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.

This is the eighteenth issue (Vol. 9, No. 2, December 2021 of the Notices of the International Congress of Chinese Mathematicians (or ICCM Notices, for short), the official periodical of the ICCM organization. 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.

This volume presents five papers based on selected lectures given at the Current Developments in Mathematics conference, held in November 2019 at Harvard University.

This is the seventeenth issue (Vol. 9, No. 1, July 2021 of the Notices of the International Congress of Chinese Mathematicians (or ICCM Notices, for short), the official periodical of the ICCM organization. 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.

This is the twelfth issue (Vol. 6, No. 2, December 2018) of the Notices of the International Congress of Chinese Mathematicians (or ICCM Notices, for short), the official periodical of the ICCM organization. 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.

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.

The 2012 discovery of the Higgs boson was a sensational triumph-the culmination of a 48-year-long search that put the finishing touches on the so-called "Standard Model" of particle physics. While the celebrations were still underway, researchers in China were making plans to continue the centuries-old quest to identify the fundamental building blocks of nature. More specifically, they began laying the groundwork for a giant accelerator-up to 100 kilometers in circumference-that would transport physics into a previously inaccessible, high-energy realm where a host of new particles, and perhaps a sweeping new symmetry, might be found. The case for such an instrument is compelling: Even though the Standard Model can describe the behavior of particles with astounding accuracy, it is incomplete. The theory has little to say about the Big Bang, gravity, dark matter, dark energy, and other far-reaching phenomena. This book explains how an ambitious new machine-on the scale of China's proposed "Great Collider"-could provide us with a fuller understanding of the origins of our universe and its most basic constituents.

This is the fifteenth issue (Vol. 8, No. 1, July 2020) of the Notices of the International Congress of Chinese Mathematicians (or ICCM Notices, for short), the official periodical of the ICCM organization.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.

A Fields medalist recounts his lifelong transnational effort to uncover the geometric shape-the Calabi-Yau manifold-that may store the hidden dimensions of our universe. Harvard geometer and Fields medalist Shing-Tung Yau has provided a mathematical foundation for string theory, offered new insights into black holes, and mathematically demonstrated the stability of our universe. In this autobiography, Yau reflects on his improbable journey to becoming one of the world's most distinguished mathematicians. Beginning with an impoverished childhood in China and Hong Kong, Yau takes readers through his doctoral studies at Berkeley during the height of the Vietnam War protests, his Fields Medal-winning proof of the Calabi conjecture, his return to China, and his pioneering work in geometric analysis. This new branch of geometry, which Yau built up with his friends and colleagues, has paved the way for solutions to several important and previously intransigent problems. With complicated ideas explained for a broad audience, this book offers readers not only insights into the life of an eminent mathematician, but also an accessible way to understand advanced and highly abstract concepts in mathematics and theoretical physics.

In the twentieth century, American mathematicians began to make critical advances in a field previously dominated by Europeans. Harvard's mathematics department was at the center of these developments. A History in Sum is an inviting account of the pioneers who trailblazed a distinctly American tradition of mathematics--in algebraic geometry and topology, complex analysis, number theory, and a host of esoteric subdisciplines that have rarely been written about outside of journal articles or advanced textbooks. The heady mathematical concepts that emerged, and the men and women who shaped them, are described here in lively, accessible prose. The story begins in 1825, when a precocious sixteen-year-old freshman, Benjamin Peirce, arrived at the College. He would become the first American to produce original mathematics--an ambition frowned upon in an era when professors largely limited themselves to teaching. Peirce's successors--William Fogg Osgood and Maxime Bocher--undertook the task of transforming the math department into a world-class research center, attracting to the faculty such luminaries as George David Birkhoff. Birkhoff produced a dazzling body of work, while training a generation of innovators--students like Marston Morse and Hassler Whitney, who forged novel pathways in topology and other areas. Influential figures from around the world soon flocked to Harvard, some overcoming great challenges to pursue their elected calling. A History in Sum elucidates the contributions of these extraordinary minds and makes clear why the history of the Harvard mathematics department is an essential part of the history of mathematics in America and beyond.

This collection of papers constitutes a wide-ranging survey of recent developments in differential geometry and its interactions with other fields, especially partial differential equations and mathematical physics. This area of mathematics was the subject of a special program at the Institute for Advanced Study in Princeton during the academic year 1979-1980; the papers in this volume were contributed by the speakers in the sequence of seminars organized by Shing-Tung Yau for this program. Both survey articles and articles presenting new results are included. The articles on differential geometry and partial differential equations include a general survey article by the editor on the relationship of the two fields and more specialized articles on topics including harmonic mappings, isoperimetric and Poincare inequalities, metrics with specified curvature properties, the Monge-Arnpere equation, L2 harmonic forms and cohomology, manifolds of positive curvature, isometric embedding, and Kraumlhler manifolds and metrics. The articles on differential geometry and mathematical physics cover such topics as renormalization, instantons, gauge fields and the Yang-Mills equation, nonlinear evolution equations, incompleteness of space-times, black holes, and quantum gravity. A feature of special interest is the inclusion of a list of more than one hundred unsolved research problems compiled by the editor with comments and bibliographical information.

This is the fourteenth issue (Vol. 7, No. 2, December 2019) of the Notices of the International Congress of Chinese Mathematicians (or ICCM Notices, for short), the official periodical of the ICCM organization. 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.

General relativity came to life in 1915, when Albert Einstein formulated his field equations. These unify space, time, and gravitation, where the latter acts through curvature. Thereby the laws of physics obtain a geometric nature. Mathematical general relativity investigates spacetimes, which are manifolds equipped with a Lorentzian metric obeying the related Einstein-matter systems of nonlinear partial differential equations. The fruitful interactions of mathematics and physics in general relativity have produced breakthroughs in all the related research fields. This volume includes 14 articles presenting aspects of the most important general relativity research of the past 100 years. Among them we find cosmological and non-cosmological questions, the Cauchy problem for the Einstein equations, stability results, black holes and their formation, gravitational waves and their memory effect, the concept of energy, and the asymptotics of spacetimes. Through geometric analysis and advanced theory of nonlinear partial differential equations, long-standing problems have been solved lately that had remained locked since the beginnings of this beautiful theory of general relativity. Many more burning questions have been formulated and are yet to be answered.

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.

String theory says we live in a ten-dimensional universe, but that only four are accessible to our everyday senses. According to theorists, the missing six are curled up in bizarre structures known as Calabi-Yau manifolds. In "The Shape of Inner Space," Shing-Tung Yau, the man who mathematically proved that these manifolds exist, argues that not only is geometry fundamental to string theory, it is also fundamental to the very nature of our universe.

Time and again, where Yau has gone, physics has followed. Now for the first time, readers will follow Yau's penetrating thinking on where we've been, and where mathematics will take us next. A fascinating exploration of a world we are only just beginning to grasp, "The Shape of Inner Space" will change the way we consider the universe on both its grandest and smallest scales.

Consists of the proceedings of the Third International Congress of Chinese Mathematicians, held at the Chinese University of Hong Kong in December 2004. This book features lectures on topics such as algebra, number theory and cryptography; algebraic geometry and algebraic topology; geometric analysis; and, complex analysis and complex geometry.

This work presents lectures from the important String Theory International Conference held in 2002 in Hangzhou, China. These include talks given by several mathematicians of particular prominence in the field, among them Stephen Hawking and Edward Witten. Interest in string theory is driven largely by the hope that it will evolve to be the ultimate 'Theory of Everything'. Work on string theory has led to advances in many branches of mathematics. This rapidly developing subject is one of the mainstream topics of mathematics in the 21st century. The current volume presents lectures from the important String Theory International Conference held in 2002 in Hangzhou, China. These include talks given by several mathematicians of particular prominence in the field, among them Stephen Hawking and Edward Witten.

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.

In 2019, mathematicians around the world celebrated the 70th birthday of Professor Shing-Tung Yau and expressed their appreciation and admiration. A number of conferences were held throughout the year for this special occasion: at Harvard University (May), at the University of Rome (May/June), at the Chinese University of Hong Kong (June), at the 8th ICCM Congress at Tsinghua University (June), at Lehigh University (November), and elsewhere. This is the second of the two Surveys in Differential Geometry volumes in honor of Professor Yau, consisting of articles by some of the speakers at those special conferences, and several additional contributions. The articles in this volume include: Tristan C. Collins and Yun Shi, "Stability and the deformed Hermitian Yang-Mills equation"; Simon Donaldson and Christopher Scaduto, "Associative submanifolds and gradient cycles"; Davide Gaiotto and Edward Witten, "Probing quantization via branes"; L. Gottsche and M. Kool, "Sheaves on surfaces and virtual invariants"; Goo Ishikawa and Stanislaw Janeczko, "Symplectic singularities of differentiable mappings"; Jun Li and Chiu-Chu Melissa Liu, "Counting curves in quintic Calabi-Yau threefolds and Landau-Ginzberg models"; Nikolai Nadirashvili et al., "Conformally maximal metrics for Laplace eigenvalues on surfaces"; Kieran G. O'Grady, "Moduli of sheaves on $K3$'s and higher dimensional HK varieties"; Richard Schoen and Shing-Tung Yau "Positive scalar curvature and minimal hypersurface singularities"; Penny Smith and Karen Uhlenbeck, "Removeability of a codimension four singular set for solutions of a Yang-Mills-Higgs equation with small energy"; and Mu-Tao Wang, "Limits of quasi-local angular momentum on an isolated gravitating system".

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.

For the past forty years, Robert Bartnik has been one of the leading mathematicians working on mathematical general relativity and geometric analysis. Since his early dissertation work on the existence of maximal hypersurfaces in general asymptotically flat spacetimes, done under the guidance of S.T. Yau at the Institute for Advanced Study at Princeton, Bartnik's work has had a major impact on a number of different areas in mathematical relativity. His careful definition of the ADM mass on asymptotically Euclidean geometries, together with his analysis of the Laplace operator on such geometries, has been highly influential in geometric analysis. This work led in turn to his insightful definition of ""quasi-local mass,"" a topic of intense interest to this day. Bartnik's collaboration with his student John McKinnon resulted in their iconic discovery of a globally regular static solution of the Einstein-Yang-Mills equations. His proof that there exist globally hyperbolic spacetime solutions of Einstein's equations, which contain no constant mean curvature Cauchy surfaces, was very surprising, and has led to a variety of further results of this nature. The procedure he developed for generating solutions of the Einstein constraint equations using a parabolic PDE system, has already led to important applications and is likely to be very useful in the future. With the publication of this volume, the editors wish to honor Robert Bartnik's great contributions to their field. Included in this collection are most of his published papers, together with short essays by friends and colleagues who have been strongly influenced by him. The editors dedicate this collection to Robert, and to all those who will greatly benefit from being introduced to his work.

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.

This is the first of two volumes consisting of lectures given at conferences held in 2019 to celebrate the seventieth birthday of Shing-Tung Yau: at Harvard University (May), at the University of Rome (May/June), at the Chinese University of Hong Kong (June), at the 8th ICCM Congress at Tsinghua University (June), at Lehigh University in Pennsylvania (November), and elsewhere. Included here are: Werner Ballmann on "Bottom of spectra and coverings." Robert J. Berman with "An invitation to Kahler-Einstein metrics and random point processes." Duong H. Phong on "Unification of the Kahler-Ricci and anomaly flows" Cumrun Vafa on "SCFTs, holography, and topological strings" Stephen S.-T. Yau on "Recent results on kk-th Yau algebras over simple elliptic singularities E~6E~6" Kefeng Liu on "Global methods of solving equations on manifolds" Chuu-Lian Terng on "The geometric airy curve flow on RnRn" Valentino Tosatti on "Collapsing Calabi-Yau manifolds."

This volume presents five papers based on selected lectures given at the Current Developments in Mathematics conference, held in November 2018 at Harvard University.

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.