Part of Institute of Mathematical Statistics Monographs
Not yet published - available from January 2025
format: Hardback isbn: 9781009566995
Brownian motion is an important topic in various applied fields where the analysis of random events is necessary. Introducing Brownian motion from a statistical viewpoint, this detailed text examines the distribution of quadratic plus linear or bilinear functionals of Brownian motion and demonstrates the utility of this approach for time series analysis. It also offers the first comprehensive guide on deriving the Fredholm determinant and the resolvent associated with such statistics. Presuming only a familiarity with standard statistical theory and the basics of stochastic processes, this book brings together a set of important statistical tools in one accessible resource for researchers and graduate students. Readers also benefit from online appendices, which provide probability density graphs and solutions to the chapter problems.
Introduces Brownian motion from a statistical viewpoint and uses statistics based on its quadratic functionals to compute distributions
Defines the Fredholm determinant in connection with Brownian motion and explains how to derive in detail using various examples
Discusses time series analysis from a distributional point of view, combining Brownian motion and the Fredholm determinant
Part I. Theory:
1. Quadratic functionals of the Brownian motion
2. Integral equations and the Fredholm determinant
3. Integral equations and the resolvent
Part II. Applications:
4. Fredholm determinants for goodness of fit tests
5. Fredholm determinants in the state space model
6. Fredholm determinants in the moving average model
7. Fredholm determinants in the autoregressive model
8. Fredholm determinants for the fractional Brownian motion
References
Author index
Subject index.
Part of London Mathematical Society Lecture Note Series
Not yet published - available from January 2025
format: Paperback isbn: 9781009519526
This work develops techniques and basic results concerning the homotopy theory of enriched diagrams and enriched Mackey functors. Presentation of a category of interest as a diagram category has become a standard and powerful technique in a range of applications. Diagrams that carry enriched structures provide deeper and more robust applications. With an eye to such applications, this work provides further development of both the categorical algebra of enriched diagrams, and the homotopy theoretic applications in K-theory spectra. The title refers to certain enriched presheaves, known as Mackey functors, whose homotopy theory classifies that of equivariant spectra. More generally, certain stable model categories are classified as modules - in the form of enriched presheaves - over categories of generating objects. This text contains complete definitions, detailed proofs, and all the background material needed to understand the topic. It will be indispensable for graduate students and researchers alike.
Includes complete definitions and detailed proofs, filling a number of gaps in the current literature and extending the basic techniques to a robust theory
Establishes uniform terminology and notation to make the text more accessible to a non-expert audience
Self-contained, with appendices containing a lighter treatment of general background and three chapters discussing specialized background from the research literature
Details open questions that arise from the material and are suitable for students
1. Motivations from equivariant topology
Part I. Background on Multicategories and K-Theory Functors:
2. Categorically enriched multicategories
3. Infinite loop space machines
4. Homotopy theory of multicategories
Part II. Homotopy Theory of Pointed Multicategories, M1-Modules, and Permutative Categories:
5. Pointed multicategories and M1-modules model all connective spectra
6. Multiplicative homotopy theory of pointed multicategories and M1-modules
Part III. Enrichment of Diagrams and Mackey Functors in Closed Multicategories:
7. Multicategorically enriched categories
8. Change of multicategorical enrichment
9. The closed multicategory of permutative categories
10. Self-enrichment and standard enrichment of closed multicategories
11. Enriched diagrams and Mackey functors of closed multicategories
Part IV. Homotopy Theory of Enriched Diagrams and Mackey Functors:
12. Homotopy equivalences between enriched diagram and Mackey functor categories
13. Applications to multicategories and permutative categories
Appendices: A. Categories
B. Enriched category theory
C. Multicategories
D. Open questions
Bibliography
Index.
Part of London Mathematical Society Lecture Note Series
Not yet published - available from March 2025
format: Paperbackisbn: 9781009498982
This is the first book to revisit the theory of rewriting in the context of strict higher categories, through the unified approach provided by polygraphs, and put it in the context of homotopical algebra. The first half explores the theory of polygraphs in low dimensions and its applications to the computation of the coherence of algebraic structures. Illustrated with algorithmic computations on algebraic structures, the only prerequisite in this section is basic category theory. The theory is introduced step-by-step, with detailed proofs. The second half introduces and studies the general notion of n-polygraph, before addressing the homotopy theory of these polygraphs. It constructs the folk model structure on the category on strict higher categories and exhibits polygraphs as cofibrant objects. This allows the formulation of higher-dimensional generalizations of the coherence results developed in the first half. Graduate students and researchers in mathematics and computer science will find this work invaluable.
The first book about polygraphs, providing detailed proofs and presentations
Introduces the theory step-by-step using low-dimensional illustrations
Provides a global overview of the topic, bringing together category theory, rewriting and homological algebra
Part of New Mathematical Monographs
Not yet published - available from April 2025
format: Hardback isbn: 9781009538770
The K-stability of Fano varieties has been a major area of research over the last decade, ever since the Yau-Tian-Donaldson conjecture was resolved. This is the first book to give a comprehensive algebraic treatment of this emerging field. It introduces all the notions of K-stability that have been used over the development of the subject, proves their equivalence, and discusses newly developed theory, including several new proofs for existing theorems. Aiming to be as self-contained as possible, the text begins with a chapter covering essential background knowledge, and includes exercises throughout to test understanding. Written by someone at the forefront of developments in the area, it will be a source of inspiration for graduate students and researchers who work in algebraic geometry.
Gives a comprehensive treatment of the emerging field of K-stability
Contains several new proofs for existing theorems
Aims to be as self-contained as possible, allowing researchers in algebraic geometry to become familiar with K-stability
Preface
Notion and conventions
Preliminaries
1. Higher dimensional geometry background
2. K-stability via test configurations
3. K-stability via filtrations
4. K-stability via valuations
5. Higher rank finite generation
6. Reduced stability
7. K-moduli stack
8. K-moduli space
9. Positivity of the CM line bundle
Appendix A. Solutions to exercises
Bibliography
Glossary
Index.