Author: Tadashi Ochiai
Publisher: American Mathematical Society
ISBN: 1470456737
Category : Mathematics
Languages : en
Pages : 228
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Book Description
Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to $p$-adic $L$-functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation is to update the classical theory for class groups, taking into account the changed point of view on Iwasawa theory. The goal of this second part of the three-part publication is to explain various aspects of the cyclotomic Iwasawa theory of $p$-adic Galois representations.
Author: Tadashi Ochiai
Publisher: American Mathematical Society
ISBN: 1470456729
Category : Mathematics
Languages : en
Pages : 167
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Book Description
Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to $p$-adic $L$-functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation of this book is an update of the classical theory for class groups taking into account the changed point of view on Iwasawa theory. The goal of this first part of the two-part publication is to explain the theory of ideal class groups, including its algebraic aspect (the Iwasawa class number formula), its analytic aspect (Leopoldt–Kubota $L$-functions), and the Iwasawa main conjecture, which is a bridge between the algebraic and the analytic aspects. The second part of the book will be published as a separate volume in the same series, Mathematical Surveys and Monographs of the American Mathematical Society.
Author: Donald Yau
Publisher: American Mathematical Society
ISBN: 1470478102
Category : Mathematics
Languages : en
Pages : 439
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Book Description
Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. The three books published by the AMS in the Mathematical Surveys and Monographs series under the title Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory (Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories, Volume II: Braided Bimonoidal Categories with Applications?this book, and Volume III: From Categories to Structured Ring Spectra) provide a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user-friendly resource for beginners and experts alike. Part 1 of this book studies braided bimonoidal categories, with applications to quantum groups and topological quantum computation. It is proved that the categories of modules over a braided bialgebra, of Fibonacci anyons, and of Ising anyons form braided bimonoidal categories. Two coherence theorems for braided bimonoidal categories are proved, confirming the Blass-Gurevich Conjecture. The rest of this part discusses braided analogues of Baez's Conjecture and the monoidal bicategorical matrix construction in Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories. Part 2 studies ring and bipermutative categories in the sense of Elmendorf-Mandell, braided ring categories, and $E_n$-monoidal categories, which combine $n$-fold monoidal categories with ring categories.
Author: Niles Johnson
Publisher: American Mathematical Society
ISBN: 1470478110
Category : Mathematics
Languages : en
Pages : 633
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Book Description
Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. The three books published by the AMS in the Mathematical Surveys and Monographs series under the title Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory (Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories, Volume II: Braided Bimonoidal Categories with Applications, and Volume III: From Categories to Structured Ring Spectra?this book) provide a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user-friendly resource for beginners and experts alike. Part 1 of this book is a detailed study of enriched monoidal categories, pointed diagram categories, and enriched multicategories. Using this machinery, Part 2 discusses the rich interconnection between the higher ring-like categories, homotopy theory, and algebraic $K$-theory. Starting with a chapter on homotopy theory background, the first half of Part 2 constructs the Segal $K$-theory functor and the Elmendorf-Mandell $K$-theory multifunctor from permutative categories to symmetric spectra. For the latter, the detailed treatment here includes identification and correction of some subtle errors concerning its extended domain. The second half applies the $K$-theory multifunctor to small ring, bipermutative, braided ring, and $E_n$-monoidal categories to obtain, respectively, strict ring, $E_{infty}$-, $E_2$-, and $E_n$-symmetric spectra.
Author: Jacques Sauloy
Publisher: American Mathematical Society
ISBN: 1470478404
Category : Mathematics
Languages : en
Pages : 693
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Book Description
The roots of the modern theories of differential and $q$-difference equations go back in part to an article by George D. Birkhoff, published in 1913, dealing with the three ?sister theories? of differential, difference and $q$-difference equations. This book is about $q$-difference equations and focuses on techniques inspired by differential equations, in line with Birkhoff's work, as revived over the last three decades. It follows the approach of the Ramis school, mixing algebraic and analytic methods. While it uses some $q$-calculus and is illustrated by $q$-special functions, these are not its main subjects. After a gentle historical introduction with emphasis on mathematics and a thorough study of basic problems such as elementary $q$-functions, elementary $q$-calculus, and low order equations, a detailed algebraic and analytic study of scalar equations is followed by the usual process of transforming them into systems and back again. The structural algebraic and analytic properties of systems are then described using $q$-difference modules (Newton polygon, filtration by the slopes). The final chapters deal with Fuchsian and irregular equations and systems, including their resolution, classification, Riemann-Hilbert correspondence, and Galois theory. Nine appendices complete the book and aim to help the reader by providing some fundamental yet not universally taught facts. There are 535 exercises of various styles and levels of difficulty. The main prerequisites are general algebra and analysis as taught in the first three years of university. The book will be of interest to expert and non-expert researchers as well as graduate students in mathematics and physics.
Author: Michael Kapovich
Publisher: American Mathematical Society
ISBN: 1470474255
Category : Mathematics
Languages : en
Pages : 295
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Book Description
This book offers an alternative proof of the Bestvina?Feighn combination theorem for trees of hyperbolic spaces and describes uniform quasigeodesics in such spaces. As one of the applications of their description of uniform quasigeodesics, the authors prove the existence of Cannon?Thurston maps for inclusion maps of total spaces of subtrees of hyperbolic spaces and of relatively hyperbolic spaces. They also analyze the structure of Cannon?Thurston laminations in this setting. Furthermore, some group-theoretic applications of these results are discussed. This book also contains background material on coarse geometry and geometric group theory.
Author: Ellen Elizabeth Eischen
Publisher: American Mathematical Society
ISBN: 1470474921
Category : Mathematics
Languages : en
Pages : 199
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Book Description
The Langlands program has been a very active and central field in mathematics ever since its conception over 50 years ago. It connects number theory, representation theory and arithmetic geometry, and other fields in a profound way. There are nevertheless very few expository accounts beyond the GL(2) case. This book features expository accounts of several topics on automorphic forms on higher rank groups, including rationality questions on unitary group, theta lifts and their applications to Arthur's conjectures, quaternionic modular forms, and automorphic forms over functions fields and their applications to inverse Galois problems. It is based on the lecture notes prepared for the twenty-fifth Arizona Winter School on “Automorphic Forms beyond GL(2)”, held March 5–9, 2022, at the University of Arizona in Tucson. The speakers were Ellen Eischen, Wee Teck Gan, Aaron Pollack, and Zhiwei Yun. The exposition of the book is in a style accessible to students entering the field. Advanced graduate students as well as researchers will find this a valuable introduction to various important and very active research areas.
Author: Renate Scheidler
Publisher:
ISBN: 9781935107026
Category :
Languages : en
Pages :
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Book Description
Author: Kazuya Kato
Publisher: American Mathematical Soc.
ISBN: 9780821808634
Category : Mathematics
Languages : en
Pages : 180
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Book Description
This is the English translation of the original Japanese book. In this volume, "Fermat's Dream", core theories in modern number theory are introduced. Developments are given in elliptic curves, $p$-adic numbers, the $\zeta$-function, and the number fields. This work presents an elegant perspective on the wonder of numbers. Number Theory 2 on class field theory, and Number Theory 3 on Iwasawa theory and the theory of modular forms, are forthcoming in the series.
Author:
Publisher:
ISBN:
Category : Monographic series
Languages : en
Pages : 1404
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Book Description
Vols. for 1980- issued in three parts: Series, Authors, and Titles.
