This book explores the work of Bernhard Riemann and its impact on mathematics, philosophy and physics. It features contributions from a range of fields, historical expositions, and selected research articles that were motivated by Riemann's ideas and demonstrate their timelessness. The editors are convinced of the tremendous value of going into Riemann's work in depth, investigating his original ideas, integrating them into a broader perspective, and establishing ties with modern science and philosophy. Accordingly, the contributors to this volume are mathematicians, physicists, philosophers and historians of science. The book offers a unique resource for students and researchers in the fields of mathematics, physics and philosophy, historians of science, and more generally to a wide range of readers interested in the history of ideas.

This book explores the work of Bernhard Riemann and its impact on mathematics, philosophy and physics. It features contributions from a range of fields, historical expositions, and selected research articles that were motivated by Riemann's ideas and demonstrate their timelessness. The editors are convinced of the tremendous value of going into Riemann's work in depth, investigating his original ideas, integrating them into a broader perspective, and establishing ties with modern science and philosophy. Accordingly, the contributors to this volume are mathematicians, physicists, philosophers and historians of science. The book offers a unique resource for students and researchers in the fields of mathematics, physics and philosophy, historians of science, and more generally to a wide range of readers interested in the history of ideas.

The concept of symmetric space is of central importance in many branches of mathematics. Compactifications of these spaces have been studied from the points of view of representation theory, geometry, and random walks. This work is devoted to the study of the interrelationships among these various compactifications and, in particular, focuses on the martin compactifications. It is the first exposition to treat compactifications of symmetric spaces systematically and to uniformized the various points of view. The work is largely self-contained, with comprehensive references to the literature. It is an excellent resource for both researchers and graduate students.

This is the thirteenth issue (Vol. 7, No. 1, July 2019) of the Notices of the International Congress of Chinese Mathematicians (or ICCM Notices, for short), the official periodical of the ICCM organization. This special issue honors Shing-Tung Yau on the occasion of his 70th birthday. Published semi-annually, the Notices bring news, research, and presentation of various perspectives, relevant to Chinese mathematics development and education. Readers of the Notices will find research papers on various topics by prominent experts from around the world, interesting and timely articles on current applications and trends, biographical and historical essays, profiles of important institutions of research and learning, and more.

Noncompact symmetric and locally symmetric spaces naturally appear in many mathematical theories, including analysis (representation theory, nonabelian harmonic analysis), number theory (automorphic forms), algebraic geometry (modulae) and algebraic topology (cohomology of discrete groups). In most applications, it is necessary to form an appropriate compactification of the space. The literature dealing with such compactifications is vast. The main purpose of this book is to introduce uniform constructions of most of the known compactifications with emphasis on their geometric and topological structures. The book is divided into three parts. Part I studies compactifications of Riemannian symmetric spaces and their arithmetic quotients. Part II is a study of compact smooth manifolds. Part III studies the compactification of locally symmetric spaces. Familiarity with the theory of semisimple Lie groups is assumed, as is familiarity with algebraic groups defined over the rational numbers in later parts of the book, although most of the pertinent material is recalled as presented. and research mathematicians interested in the applications of Lie theory and representation theory to diverse fields of mathematics.

The concept of symmetric space is of central importance in many branches of mathematics. Compactifications of these spaces have been studied from the points of view of representation theory, geometry, and random walks. This work is devoted to the study of the interrelationships among these various compactifications and, in particular, focuses on the martin compactifications. It is the first exposition to treat compactifications of symmetric spaces systematically and to uniformized the various points of view.

Key features:

* definition and detailed analysis of the Martin compactifications

* new geometric Compactification, defined in terms of the Tits building, that coincides with the Martin Compactification at the bottom of the positive spectrum.

* geometric, non-inductive, description of the Karpelevic Compactification

* study of the well-know isomorphism between the Satake compactifications and the Furstenberg compactifications

* systematic and clear progression of topics from geometry to analysis, and finally to random walks

The work is largely self-contained, with comprehensive references to the literature. It is an excellent resource for both researchers and graduate students.

Groups and group actions are probably the most central objects in mathematics. Comprising volumes 31, 32, 40 and 41 of the ALM series, the Handbook of Group Actions presents survey articles on the topic of group actions and how they appear in several mathematical contexts. The general subject matter is organized under the following sections: geometry, mapping class groups, knot groups, topology, representation theory, deformation theory, and discrete groups. The various articles deal with both classical material and modern developments. They are written by specialists in their respective subject areas, and addressed to graduate students who want to learn the theory, as well as to specialists as a reference. This is the fourth volume of the Handbook of Group Actions.

Comprising volumes 28 and 29 of the ALM series, this outstanding collection presents all the survey papers of Shing-Tung Yau published to date (through 2013), each with Yau's own commentary. Among these are several papers not otherwise easily accessible. Also presented are several commentaries on Yau's work written by outstanding scholars from around the world especially for publication here. Shing-Tung Yau's work is mainly in differential geometry, and he is one of the originators of the broad subject of geometric analysis - in which he remains one of the most active participants. His contributions have had an influence on both physics and mathematics, and he has long been active at the interface between geometry and theoretical physics. His proof of the positive energy theorem in general relativity demonstrated - sixty years after its discovery - that Einstein's theory is consistent and stable. His proof of the Calabi conjecture allowed physicists - using Calabi-Yau compactification - to show that string theory is a viable candidate for a unified theory of nature. Calabi-Yau manifolds are part of the standard toolkit of string theorists today. He was awarded the Fields Medal in 1982 and the Wolf Prize in 2010, and has received many other honors. The Selected Expository Works of Shing-Tung Yau with Commentary provides the reader with systematic commentary on all aspects of mathematics by a contemporary master. The reader can thereby see the world of mathematics through his particular perspective, and gain understanding of the motivation and evolution of mathematical ideas.

This two-part volume represents the proceedings of the Fifth International Congress of Chinese Mathematicians, held at Tsinghua University, Beijing, in December 2010. The Congress brought together eminent Chinese and overseas mathematicians to discuss the latest developments in pure and applied mathematics. Included are 60 papers based on lectures given at the conference.

This volume presents a selection of work based upon lectures given by distinguished mathematicians at the Yau Mathematical Sciences Center at Tsinghua University, and at the Tsinghua Sanya International Mathematics Forum. Mathematics is very diversified and broad, but there are some basic underlying themes. The editors hope that the diversity of material presented here conveys the richness and unity of mathematics.

This book consists of five expository papers on moduli spaces and locally symmetric spaces based on lecture notes given by the authors at two instructional workshops held at the Morningside Center of Mathematics, Beijing, in February 2017 and March 2019. They give accessible and systematic introductions to moduli spaces of Riemann surfaces, algebraic curves, moduli spaces of vector bundles on Riemann surfaces, moduli spaces of singularities, and compactification of a natural class of locally symmetric spaces. They should serve as good instructions to some important aspects of these important spaces.

The International Consortium of Chinese Mathematicians was founded in 2016, with the purpose of promoting and advancing mathematics within the extended Chinese community, and building good relationships between Chinese mathematicians and other mathematicians throughout the world. The first meeting of the Consortium was held in December 2017 at Sun Yat-sen University in Guanzhou, Guangdong, with about 150 mathematicians presenting their outstanding works in various branches of mathematics, ranging from number theory to geometry, analysis, and applied mathematics. This volume presents 30 expository papers by speakers at the meeting, based upon their lectures given there.

The International Congress of Chinese Mathematicians (ICCM) is an important event among the large international community of mathematicians of Chinese descent. The Seventh Congress was held at Beijing, China in August 2016. This is the two-volume set comprising the Proceedings of the Seventh Congress of Chinese Mathematicians. It presents four Morningside Lectures, 16 Plenary Lectures, and 31 Invited Lectures-all dealing with the latest developments in mathematics. The Proceedings of the Seventh Congress of Chinese Mathematicians are an important reference for researchers in all fields of mathematics, and make a valuable addition to the collection of a university library.

Complex geometry has been extensively studied and developed since the 19th century. This volume examines the subject from a global, historical perspective. It begins with an essay on the historical development of complex geometry, with extensive quotations of experts past and present, followed by a discussion of Calabi–Yau manifolds, and a look at Shing-Tung Yau as a writer and mathematician. Next, it presents a series of selected papers, beginning with Bernhard Riemann’s thesis on the foundation of complex analysis, and his masterpiece on the foundations of geometry, followed by numerous seminal papers of modern practitioners: Atiyah, Bott, Chern, Calabi, Chow, Donaldson, Hirzebruch, Kähler, Kodaira, Siu, Uhlenbeck, and Yau. The volume concludes with a set of commentaries written by Yau, from his personal perspective, on the broad subject of complex geometry and its applications; and finally with an extensive list of papers on complex geometry from the 20th and 21st centuries, categorized by topic.

Groups and group actions are probably the most central objects in mathematics. Comprising volumes 31, 32, 40 and 41 of the ALM series, the Handbook of Group Actions presents survey articles on the topic of group actions and how they appear in several mathematical contexts. The general subject matter is organized under the following sections: geometry, mapping class groups, knot groups, topology, representation theory, deformation theory, and discrete groups. The various articles deal with both classical material and modern developments. They are written by specialists in their respective subject areas, and addressed to graduate students who want to learn the theory, as well as to specialists as a reference. This is the third volume of the Handbook of Group Actions.

Groups and group actions are probably the most central objects in mathematics. Comprising volumes 31, 32, 40 and 41 of the ALM series, the Handbook of Group Actions presents survey articles on the topic of group actions and how they appear in several mathematical contexts. The general subject matter is organized under the following sections: geometry, mapping class groups, knot groups, topology, representation theory, deformation theory, and discrete groups. The various articles deal with both classical material and modern developments. They are written by specialists in their respective subject areas, and addressed to graduate students who want to learn the theory, as well as to specialists as a reference. This is the four-volume set.

This volume consists of survey papers and introductions pertaining to Hodge theory, variation of Hodge structures, L(2)-methods in complex analysis and geometry, and related results in algebraic geometry. Contributors include some of the world's leading experts: Ayoub, Bierstone, Griffiths, M. Green, Hain, and Ohsawa.

The concept of Riemann surfaces was introduced in Riemann's thesis, and the moduli space of Riemann surfaces was defined by Riemann in a masterpiece a few years later. Due to a broad connection with many subjects in mathematics and physics, Riemann surfaces and their moduli spaces have been intensively studied and should continue to attract attention in years to come. Recently, there has been an explosion of interest in and work on tropical algebraic curves-analogues of algebraic curves over the complex numbers and hence of Riemann surfaces. This book is an accessible introduction to all these topics, with special emphasis given to their many connections with subjects such as algebraic geometry, complex analysis, hyperbolic geometry, topology, geometric group theory, and mathematical physics.

This volume presents a set of never-before-published notes from lectures given by S.-S. Chern in 1951 on the topic of “Minimal Submanifolds in a Riemannian Manifold.� Also presented are five of Chern’s expository papers which complement the lecture notes and provide an overview of the scope and power of differential geometry: From Triangles to Manifolds, Curves and Surfaces in Euclidean Space, Characteristic Classes and Characteristic Forms, Geometry and Physics, and The Geometry of G-Structures.

This volume presents two sets of never-before-published notes from lectures given by S.-S. Chern. The set entitled Differential Manifolds" gives a smooth and rapid introduction to differential manifolds and differential geometry, while the set entitled Lectures on Integral Geometry is an accessible introduction to that subject. Also presented is a paper by Chern that serves as a gentle introduction to differential geometry, and reviews the status of global differential geometry in 1971.

This is the first of two volumes constituting The Legacy of Bernhard Riemann After One Hundred and Fifty Years. Bernhard Riemann (1826-1866) possessed an original and broad vision of mathematics together with powerful skill. His work continues to influence almost all major branches of mathematics. The twenty-three papers in the two-volume set examine Riemann, his work, and his significance in the context of modern mathematical developments. Contributing authors (to the two-volume set): Michael F. Atiyah, M. V. Berry, Ching-Li Chai, Brian Conrey, Jean-Pierre Demailly, F. T. Farrell, James Glimm, David Harbater, Lizhen Ji, Jurgen Jost, Wolfgang Luck, Dan Marchesin, William H. Meeks III, Peter W. Michor, James S. Milne, Frans Oort, Joaquin Perez, Bradley Plohr, Herman J. J. te Riele, Michael Rubinstein, Norbert Schappacher, Chi-Wang Shu, Dennis Sullivan, Claire Voisin, Shing-Tung Yau.

This is the second of two volumes constituting The Legacy of Bernhard Riemann After One Hundred and Fifty Years. Bernhard Riemann (1826-1866) possessed an original and broad vision of mathematics together with powerful skill. His work continues to influence almost all major branches of mathematics. The twenty-three papers in the two-volume set examine Riemann, his work, and his significance in the context of modern mathematical developments. Contributing authors (to the two-volume set): Michael F. Atiyah, M. V. Berry, Ching-Li Chai, Brian Conrey, Jean-Pierre Demailly, F. T. Farrell, James Glimm, David Harbater, Lizhen Ji, Jurgen Jost, Wolfgang Luck, Dan Marchesin, William H. Meeks III, Peter W. Michor, James S. Milne, Frans Oort, Joaquin Perez, Bradley Plohr, Herman J. J. te Riele, Michael Rubinstein, Norbert Schappacher, Chi-Wang Shu, Dennis Sullivan, Claire Voisin, Shing-Tung Yau.

Groups and group actions are probably the most central objects in mathematics. Comprising volumes 31 and 32 of the ALM series (with further volumes forthcoming), the Handbook of Group Actions presents survey articles on the topic of group actions and how they appear in several mathematical contexts. The general subject matter is organized under the following sections: geometry, mapping class groups, knot groups, topology, representation theory, deformation theory, and discrete groups. The various articles deal with both classical material and modern developments. They are written by specialists in their respective subject areas, and addressed to graduate students who want to learn the theory, as well as to specialists as a reference. This is the first volume of the Handbook of Group Actions.

Groups and group actions are probably the most central objects in mathematics. Comprising volumes 31 and 32 of the ALM series (with further volumes forthcoming), the Handbook of Group Actions presents survey articles on the topic of group actions and how they appear in several mathematical contexts. The general subject matter is organized under the following sections: geometry, mapping class groups, knot groups, topology, representation theory, deformation theory, and discrete groups. The various articles deal with both classical material and modern developments. They are written by specialists in their respective subject areas, and addressed to graduate students who want to learn the theory, as well as to specialists as a reference. This is the second volume of the .

Mathematics has a long history, yet it is forever new. The concepts, theories, theorems, and conjectures of the past are the foundation of modern mathematics. How can one best learn them? One effective way is to read the great books on mathematics. The present volume serves as a guide to the most important books among the vast mathematical literature of the twentieth century, organized by subject. In doing so, it provides concise summaries of all major subjects in contemporary mathematics. Also included are brief one-paragraph introductions to numerous works of mathematics from throughout the ages, by old masters such as Euler, Galileo, Gauss, Kepler, Leibniz, Newton, Poincare, et al, accompanied by many images of early printed editions.