessary calculational
skills to tackle more sophisticated questions in theoretical physics.
Most of the topics covered in this book have previously only been scattered across many different sources
of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive
volume.
Contains a modern, streamlined presentation of classical topics, which are normally taught separately
Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem,
the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity
Focuses on the clear presentation of the mathematical notions and calculational technique
Chapter 1. Manifolds and Tensors.
Chapter 2. Geometry and Integration on Manifolds.
Chapter 3. Symmetries of Manifolds.
Chapter 4. Newtonian Mechanics.
Chapter 5. Lagrangian Methods and Symmetry.
Chapter 6. Relativistic Mechanics.
Chapter 7. Lie Groups.
Chapter 8. Lie Algebras.
Chapter 9. Representations.
Chapter 10. Rotations and Euclidean Symmetry.
ISBN 9781032108988
February 23, 2022 Forthcoming by Chapman and Hall/CRC
440 Pages 66 B/W Illustrations
Introduction to Linear Algebra: Computation, Application and Theory is designed for students who have
never been exposed to the topics in a linear algebra course. The text is ?lled with interesting and diverse
application sections but is also a theoretical text which aims to train students to do succinct computation
in a knowledgeable way. After completing the course with this text, the student will not only know the best
and shortest way to do linear algebraic computations but will also know why such computations are both
e?ective and successful.
Includes cutting edge applications in machine learning and data analytics.
Suitable as a primary text for undergraduates studying linear algebra.
Requires very little in the way of pre-requisites.
1. Examples of Vector Spaces. 1.1. First Vector Space: Tuples. 1.2. Dot Product. 1.3. Application: Geometry. 1.4. Second Vector Space: Matrices. 1.5. Matrix Multiplication.
2. Matrices and Linear Systems. 2.1. Systems of Linear Equations. 2.2. Gaussian Elimination. 2.3. Application: Markov Chains. 2.4. Application: The Simplex Method.
2.5. Elementary Matrices and Matrix Equivalence. 2.6. Inverse of a Matrix. 2.7. Application: The Simplex Method Revisited. 2.8. Homogeneous/Nonhomogeneous Systems and Rank.
2.9. Determinant. 2.10. Applications of the Determinant. 2.11. Application: Lu Factorization.
3. Vector Spaces. 3.1. Definition and Examples. 3.2. Subspace. 3.3. Linear Independence. 3.4. Span. 3.5. Basis and Dimension. 3.6. Subspaces Associated with a Matrix.
3.7. Application: Dimension Theorems.
4. Linear Transformations. 4.1. Definition and Examples. 4.2. Kernel and Image. 4.3. Matrix Representation. 4.4. Inverse and Isomorphism. 4.5. Similarity of Matrices.
4.6. Eigenvalues and Diagonalization. 4.7. Axiomatic Determinant. 4.8. Quotient Vector Space. 4.9. Dual Vector Space.
5. Inner Product Spaces. 5.1. Definition, Examples and Properties. 5.2. Orthogonal and Orthonormal. 5.3. Orthogonal Matrices. 5.4. Application: QR Factorization.
5.5. Schur Triangularization Theorem. 5.6. Orthogonal Projections and Best Approximation. 5.7. Real Symmetric Matrices. 5.8. Singular Value Decomposition.
5.9. Application: Least Squares Optimization.
6. Applications in Data Analytics. 6.1. Introduction. 6.2. Direction of Maximal Spread. 6.3. Principal Component Analysis. 6.4. Dimensionality Reduction.
6.5. Mahalanobis Distance. 6.6. Data Sphering. 6.7. Fisher Linear Discriminant Function. 6.8. Linear Discriminant Functions in Feature Space.
6.9. Minimal Square Error Linear Discriminant Function.
7. Quadratic Forms. 7.1. Introduction to Quadratic Forms. 7.2. Principal Minor Criterion. 7.3. Eigenvalue Criterion. 7.4. Application: Unconstrained Nonlinear Optimization.
7.5. General Quadratic Forms. Appendix A. Regular Matrices. Appendix B. Rotations and Reflections in Two Dimensions. Appendix C. Answers to Selected Exercises.