Part of London Mathematical Society Lecture Note Series
DATE PUBLISHED: January 2023
FORMAT: Paperback ISBN: 9781009288088
Rectifiable sets, measures, currents and varifolds are foundational concepts in geometric measure theory. The last four decades have seen the emergence of a wealth of connections between rectifiability and other areas of analysis and geometry, including deep links with the calculus of variations and complex and harmonic analysis. This short book provides an easily digestible overview of this wide and active field, including discussions of historical background, the basic theory in Euclidean and non-Euclidean settings, and the appearance of rectifiability in analysis and geometry. The author avoids complicated technical arguments and long proofs, instead giving the reader a flavour of each of the topics in turn while providing full references to the wider literature in an extensive bibliography. It is a perfect introduction to the area for researchers and graduate students, who will find much inspiration for their own research inside.
Surveys a wide range of topics related to rectifiability
Focuses on the basic ideas rather than complicated technical arguments while providing extensive references to the wider literature
Includes plenty of new research to help newcomers to the field find interesting research topics
Introduction
1. Preliminaries
2. Rectifiable curves
3. One-dimensional rectifiable sets
4. Higher dimensional rectifiable sets
5. Uniform rectifiability
6. Rectifiability of measures
7. Rectifiable sets in metric spaces
8. Heisenberg and Carnot groups
9. Bounded analytic functions and the Cauchy transform
10. Singular integrals
11. Harmonic measure and elliptic measures
12. Sets of finite perimeter and functions of bounded variation
13. Currents and varifolds
14. Minimizers and quasiminimizers
15. Rectifiability of singularities
16. Miscellaneous topics related to rectifiability
References
Index.
Part of New Mathematical Monographs
PUBLICATION PLANNED FOR: February 2023
FORMAT: HardbackISBN: 9781009215916
This text, the first of two volumes, provides a comprehensive and self-contained introduction to a wide range of fundamental results from ergodic theory and geometric measure theory. Topics covered include: finite and infinite abstract ergodic theory, Young's towers, measure-theoretic Kolmogorov-Sinai entropy, thermodynamics formalism, geometric function theory, various kinds of conformal measures, conformal graph directed Markov systems and iterated functions systems, semi-local dynamics of analytic functions, and nice sets. Many examples are included, along with detailed explanations of essential concepts and full proofs, in what is sure to be an indispensable reference for both researchers and graduate students.
Largely self-contained, with detailed proofs
Highlights the connections between various subfields and aspects of dynamical systems ergodic theory, and other areas of mathematics
Brings together research previously scattered across many different papers
PUBLICATION PLANNED FOR: February 2023
FORMAT: Paperbac kISBN: 9781009166157
Optimization on Riemannian manifolds-the result of smooth geometry and optimization merging into one elegant modern framework-spans many areas of science and engineering, including machine learning, computer vision, signal processing, dynamical systems and scientific computing. This text introduces the differential geometry and Riemannian geometry concepts that will help students and researchers in applied mathematics, computer science and engineering gain a firm mathematical grounding to use these tools confidently in their research. Its charts-last approach will prove more intuitive from an optimizer's viewpoint, and all definitions and theorems are motivated to build time-tested optimization algorithms. Starting from first principles, the text goes on to cover current research on topics including worst-case complexity and geodesic convexity. Readers will appreciate the tricks of the trade for conducting research and for numerical implementations sprinkled throughout the book.
Provides readers rigorous, versatile tools motivated by applicative goals
Takes a charts-last approach to differential geometry that is more intuitive to an optimization researcher
Discusses topics of new importance, including worst-case complexity and geodesic convexity, getting readers up to speed with current research trends
Includes finer points and tricks of the trade that would normally require mentorship or years of study to pick up
Notation
1. Introduction
2. Simple examples
3. Embedded geometry: first order
4. First-order optimization algorithms
5. Embedded geometry: second order
6. Second-order optimization algorithms
7. Embedded submanifolds: examples
8. General manifolds
9. Quotient manifolds
10. Additional tools
11. Geodesic convexity
References
Index.
Part of Encyclopedia of Mathematics and its Applications
PUBLICATION PLANNED FOR: February 2023
FORMAT: Hardback ISBN: 9781316510865
Constructive mathematics ? mathematics in which 'there exists' always means 'we can construct' ? is enjoying a renaissance. fifty years on from Bishop's groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject's myriad aspects. Major themes include: constructive algebra and geometry, constructive analysis, constructive topology, constructive logic and foundations of mathematics, and computational aspects of constructive mathematics. A series of introductory chapters provides graduate students and other newcomers to the subject with foundations for the surveys that follow. Edited by four of the most eminent experts in the field, this is an indispensable reference for constructive mathematicians and a fascinating vista of modern constructivism for the increasing number of researchers interested in constructive approaches.
Gives a complete overview of modern constructive mathematics, including pure constructive mathematics, foundational issues, and applications to computing and probability
Each chapter is a commissioned survey from leading specialists of a particular sub-topic within constructive mathematics
The book begins with introductory chapters giving graduate students and other newcomers the foundations they need to engage with the rest
Part of New Mathematical Monographs
PUBLICATION PLANNED FOR: February 2023
FORMAT: Hardback ISBN: 9781009215978
This text, the second of two volumes, builds on the foundational material on ergodic theory and geometric measure theory provided in Volume I, and applies all the techniques discussed to describe the beautiful and rich dynamics of elliptic functions. The text begins with an introduction to topological dynamics of transcendental meromorphic functions, before progressing to elliptic functions, discussing at length their classical properties, measurable dynamics and fractal geometry. The authors then look in depth at compactly non-recurrent elliptic functions. Much of this material is appearing for the first time in book or paper form. Both senior and junior researchers working in ergodic theory and dynamical systems will appreciate what is sure to be an indispensable reference.
Largely self-contained, with detailed proofs
Highlights the connections between various subfields and aspects of dynamical systems ergodic theory, and other areas of mathematics
Brings together research previously scattered across many different papers, and contains new material not previously published elsewhere