本篇精選18本美國數學學會出版的「AMS/IP Studies in Advanced Mathematics 系列」叢書。
該系列叢書由美國數學學會（AMS）和國際出版社（International Press）聯合出版，哈佛大學數學系丘成桐教授主編，涵蓋高等數學研究領域重要主題，包括專著、講義、文集和會議論文集。 讀者可透過 Ebook Central 平臺查閱、利用這些書籍。
1. Differential Equations and Mathematical Physics
《微分方程和數學物理》
作者：Rudi Weikard, University of Alabama; Gilbert Weinstein, University of Alabama
內容簡介: This volume contains the proceedings of the 1999 International Conference on Differential Equations and Mathematical Physics. The contributions selected for this volume represent some of the most important presentations by scholars from around the world on developments in this area of research. The papers cover topics in the general area of linear and nonlinear differential equations and their relation to mathematical physics, such as multiparticle Schrödinger operators, stability of matter, relativity theory, fluid dynamics, spectral and scattering theory including inverse problems.
2. Extensions of the Stability Theorem of the Minkowski Space in General Relativity
《廣義相對論中閔可夫斯基空間穩定性定理的擴展》
作者: Lydia Bieri, Harvard University; Nina Zipser, Harvard University
内容簡介: This book consists of two independent works： Part I is “Solutions of the Einstein Vacuum Equations”, by Lydia Bieri. Part II is “Solutions of the Einstein-Maxwell Equations”, by Nina Zipser. A famous result of Christodoulou and Klainerman is the global nonlinear stability of Minkowski spacetime. In this book, Bieri and Zipser provide two extensions to this result. In the first part, Bieri solves the Cauchy problem for the Einstein vacuum equations with more general, asymptotically flat initial data, and describes precisely the asymptotic behavior. In particular, she assumes less decay in the power of r and one less derivative than in the Christodoulou–Klainerman result. She proves that in this case, too, the initial data, being globally close to the trivial data, yields a solution which is a complete spacetime, tending to the Minkowski spacetime at infinity along any geodesic. In contrast to the original situation, certain estimates in this proof are borderline in view of decay, indicating that the conditions in the main theorem on the decay at infinity on the initial data are sharp.
3. Foundations of $p$-adic Teichmüller Theory
《p 進Teichmüller 理論基礎》
作者: Shinichi Mochizuki （望月新一）, Research Institute for the Mathematical Sciences, Kyoto, Japan
內容簡介: This book lays the foundation for a theory of uniformization of p-adic hyperbolic curves and their moduli. On one hand, this theory generalizes the Fuchsian and Bers uniformizations of complex hyperbolic curves and their moduli to nonarchimedian places. That is why in this book, the theory is referred to as p-adic Teichmüller theory, for short. On the other hand, the theory may be regarded as a fairly precise hyperbolic analog of the Serre-Tate theory of ordinary abelian varieties and their moduli.
The theory of uniformization of p-adic hyperbolic curves and their moduli was initiated in a previous work by Mochizuki. And in some sense, this book is a continuation and generalization of that work. This book aims to bridge the gap between the approach presented and the classical uniformization of a hyperbolic Riemann surface that is studied in undergraduate complex analysis.
4. Fourth International Congress of Chinese Mathematicians
《第四屆世界華人數學家大會論文集》
作者: Lizhen Ji（季理真）, University of Michigan; Kefeng Liu（劉克峰）, University of California， Los Angeles; Lo Yang（楊樂）, Chinese Academy of Sciences; Shing-Tung Yau（丘成桐）, Harvard University
內容簡介: This volume represents selected proceedings of the Fourth International Congress of Chinese Mathematicians, held in Hangzhou, China. The Congress brought together eminent Chinese and overseas mathematicians to discuss the latest developments in pure and applied mathematics. Approximately fifteen hundred mathematicians participated in the Congress. Included in this volume are the complete Morningside Lectures, the complete plenary lectures, and selected invited lectures.
5. Geometric Analysis on the Heisenberg Group and Its Generalizations
《海森堡群的幾何分析及其推廣》
作者: Ovidiu Calin, Eastern Michigan University; Der-Chen Chang（張德健）, Georgetown University; Peter Greiner, University of Toronto
內容簡介: ：The theory of subRiemannian manifolds is closely related to Hamiltonian mechanics. In this book, the authors examine the properties and applications of subRiemannian manifolds that automatically satisfy the Heisenberg principle, which may be useful in quantum mechanics. In particular, the behavior of geodesics in this setting plays an important role in finding heat kernels and propagators for Schrödinger’s equation. One of the novelties of this book is the introduction of techniques from complex Hamiltonian mechanics.
6. Geometric topology:1993 Georgia international topology conference
《幾何拓撲：1993年喬治亞國際拓撲會議論文集》
作者: William H. Kazez
內容簡介: This is Part 1 of a two-part volume reflecting the proceedings of the 1993 Georgia International Topology Conference held at the University of Georgia during the month of August. The texts include research and expository articles and problem sets. The conference covered a wide variety of topics in geometric topology.
7. Geometry and Nonlinear Partial Differential Equations
《幾何與非線性偏微分方程》
作者: Shuxing Chen（陳恕行）, Fudan University; S.-T. Yau（丘成桐）, Harvard University
內容簡介: This book presents the proceedings of a conference on geometry and nonlinear partial differential equations dedicated to Professor Buqing Su in honor of his one-hundredth birthday. It offers a look at current research by Chinese mathematicians in differential geometry and geometric areas of mathematical physics. It is suitable for advanced graduate students and research mathematicians interested in geometry, topology, differential equations, and mathematical physics.
8. Heat Kernel and Analysis on Manifolds
《流形上的熱核和分析》
作者: Alexander Grigor’yan, University of Bielefeld
內容簡介: The heat kernel has long been an essential tool in both classical and modern mathematics but has become especially important in geometric analysis as a result of major innovations beginning in the 1970s. The methods based on heat kernels have been used in areas as diverse as analysis, geometry, and probability, as well as in physics. This book is a comprehensive introduction to heat kernel techniques in the setting of Riemannian manifolds, which inevitably involves analysis of the Laplace–Beltrami operator and the associated heat equation.
The first ten chapters cover the foundations of the subject, while later chapters deal with more advanced results involving the heat kernel in a variety of settings. The exposition starts with an elementary introduction to Riemannian geometry, proceeds with a thorough study of the spectral-theoretic, Markovian, and smoothness properties of the Laplace and heat equations on Riemannian manifolds, and concludes with Gaussian estimates of heat kernels.
Grigor’yan has written this book with the student in mind, in particular by including over 400 exercises. The text will serve as a bridge between basic results and current research.
9. Higher Franz-Reidemeister Torsion
《更高維的Reidemeister-Franz撓率》
作者: Kiyoshi Igusa, Brandeis University
內容簡介: The book is devoted to the theory of topological higher Franz-Reidemeister torsion in K-theory. The author defines the higher Franz-Reidemeister torsion based on Volodin’s K-theory and Borel’s regulator map. He describes its properties and generalizations and studies the relation between the higher Franz-Reidemeister torsion and other torsions used in K-theory: Whitehead torsion and Ray-Singer torsion. He also presents methods of computing higher Franz-Reidemeister torsion, illustrates them with numerous examples, and describes various applications of higher Franz-Reidemeister torsion, particularly for the study of homology of mapping class groups. Packed with up-to-date information, the book provides a unique research and reference tool for specialists working in algebraic topology and K-theory.
10. Integrable Systems, Geometry, and Topology
《可積系統、幾何和拓撲》
作者: Chuu-Lian Terng（滕楚蓮）, University of California, Irvine
內容簡介: As is well-known, for many soliton equations, the solutions have interpretations as differential geometric objects, and thereby techniques of soliton equations have been successfully applied to the study of geometric problems.
The article by Burstall gives a beautiful exposition on isothermic surfaces and their relations to integrable systems, and the two articles by Guest give an introduction to quantum cohomology, carry out explicit computations of the quantum cohomology of flag manifolds and Hirzebruch surfaces, and give a survey of Givental’s quantum differential equations. The article by Heintze, Liu, and Olmos is on the theory of isoparametric submanifolds in an arbitrary Riemannian manifold, which is related to the n-wave equation when the ambient manifold is Euclidean. Mukai-Hidano and Ohnita present a survey on the moduli space of Yang-Mills-Higgs equations on Riemann surfaces. The article by Terng and Uhlenbeck explains the gauge equivalence of the matrix non-linear Schrödinger equation, the Schrödinger flow on Grassmanian, and the Heisenberg Feromagnetic model.
The book provides an introduction to integrable systems and their relation to differential geometry. It is suitable for advanced graduate students and research mathematicians.
11. Introduction to $p$-adic Analytic Number Theory
《P進數解析數論導論》
作者: M. Ram Murty, Queen’s University
內容簡介: This book is an elementary introduction to p-adic analysis from the number theory perspective. With over 100 exercises, it will acquaint the non-expert with the basic ideas of the theory and encourage the novice to enter this fertile field of research.
The main focus of the book is the study of p-adic L-functions and their analytic properties. It begins with a basic introduction to Bernoulli numbers and continues with establishing the Kummer congruences. These congruences are then used to construct the p-adic analog of the Riemann zeta function and p-adic analogs of Dirichlet’s L-functions. Featured is a chapter on how to apply the theory of Newton polygons to determine Galois groups of polynomials over the rational number field. As motivation for further study, the final chapter introduces Iwasawa theory.
The book treats the subject informally, making the text accessible to non-experts. It would make a nice independent text for a course geared toward advanced undergraduates and beginning graduate students.
12. Knots, Braids, and Mapping Class Groups—Papers Dedicated to Joan S. Birman
《扭結、辮子和映射類群 | 獻給瓊·伯曼的論文集》
作者: Jane Gilman, Rutgers University; William W. Menasco, State University of New York; Xiao-Song Lin（林曉松）, University of California， Riverside
內容簡介: There are a number of specialties in low-dimensional topology that can find in their “family tree” a common ancestry in the theory of surface mappings. These include knot theory as studied through the use of braid representations and 3-manifolds as studied through the use of Heegaard splittings. The study of the surface mapping class group (the modular group) is of course a rich subject in its own right, with relations to many different fields of mathematics and theoretical physics. But its most direct and remarkable manifestation is probably in the vast area of low-dimensional topology. Although the scene of this area has been changed dramatically and experienced significant expansion since the original publication of Professor Joan Birman’s seminal work, Braids, Links, and Mapping Class Groups (Princeton University Press), she brought together mathematicians whose research span many specialties, all of common lineage.
The topics covered are quite diverse. Yet they reflect well the aim and spirit of the conference in low-dimensional topology held in honor of Joan S.Birman’s 70th birthday at Columbia University (New York, NY), which was to explore how these various specialties in low-dimensional topology have diverged in the past 20–25 years, as well as to explore common threads and potential future directions of development.
13. Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part I
《拉格朗日相交Floer理論：異常和障礙（第一卷）》
作者: Kenji Fukaya（深谷賢治）, Kyoto University; Yong-Geun Oh, University of Wisconsin, Madison; Hiroshi Ohta, Nagoya University; Kaoru Ono, Hokkaido University
內容簡介: This is a two-volume series research monograph on the general Lagrangian Floer theory and on the accompanying homological algebra of filtered A∞-algebras. This book provides the most important step towards a rigorous foundation of the Fukaya category in general context. In Volume I, general deformation theory of the Floer cohomology is developed in both algebraic and geometric contexts. An essentially self-contained homotopy theory of filtered A∞algebras and A∞bimodules and applications of their obstruction-deformation theory to the Lagrangian Floer theory are presented. Volume II contains detailed studies of two of the main points of the foundation of the theory: transversality and orientation. The study of transversality is based on the virtual fundamental chain techniques (the theory of Kuranishi structures and their multisections) and chain level intersection theories. A detailed analysis comparing the orientations of the moduli spaces and their fiber products is carried out. A self-contained account of the general theory of Kuranishi structures is also included in the appendix of this volume.
14. Lagrangian Intersection Floer Theory:Anomaly and Obstruction, Part II
《拉格朗日相交Floer理論：異常和障礙（第二卷）》
作者: Kenji Fukaya（深谷賢治）, Kyoto University; Yong-Geun Oh, University of Wisconsin, Madison; Hiroshi Ohta, Nagoya University; Kaoru Ono, Hokkaido University
內容簡介: This is a two-volume series research monograph on the general Lagrangian Floer theory and on the accompanying homological algebra of filtered A∞-algebras. This book provides the most important step towards a rigorous foundation of the Fukaya category in general context. In Volume I, general deformation theory of the Floer cohomology is developed in both algebraic and geometric contexts. An essentially self-contained homotopy theory of filtered A∞algebras and A∞bimodules and applications of their obstruction-deformation theory to the Lagrangian Floer theory are presented. Volume II contains detailed studies of two of the main points of the foundation of the theory: transversality and orientation. The study of transversality is based on the virtual fundamental chain techniques (the theory of Kuranishi structures and their multisections) and chain level intersection theories. A detailed analysis comparing the orientations of the moduli spaces and their fiber products is carried out. A self-contained account of the general theory of Kuranishi structures is also included in the appendix of this volume.
15. Laguerre Calculus and Its Applications on the Heisenberg Group
《拉蓋爾積分公式及其在海森堡群上的應用》
作者: Carlos Berenstein, University of Maryland; Der-Chen Chang（張德健）, Georgetown University; Jingzhi Tie, University of Georgia
內容簡介: For nearly two centuries, the relation between analytic functions of one complex variable, their boundary values, harmonic functions, and the theory of Fourier series has been one of the central topics of study in mathematics. The topic stands on its own, yet also provides very useful mathematical applications.
This text provides a self-contained introduction to the corresponding questions in several complex variables: namely, analysis on the Heisenberg group and the study of the solutions of the boundary Cauchy-Riemann equations. In studying this material, readers are exposed to analysis in non-commutative compact and Lie groups, specifically the rotation group and the Heisenberg groups—both fundamental in the theory of group representations and physics.
Introduced in a concrete setting are the main ideas of the Calderón-Zygmund-Stein school of harmonic analysis. Also considered in the book are some less conventional problems of harmonic and complex analysis, in particular, the Morera and Pompeiu problems for the Heisenberg group, which relates to questions in optics, tomography, and engineering.
The book was borne of graduate courses and seminars held at the University of Maryland (College Park), the University of Toronto (ON), Georgetown University (Washington, DC), and the University of Georgia (Athens). Readers should have an advanced undergraduate understanding of Fourier analysis and complex analysis in one variable.
16. Lectures on Chaotic Dynamical Systems
《混沌動力學系統講義》
作者: Valentin Afraimovich, Sze-Bi Hsu（許世壁）等
內容簡介: This book is devoted to chaotic nonlinear dynamics. It presents a consistent, up-to-date introduction to the field of strange attractors, hyperbolic repellers, and nonlocal bifurcations. The authors keep the highest possible level of “physical” intuition while staying mathematically rigorous. In addition, they explain a variety of important nonstandard algorithms and problems involving the computation of chaotic dynamics.
The book will help readers who are not familiar with nonlinear dynamics to understand and appreciate sophisticated modern dynamical systems and chaos. Intended for courses in either mathematics, physics, or engineering, prerequisites are calculus, differential equations, and functional analysis.
17. Lectures on Mean Curvature Flows
《平均曲率流講義》
作者: Xi-Ping Zhu（朱熹平）, Zhongshan University
內容簡介: “Mean curvature flow” is a term that is used to describe the evolution of a hypersurface whose normal velocity is given by the mean curvature. In the simplest case of a convex closed curve on the plane, the properties of the mean curvature flow are described by Gage-Hamilton’s theorem. This theorem states that under the mean curvature flow, the curve collapses to a point, and if the flow is diluted so that the enclosed area equals π, the curve tends to the unit circle.
In this book, the author gives a comprehensive account of fundamental results on singularities and the asymptotic behavior of mean curvature flows in higher dimensions. Among other topics, he considers in detail Huisken’s theorem (a generalization of Gage-Hamilton’s theorem to higher dimension), evolution of non-convex curves and hypersurfaces, and the classification of singularities of the mean curvature flow.
Because of the importance of the mean curvature flow and its numerous applications in differential geometry and partial differential equations, as well as in engineering, chemistry, and biology, this book can be useful to graduate students and researchers working in these areas. The book would also make a nice supplementary text for an advanced course in differential geometry.
Prerequisites include basic differential geometry, partial differential equations, and related applications.
18. Lectures on Systems, Control, and Information: Lectures at the Morningside Center of Mathematics
《系統、控制和資訊講義：晨興數學中心講座》
作者: Lei Guo（郭雷）, Chinese Academy of Sciences; Stephen S.-T. Yau（丘成桐）, University of Illinois
內容簡介: This volume presents lectures delivered at a workshop held at the Chinese Academy of Sciences (Bejing). The following articles are included: “Nonlinear Systems Control” by R. Brockett, “Adaptive Control of Discrete-Time Nonlinear Systems with Structural Uncertainties” by L.-L. Xie and L. Guo, “Networks and Learning” by P. R. Kumar, “Mathematical Aspects of the Power Control Problem in Mobile Communication Systems” by C. W. Sung and W. S. Wong, and “Brockett’s Problem on Nonlinear Filtering Theory” by S. S.-T. Yau.
Basic concepts and current research are both presented in this book. The volume offers a comprehensive and easy-to-follow account of many fundamental issues in this diverse field. It would be a suitable text for a graduate course on wireless communication.
美國數學學會及其電子書簡介
美國數學會（American Mathematical Society，AMS）是一個由專業數學家組成的協會，致力於數學研究和學術研究。
美國數學學會出版的圖書是全世界數學文獻領域最受尊敬的合集之一。 AMS出版具有突破性的專刊，本科生和研究生的教材，會議集，翻譯，數學普及的作品。
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