Format: Paperback / softback, 390 pages, height x width: 235x155 mm, Approx. 390 p.,
Series: Lecture Notes in Mathematics 2345
Pub. Date: 15-Jun-2024
ISBN-13: 9783031559136
This book delves into the p-adic Simpson correspondence, its construction, and development. Offering fresh and innovative perspectives on this important topic in algebraic geometry, the text serves a dual purpose: it describes an important tool in p-adic Hodge theory, which has recently attracted significant interest, and also provides a comprehensive resource for researchers. Unique among the books in the existing literature in this field, it combines theoretical advances, novel constructions, and connections to Hodge-Tate local systems.
This exposition builds upon the foundation laid by Faltings, the collaborative efforts of the two authors with T. Tsuji, and contributions from other researchers. Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence, whose construction has been taken up in several different ways. Following the approach they initiated with T. Tsuji, the authors develop new features of the p-adic Simpson correspondence, inspired by their construction of the relative Hodge-Tate spectral sequence. First, they address the connection to Hodge-Tate local systems. Then they establish the functoriality of the p-adic Simpson correspondence by proper direct image. Along the way, they expand the scope of their original construction.
The book targets a specialist audience interested in the intricate world of p-adic Hodge theory and its applications, algebraic geometry and related areas. Graduate students can use it as a reference or for in-depth study. Mathematicians exploring connections between complex and p-adic geometry will also find it valuable.
Preface.- An overview.- Preliminaries.- Local Study.- Global Study.- Relative cohomologies of Higgs-Tate algebras.Local Study.- Relative cohomology of Dolbeault modules.- References.- Index..
Format: Paperback / softback, 210 pages, height x width: 235x155 mm,
1 Illustrations, black and white; X, 210 p., 1
Series: Lecture Notes in Mathematics 2349
Pub. Date: 09-Jul-2024
ISBN-13: 9783031572005
This book concentrates on the famous Grothendieck inequality and the continued search for the still unknown best possible value of the real and complex Grothendieck constant (an open problem since 1953). It describes in detail the state of the art in research on this fundamental inequality, including Krivine's recent contributions, and sheds light on related questions in mathematics, physics and computer science, particularly with respect to the foundations of quantum theory and quantum information theory. Unifying the real and complex cases as much as possible, the monograph introduces the reader to a rich collection of results in functional analysis and probability. In particular, it includes a detailed, self-contained analysis of the multivariate distribution of complex Gaussian random vectors. The notion of Completely Correlation Preserving (CCP) functions plays a particularly important role in the exposition.The prerequisites are a basic knowledge of standard functional analysis, complex analysis, probability, optimisation and some number theory and combinatorics. However, readers missing some background will be able to consult the generous bibliography, which contains numerous references to useful textbooks.
The book will be of interest to PhD students and researchers in functional analysis, complex analysis, probability, optimisation, number theory and combinatorics, in physics (particularly in relation to the foundations of quantum mechanics) and in computer science (quantum information and complexity theory).
Format: Hardback, 345 pages, height x width: 254x178 mm, 2 Illustrations,
color; 60 Illustrations, black and white; X, 345 p. 62 illus., 2 illus. in color.
Series: Springer Texts in Statistics
Pub. Date: 02-Jul-2024
ISBN-13: 9783031558542
This book continues the mission of the previous text by the author, Lectures on Categorical Data Analysis, by expanding on the introductory concepts from that volume and providing a mathematically rigorous presentation of advanced topics and current research in statistical techniques which can be applied in the social, political, behavioral, and life sciences. It presents an intuitive and unified discussion of an array of themes in categorical data analysis, and the emphasis on structure over stochastics renders many of the methods applicable in machine learning environments and for the analysis of big data.
The book focuses on graphical models, their application in causal analysis, the analytical properties of parameterizations of multivariate discrete distributions, marginal models, and coordinate-free relational models. To guide the readers in future research, the volume provides references to original papers and also offers detailed proofs of most of the significant results. Like the previous volume, it features exercises and research questions, making it appropriate for graduate students, as well as for active researchers.
1. Introduction.-
2. Undirected graphical models.-
3. Directed graphical models.-
4. Marginal models: definition.-
5. Marginal log-linear models: applications.- ?6. Path models.-
7. Relational models: definition and interpretation.-
8. Relational models as exponential families.-
9. Relational models: estimation and testing.-
10. Model testing.-
11. The mixture index of fit.
Format: Paperback / softback, 350 pages, height x width: 235x155 mm,
100 Illustrations, color; Approx. 350 p. 100 illus. in color.,
Series: UNITEXT 159
Pub. Date: 28-Jul-2024
ISBN-13: 9783031569098
This book provides a first introduction to the fundamental concepts of commutative algebra. What sets it apart from other textbooks is the extensive collection of 400 solved exercises, providing readers with the opportunity to apply theoretical knowledge to practical problem solving, fostering a deeper and more thorough understanding of the subject.
The topics presented here are not commonly found in a single text. Consequently, the first part presents definitions, properties, and results crucial for understanding and solving the exercises, serving also as a valuable reference. The second part contains the exercises and a section titled with "True or False?" questions, which serves as a valid self-assessment test. Considerable effort has been invested in crafting solutions that provide the essential details, aiming for a well-balanced presentation. We intend to guide students systematically through the challenging process of writing mathematical proofs with formal correctness and clarity.
Our approach is constructive, aiming to illustrate concepts by applying them to the analysis of multivariate polynomial rings and modules over a principal ideal domain (PID) whenever feasible. Algorithms for computing these objects facilitate the generation of diverse examples. In particular, the structure of finitely generated modules over a PID is analyzed using the Smith canonical form of matrices. Furthermore, various properties of polynomial rings are investigated through the application of Buchbergers Algorithm for computing Grobner bases.
This book is intended for advanced undergraduates or masters students, assuming only basic knowledge of finite fields, Abelian groups, and linear algebra. This approach aims to inspire the curiosity of readers and encourages them to find their own proofs while providing detailed solutions to support their learning. It also provides students with the necessary tools to pursue more advanced studies in commutative algebra and related subjects.
Part I Theory.- 1 Rings.- 2 The Ring K[ x1, . . . , xn].- 3 Affine
Algebraic Varieties.- 4 Modules.- 5 Tensor Product.- 6 Localization.- 7
Noetherian and Artinian Rings. Primary Decomposition.- Part II Exercises.-
8 Rings and Ideals.- 9 Polynomials, Grobner Bases, Resultant, and Varieties.-
10 Modules.- 11 Tensor Product.- 12 Localization.- 13 Noetherian and Artinian
Modules.- 14 True or False?.- 15 Review Exercises.- Part III Proofs and
Solutions.- 16 Proofs of Theoretical Results.- 17 Solutions to the Exercises.
Format: Paperback / softback, 461 pages, height x width: 240x168 mm,
23 Illustrations, color; 5 Illustrations, black and white; XIV, 461 p. 28 illus., 23 illus. in color.,
Series: Mathematics Study Resources 10
Pub. Date: 15-Sep-2024
ISBN-13: 9783662689226
This textbook, which is based on the second edition of a book that has been previously published in German language, provides a comprehension-oriented introduction to asymptotic stochastics. It is aimed at the beginning of a master's degree course in mathematics and covers the material that can be taught in a four-hour lecture with two-hour exercises. Individual chapters are also suitable for seminars at the end of a bachelor's degree course.
In addition to more basic topics such as the method of moments in connection with the convergence in distribution or the multivariate central limit theorem and the delta method, the book covers limit theorems for U-statistics, the Wiener process and Donsker's theorem, as well as the Brownian bridge, with applications to statistics. It concludes with a central limit theorem for triangular arrays of Hilbert space-valued random elements with applications to weighted L2 statistics.
The book is deliberately designed forself-study. It contains 138 self-questions, which are answered at the end of each chapter, as well as 194 exercises with solutions.
This book is a translation of an original German edition. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation.
Preface.- List of Symbols.- 1 Prerequisites from Probability Theory.- 2
A Poisson Limit Theorem for Triangular Arrays .- 3 The Method of Moments .- 4
A Central Limit Theorem for Stationary m-Dependent Sequences.- 5 The
multivariate normal distribution .- 6 Convergence in Distribution and Central
Limit Theorem in Rd .- 7 Empirical Distribution Function.- 8 Limit Theorems
for U-Statistics.- 9 Basic Concepts of Estimation Theory.- 10 Maximum
Likelihood Estimation.- 11 Asymptotic (relative) efficiency of estimators.-
12 Likelihood Ratio Tests.- 13 Probability Measures on Metric Spaces.- 14
Convergence of Distributions in Metric Spaces.- 15 Wiener Process, Donskers
Theorem, and Brownian Bridge.- 16 The Space D[ 0,1], Empirical Processes.- 17
Random Elements in Separable Hilbert Spaces.- Afterword.- Solutions to the
Problems.- Bibliography.- Index.