Fifth International Congress of Chinese Mathematicians Part 1

Home , Gilbert Strang, Kefeng Liu, Lizhen Ji, Sijue Wu, Tian Jun, Winnie Li

AMS/IP Studies in Advanced Mathematics S.-T. Yau, Series Editor

Lizhen Ji Yat Sun Poon Lo Yang Shing-Tung Yau Editors

American Mathematical Society • International Press Fifth International Congress of Chinese Mathematicians

https://doi.org/10.1090/amsip/051.1 AMS/IP Studies in Advanced Mathematics

Volume 51, Part 1

Fifth International Congress of Chinese Mathematicians

Lizhen Ji Yat Sun Poon Lo Yang Shing-Tung Yau Editors

American Mathematical Society • International Press Shing-Tung Yau, General Editor

2000 Mathematics Subject Classiﬁcation. Primary 05–XX, 08–XX, 11–XX, 14–XX, 22–XX, 35–XX, 37–XX, 53–XX, 58–XX, 62–XX, 65–XX, 20–XX, 30–XX, 80–XX, 83–XX, 90–XX. All photographs courtesy of International Press.

Library of Congress Cataloging-in-Publication Data International Congress of Chinese Mathematicians (5th : 2010 : Beijing, China) p. cm. (AMS/IP studies in advanced mathematics ; v. 51) Includes bibliographical references. ISBN 978-0-8218-7555-1 (set : alk. paper)—ISBN 978-0-8218-7586-5 (pt. 1 : alk. paper)— ISBN 978-0-8218-7587-2 (pt. 2 : alk. paper) 1. Mathematics—Congresses. I. Ji, Lizhen, 1964– II. Title. III. Title: 5th International Congress of Chinese Mathematicians. QA1.I746 2010 510—dc23 2011048032

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In Part One

Preface...... ix

噾ⅉ㠇⸇⮶㦒ⓜ岏 xi Welcome Letter from the Congress President...... xii Past Congresses of ICCM...... xiii Past Morningside Medalists, Chern Prize Recipients, and ICCM International Cooperation Award Recipients...... xiii On the Works of the 2010 Morningside Medalists ...... xv About the Medals and Awards of ICCM ...... xviii The Committees of the Congress...... xix Sponsors, Donors and Contributors ...... xxi Photographs ...... xxii

Papers on Pure Mathematics Gradient and oscillation estimates and their applications in geometric PDE ···················3 BEN ANDREWS On canonical and explicit classification of algebraic threefolds·····································21 JUNGKAI A. CHEN AND MENG CHEN Canonical pseudonorms on pluricanonical spaces························································33 CHEN-YU CHI The enigmatic Tate-Shafarevich group ·········································································43 JOHN COATES Eta invariants for even-dimensional manifolds ·····························································51 XIANZHE DAI

v vi CONTENTS

Ricci flow on 4-manifolds and Seiberg-Witten equations ·············································61 FUQUAN FANG AND ZHENLEI ZHANG Vector bundles and p-adic Galois representations ························································77 LAURENT FARGUES AND JEAN-MARC FONTAINE Geometry of nilpotent orbits: results and conjectures·················································115 BAOHUA FU Integrable connections and Galois representations ·····················································127 LEI FU Asymptotic Chow polystability in Kähler geometry····················································139 AKITO FUTAKI Representations of metaplectic groups········································································155 WEE TECK GAN Notes on partial conjugation ······················································································171 CHUYING FANG AND XUHUA HE Geometric modular forms and the cohomology of torsion automorphic sheaves········183 KAI-WEN LAN SYZ transformations for toric varieties········································································209 NAICHUNG CONAN LEUNG Some variational problems in conformal geometry·····················································221 BIN GUO AND HAIZHONG LI Geometry of tamed almost complex structures on 4-dimensional manifolds ··············233 TEDI DRAGHICI, TIAN-JUN LI, AND WEIYI ZHANG The arithmetic of noncongruence modular forms·······················································253 WEN-CHING WINNIE LI Analytic continuations of quantum cohomology ························································269 YUAN-PIN LEE, HUI-WEN LIN, AND CHIN-LUNG WANG Mathematical aspects of string duality ········································································283 KEFENG LIU AND PAN PENG The tame kernels of number fields ·············································································293 XUEJUN GUO AND HOURONG QIN Notes on MVW-extensions·························································································305 BINYONG SUN Vertex operator representations for a class of BC*-graded Lie algebras·······················315 FULIN CHEN AND SHAOBIN TAN Notes on orbifold Gromov-Witten theory ··································································327 HSIAN-HUA TSENG Cohomologies and elliptic operators on symplectic manifolds ···································341 LI-SHENG TSENG Quasilocal mass from a mathematical perspective ······················································355 MU-TAO WANG On dimension data and local vs. global conjugacy ·····················································365 SONG WANG Variational problems of Monge-Ampère type ·····························································383 XU-JIA WANG AND BIN ZHOU Wellposedness of the two and three dimensional full water wave problem ················397 SIJUE WU CONTENTS vii

Recent developments in differentiable sphere theorems ·············································415 HONG-WEI XU New invariants for complex manifolds, singularities, and CR manifolds with applications·········································································································431 RONG DU AND STEPHEN YAU Gross-Zagier formula and arithmetic fundamental lemma··········································447 WEI ZHANG Integrality properties of mirror maps··········································································461 JIAN ZHOU Ohsawa-Takegoshi L² extension theorem: revisited ····················································475 XIANGYU ZHOU AND LANGFENG ZHU

In Part Two

Papers on Applied Mathematics Regularity of Einstein spacetimes················································································493 BING-LONG CHEN A survey on retrograde and prograde orbits of the three-body problem by variational methods·····························································································499 KUO-CHANG CHEN Recent development of computational conformal geometry ·······································515 DAVID XIANFENG GU, WEI ZENG, LOK MING LUI, FENG LUO, AND SHING-TUNG YAU Some developments in spectral methods····································································561 BEN-YU GUO, CHAO ZHANG, AND TAO SUN On the center of mass in general relativity··································································575 LAN-HSUAN HUANG Tailored finite point method for numerical simulation of partial differential equations·············································································································593 ZHONGYI HUANG Loop symmetric functions and factorizing matrix polynomials···································609 THOMAS LAM Spectral inequalities for partial differential equations and their applications···············629 ARI LAPTEV Structured doubling algorithms for solving g-palindromic quadratic eigenvalue problems ·············································································································645 ERIC KING-WAH CHU, TSUNG-MING HUANG, AND WEN-WEI LIN Ricci curvature and functional inequalities on graphs·················································663 YONG LIN Complexity dichotomies of counting problems ··························································677 PINYAN LU A survey on recent development in computational quasi-conformal geometry and its applications ····································································································697 LOK MING LUI, TSZ WAI WONG, WEI ZENG, XIANFENG GU, PAUL M. THOMPSON, TONY F. CHAN, SHING-TUNG YAU viii CONTENTS

Dynamics of shock fronts for some hyperbolic systems··············································719 TAO LUO Time-stepping methods for linear complementarity systems ······································731 LANSHAN HAN AND JONG-SHI PANG A brief survey of high order accurate maximum principle satisfying and positivity preserving discontinuous Galerkin and finite volume schemes for conservation laws ································································································747 CHIWANG SHU A one parameter family of expanding wave solutions of the Einstein equations that induces an anomalous acceleration into the standard model of cosmology··········759 JOEL SMOLLER AND BLAKE TEMPLE Banded matrices with banded inverses and A = LPU ··················································771 GILBERT STRANG Dissimilarity data in statistical model building and machine learning·························785 GRACE WAHBA Some progress on computational geometry ································································811 REN-HONG WANG Geometrization program of semilinear elliptic equations············································831 JUNCHENG WEI Fast and near-optimal matrix completion via randomized basis pursuit ·····················859 ZHISU ZHU, ANTHONY MAN-CHO SO, AND YINYU YE Mathematical questions of quantum dilute gases························································883 JUN YIN Algebraic dynamics, canonical heights and Arakelov geometry ··································893 XINYI YUAN Well-posedness and control of the Korteweg-de Vries equation on a bounded domain················································································································931 BING-YU ZHANG Statistical analysis in genetic association studies of mental illnesses····························957 YUAN JIANG, HEPING ZHANG, AND WENSHENG ZHU Compressible Navier-Stokes equations with large density oscillation ·························971 HUIJIANG ZHAO Some results on variational and topological methods ·················································987 WENMING ZOU Preface

The International Congress of Chinese Mathematicians, or ICCM, is a major event in the mathematics community, especially among Chinese mathematicians. The First Con- gress was held in Beijing in 1998. Subsequent congresses were held in Taipei (2001), Hong Kong (2004), and Hangzhou (2007). In 2010, after twelve years, the Congress re- turned to Beijing. The Fifth International Congress of Chinese Mathematicians was held on December 17 to 22, 2010. The opening ceremony was held in the Great Hall of the People, with two thousand in attendance, and the proceedings took place on the campus of Tsinghua University. ICCM 2010 was co-hosted by the Academy of Mathematics and Systems Sci- ence of the Chinese Academy of Sciences, and by Tsinghua University. ICCM 2010 was specially dedicated to the 100th anniversary of the birth of Loo-Keng Hua, and to the 99th anniversary of the birth of Shiing-Shen Chern—two of the greatest Chinese mathematicians of the twentieth century. While teaching at Tsinghua University, Hua and Chern trained many outstanding mathematicians (among them H. C. Wang, H. Wang, and Z. X. Wan) who went on to become leading mathematicians in China. Hua was a dominating figure in the development of mathematics in modern China, and was instrumental in the formation of the Academy of Mathematics and Sys- tems Science. The world-class applied mathematician C. C. Lin, and statistician P. L. Hsu, were also from Tsinghua. Lin, Hsu, Chern, and Hua represent four major areas of mathematical sciences. It is therefore quite fitting that ICCM 2010 was co-organized by the Acad- emy of Mathematics and Systems Science and by Tsinghua University. Several mathematic awards were given at the opening ceremony—including the Morningside Medal of Mathematics, the Chern Prize, and the International Cooperation Award The Morningside Medal of Mathematics is awarded to exceptional mathematicians of Chinese descent under the age of forty-five, for seminal achievement in mathematics and applied mathematics. The 2010 Morningside Gold Medal of Mathematics was awarded to Mu-Tao Wang of Columbia University and to Sijue Wu of the University of Michigan. The 2010 Morningside Gold Medal of Applied Mathematics was awarded to Jun S. Liu, of Harvard University. The 2010 Morningside Silver Medal of Mathematics was awarded to Jixiang Fu of Fudan University, Juncheng Wei of the Chinese University of Hong Kong, Jungkai Alfred Chen of Taiwan University and Meng Chen of Fudan University, as a col- laborative team.

ix x PREFACE

The Chern Prize in Mathematics was established in 2001 in honor of Shiing-Shen Chern, one of the greatest geometers and Chinese mathematicians of the twentieth century. It is presented to mathematicians of Chinese descent who have made exceptional contributions to mathematical research or to public service activities which support mathematics. The 2010 Chern Prize was awarded to Jiaxing Hong of Fudan University, Conan Nai-Chung Leung of the Chinese University of Hong Kong, and to Wen-Ching Winnie Li, of the National Center of Theoretical Sciences in Taiwan and Penn State Uni- versity. First presented in 2004, the ICCM International Cooperation Award is given to an individual who has promoted the development of mathematics in China, Hong Kong, and Taiwan through collaboration, teaching, and support of Chinese mathematicians. The 2010 ICCM International Cooperation Award was presented to Richard M. Schoen, of Stanford University. Seven outstanding mathematicians were invited to give Morningside Lectures. Also presented were thirty-one hour-long talks and nearly two hundred 45-minute talks, by some of the outstanding mathematicians in the world today. The talks covered a broad spectrum of subjects in mathematical sciences—including algebra, number theory, geometry, analysis, dynamical systems, functional analysis, partial differential equations, numerical analysis, bio-mathematics, probability, and statistics. For the present volume, we solicited sixty distinguished speakers from the 2010 Congress, to write papers which explain their topics in a manner both accessible and comprehensive. The result is the book in your hand. The sixty papers presented herein do justice to the high quality and mathematical innovation of the lectures and talks given at the Congress. They are interesting and useful to both experts and general mathematicians. Preparation of this book entailed much work, and it could not have been finished in such a timely fashion without generous help from many people. Firstly, we would like to thank all the contributors for their hard work in writing such high-quality papers, and the referees for reviewing the papers. Secondly, we would like to thank Ms. Yan Zhang and Prof. Yunbo Zeng for their efforts in collecting and putting the papers into the proper format. Finally, we wish to take this opportunity to thank the Morningside Group, the Acad- emy of Mathematics and Systems Science in the Chinese Academy of Sciences, and the staff and faculty of both the Department of Mathematical Sciences and the Mathematical Sciences Center of Tsinghua University, for making ICCM 2010 such a success.

The Editors: Lizhen Ji Yat Sun Poon Lo Yang Shing-Tung Yau

September 2, 2011 噾ⅉ㠇⸇⮶㦒ⓜ岏

噾季㠇⸇⹅⃚㦒᧨ⱚ㡋℻ズ᧨め㸆₏侏ᇭ截◾⃚㸁᧨ℛ◒仛ⅉ᧨ ㈸勩㡋℻ズ⮶㦒⪑ᇭ㣑⋋ⷮ␻᧨␯坚䆎抣᧨≙䱏❇楕᧨嶪⸇≽䧵᧨≰ ♾㲑⃮ᇭ 摠⭖噾㺞᧨⢘咗⏗ズ᧨䤍ⅲ⃚喀᧨栚仛⸇嚻㄃⃚⪉᧨┸噾⮞沿歪 ⃚╱ᇭ⋋ℛ⏻䤍㸁␴房᧨廗♾䎰戼ⅴ侏₼噾仛ⅉₜ㉧䯥⃝ᇭ 拑⇫戼㥿᧶ 勩嶇影⏽㦒⪑⃚₼᧨ 㒆㈏➁⏽⠓␅ⱚ俑ᇭ ⻤剔⏽⇨㒆᧨ 灃澃⏽⃬䤚ᇭ ㉒歪㙩⏽㿸埸᧨ 䧍⚛拙⏽⸇䂀ᇭ ⻤㠇⃚初⏽浧㷛᧨ 㑘⮸⃚䦮⏽⚗乏ᇭ 㡴⺖⒉⏽⦚₼᧨ ⮫䤝䤝⏽㡱㢝ᇭ

xi xii

Welcome Letter from the Congress President

December 17, 2010

Distinguished guests, fellow mathematicians and friends:

It gives me honor and great pleasure to welcome you to the Fifth International Con- gress of Chinese Mathematicians (ICCM 2010) held in Beijing, China this year. Twelve years ago, the First International Congress of Chinese Mathematicians (ICCM 1998) was inaugurated here in the same city, and it is gratifying to see that ICCM has grown from strength to strength since its inception, and has today attained international acclaim. It is an internationally highly-regarded conference. More than a thousand mathematicians, university presidents, heads of leading scientific bodies, and students will gather at this special event. Mathematicians will present their research and discuss the latest developments, with a view to promoting and stimulating research in mathematics as a thriving science, both in China and around the globe. On the morning of the first day of the conference, there will be an opening ceremony at which we will announce and recognize the winners of the Morningside Medal of Mathematics for their outstanding attainments in pure and applied mathematics. These medalists have demonstrated exceptional achievement in their fields of study, and I am confident that their relentless effort and dedication will further inspire young Chinese tal- ents to explore the world of mathematics through research and publications. At the ceremony, we will also present the Chern Prize and the ICCM International Cooperation Award to individuals for their long-term commitments and contributions to mathematics and for their promotion of international exchanges. This year marks the one-hundredth birthday of the late Hua Loo-Keng and the ninety-ninth birthday of the late Shiing-Shen Chern, both world-renowned Chinese mathematicians of the twentieth century. We will have special sessions in the afternoon to pay tribute to these two great scientists, which kick off the celebration of the 100th Anniversary of Tsinghua University forthcoming in 2011. After that, we will have a panel discussion featuring business leaders who will share their views on the increasingly im- portant application of mathematics in internet technology and business development, and explore the role played by mathematics education in the technology evolution of the twenty-first century. Last but not least, my deepest appreciation goes to brothers Ronnie and Gerald Chan, founders of Morningside, who have been most supportive of the Morningside Medal of Mathematics and the ICCM since their inception. It goes without saying that the generosity and continuous support of the Chan brothers have contributed much to the success of the ICCM. Apart from the tremendous support given by Morningside, this year’s conference could not have been made possible without the sponsorship of the Mathematical Sciences Center of Tsinghua University, the Morningside Center of Mathe- matics, and the School of Mathematics of the Academy of Sciences. I would like to ac- knowledge the leadership of Professor Yat Sun Poon and Professor Jie Xiao at Tsinghua University, and of Professors Lo Yang, Guo Lei, and Wang Yuefei, at the Chinese Acad- emy of Sciences, in the organizing of this triennial event. My heart-felt thanks also go to the members of the various ICCM 2010 committees, to the staff at the Mathematical Sci- ences Center of Tsinghua University, to the Morningside Center of Mathematics at the Chinese Academy of Sciences, and to Morningside and Hang Lung Properties, for their hard work and commitment to making this Congress a success. As we enter the Fifth ICCM, I am delighted to see that the Congress has helped strengthen the cooperation among Chinese and non-Chinese mathematicians, which is of PAST AWARD RECIPIENTS xiii vital importance for the enhancement of the development of mathematics in China as well in other parts of the world. It is hoped that the Congress will continue to thrive and to be a stimulant which inspires more young people to be interested in mathematical research, as this will have a profound impact on the future development of the study of mathematics as a living science. I am sure that everyone will enjoy ICCM 2010 and find it to be a rewarding experience.

With best regards,

Shing-Tung Yau President International Congress of Chinese Mathematicians

Past Congresses of ICCM

1998 Beijing 2001 Taipei 2004 Hong Kong 2007 Hangzhou 2010 Beijing

Past Morningside Medalists, Chern Prize Recipients, and ICCM International Cooperation Award Recipients

Morningside Medalists 2010 Gold Medalists Mu-Tao Wang Columbia University Sijue Wu University of Michigan at Ann Arbor Jun S. Liu Harvard University Silver Medalists Jungkai Alfred Chen, Meng Chen (jointly) National Taiwan University & Fudan University Jixiang Fu Fudan University Juncheng Wei The Chinese University of Hong Kong xiv PAST AWARD RECIPIENTS

Morningside Medalists 2007 Gold Medalists Xu-Jia Wang Australian National University Jianqing Fan Princeton University Silver Medalists Melissa Chiu-Chu Liu Columbia University Lizhen Ji University of Michigan at Ann Arbor Shi Jin University of Wisconsin at Madison Chiun-Chuan Chen National Taiwan University Ye Tian Chinese Academy of Science

Morningside Medalists 2004 Gold Medalists Yizhao Hou California Institute of Technology Kefeng Liu Zhejiang University Zhouping Xin The Chinese University of Hong Kong Ziliang Ying Columbia University Silver Medalists Jinyi Cai University of Wisconsin at Madison Aiko Liu University of California at Berkeley Xiping Zhu Zhongshan University

Morningside Medalists 2001 Gold Medalists Jun Li Stanford University Horng-Tzer Yau New York University Silver Medalists Daqing Wan University of California at Irvine Chin-Lung Wang Tsing Hua University (Taiwan) Sijue Wu University of Maryland, College Park Nanhua Xi The Chinese Academy of Sciences

Morningside Medalists 1998 Gold Medalists Chang-Shou Lin Chung Cheng University (Taiwan) Shouwu Zhang Columbia University Silver Medalists Raymond Chan The Chinese University of Hong Kong Chong-Qing Cheng Nanjing University Kefeng Liu University of California at Los Angeles Tong Yang City University of Hong Kong ON THE WORKS OF THE 2010 MORNINGSIDE MEDALISTS xv

Chern Prize Recipients 2010 Jiaxing Hong Fudan University Conan Nai-Chung Leung The Chinese University of Hong Kong Wen-Ching Winnie Li National Tsing Hua University & Pennsylvania State University 2007 Shiu-Yuen Cheng Hong Kong University of Science and Technology Mu-Tao Wang Columbia University 2004 Fanghua Lin New York University Lo Yang The Chinese Academy of Sciences 2001 Song-Sun Lin Chiao Tung University (Taiwan) Jiu-Kang Yu Purdue University

ICCM International Cooperation Award Recipients 2010 Richard M. Schoen Stanford University 2007 Stanley J. Osher University of California at Los Angeles 2004 John Coates University of Cambridge

On the Works of the 2010 Morningside Medalists

2010 Morningside Gold Medal of Mathematics

Mu-Tao Wang Professor Wang is awarded the 2010 Morningside Gold Medal of Mathematics for his profound work in differential geometry and geometric PDE. Wang’s work on higher co- dimension mean curvature flow established sharp criteria for the long time existence, regularity, and convergence of the flow. His work on quasilocal mass in General Relativ- ity has produced a remarkably successful definition for arbitrary two surfaces in space- time. Both of these contributions are groundbreaking, and will be of lasting importance in the field. Professor Wang is a Professor of Mathematics at Columbia University. His research interests include differential geometry, discrete groups, partial differential equations and general relativity. He received his B.S. and M.S. in mathematics from Taiwan xvi ON THE WORKS OF THE 2010 MORNINGSIDE MEDALISTS

University and his Ph.D. in mathematics from Harvard University. Prior to joining Co- lumbia University, Professor Wang has taught at Stanford University. Among his awards and honors are the Chern Prize, awarded at ICCM 2007, a Kavli Fellowship in 2007, a Sloan Research Fellowship for 2003-2005, the Harold M. Bacon Memorial Teaching Award from Stanford University in 2000, and a U.S. National Science Foundation Grant since 2001.

Sijue Wu Professor Wu is awarded the 2010 Morningside Gold Medal of Mathematics for her outstanding contributions to the global well-posedness of small amplitude solutions to the water wave problems in both 2-D and 3-D. She is the Robert W. and Lynne H. Browne Professor of Mathematics at the Univer- sity of Michigan. Professor Wu received her B.S. and M.S. from Peking University and her Ph.D. from Yale University. Prior to joining the University of Michigan, Professor Wu held appointments at the Courant Institute, the Institute for Advanced Study at Prince- ton, Northwestern University, the University of Iowa and the University of Maryland at College Park. In 2001 she was awarded a Morningside Silver Medal of Mathematics and a Ruth Lyttle Satter Prize by the American Mathematics Society. The prize is awarded every two years to recognize an outstanding contribution to mathematics research by a woman in the previous five years. Other academic honors include a Radcliffe Institute for Ad- vanced Study Fellowship at Harvard University for 2002-2003, and a U.S. National Sci- ence Foundation Grant, for 2004-2009.

2010 Morningside Gold Medal of Applied Mathematics

Jun S. Liu Professor Liu is awarded the 2010 Morningside Gold Medal of Applied Mathematics for his fundamental contributions to Monte Carlo inference. He has clarified the properties of Rao-Blackwellization and the collapsed Gibbs sampler, and co-introduced sequen- tial importance sampling into general Bayesian computation. By developing the Gibbs motif sampler, his work also has a profound influence on computational biology. Profes- sor Liu is a Professor of Statistics at Harvard University and a Professor of Biostatistics at the Harvard School of Public Health. He received his B.S. in mathematics from Peking University, continued his studies at Rutgers University and received his Ph.D. in statistics from the University of Chicago. Prior to re-joining Harvard, Professor Liu has held appointments at the National Center for Biotechnology Information, Harvard University, the National University of Singapore, the University of California at Los Angeles, Stanford University, and Peking University. His honors include being a Bernoulli Lecturer and an Institute of Mathematical Statistics (IMS) Fellow in 2004, being a IMS Medallion Lecturer and receiving the COPSS President’s Award in 2002, the Mitchell Prize in 2000, and a Terman Fellowship, for 1995-1998.

2010 Morningside Silver Medal of Mathematics

Jungkai Alfred Chen and Meng Chen Professor Jungkai Alfred Chen and Professor Meng Chen are jointly awarded the 2010 Morningside Silver Medal of Mathematics for their breakthrough in the explicit birational classification of algebraic threefolds. In particular, they determined the effective optimal bound for the pluri-canonical system to give rise to birational maps for threefolds of general type. This is the best and in fact the only complete result after Bombieri’s fa- mous similar work for surfaces. They also successfully applied their theory to the study of ON THE WORKS OF THE 2010 MORNINGSIDE MEDALISTS xvii

Fano threefolds to get optimal boundedness. Professor Jungkai Alfred Chen is a Professor of Mathematics at Taiwan University. He previously taught at Chung Cheng University in Taiwan. He serves as a member of the Taida Institute for Mathematical Sciences and the Center for Theoretic Sciences in Tai- wan. Professor Meng Chen is the Chang Jiang Scholars Program Chair Professor at the School of Mathematical Sciences at Fudan University. He received his B.S., M.S. and D.S. degrees from East China Normal University. Prior to joining the Fudan faculty, Professor Chen taught at Tonji University.

Jixiang Fu Professor Fu is awarded the 2010 Morningside Silver Medal of Mathematics for his pioneering work on the existence of Strominger’s system on a non-Kähler Calabi-Yau threefolds; and for his proof of the existence of balance metrics on Calabi-Yau threefolds under conifold transition. Professor Fu’s works are significant contributions to the study of non-Kähler Calabi-Yau manifolds. Professor Fu is a Professor at the School of Mathe- matical Sciences at Fudan University. He was an invited speaker at the 2010 International Congress of Mathematicians in Hyderabad, India.

Juncheng Wei Professor Wei is awarded the 2010 Morningside Silver Medal of Mathematics for his achievements in semi-linear elliptic equations. An outstanding one is the counter- example to the De Giorgi conjecture when the space dimension is larger than eight. An- other significant one is the existence of multiple peak solutions for a singularly perturbed Neumann problem. He also obtained the existence of infinitely many solutions to the prescribing scalar curvature equation on the unit sphere. Professor Wei is a Professor of Mathematics at The Chinese University of Hong Kong. He obtained his B.S. in mathematics from Wuhan University and his Ph.D. from the University of Minnesota. Professor Wei was a 2005 Croucher Senior Research Fellow, and has published more than 120 papers and has spoken at more than 40 international conferences.

The Chern Prize

Jiaxing Hong Professor Hong is awarded the 2010 Chern Prize for his systematic study of the iso- metric embedding of surfaces into Euclidean three-space and his proof of a series of existence and regularity that have far reaching consequences. He is widely recognized for his contribution to non-linear PDE. Professor Hong cares deeply about the young generation, making every effort to help and nurture young mathematicians in China. Professor Hong is a Professor at the School of Mathematical Sciences at Fudan University and the Super- intendent of the University’s Institute of Mathematics. He is also the Director of the Labo- ratory of Nonlinear Mathematical Models and Methods under the Ministry of Education, and serves as an editor for the Chinese Annals of Mathematics and the Asian Journal of Mathematics. He is a member of the Chinese Academy of Sciences.

Conan Nai-Chung Leung Professor Leung is awarded the 2010 Chern Prize for his significant research contributions in the study of mirror symmetry and quantum cohomology, in particular for his proof of the Yau-Zaslow conjecture, for introducing new methods in studying various geometric invariants and the geometry of manifolds with different holonomy. Professor Leung has helped organize many international mathematics conferences, including the xviii ABOUT THE MEDALS AND AWARDS OF ICCM

2007 International conference on Geometric Analysis held at Taiwan University and the fourth Yamabe Symposium held at the University of Minnesota in 2008. He is an editor of the New York Journal of Mathematics and The Asian Journal of Mathematics. Professor Leung is a Professor of Mathematics at The Chinese University of Hong Kong. He received his M.S. in mathematics from the University of California at San Diego, and his Ph.D. in mathematics from the Massachusetts Institute of Technology.

Wen-Ching Winnie Li The career of Dr. Wen-Ching Winnie Li began with a brilliant 1974 dissertation ex- tending the seminal 1970 paper by Atkin and Lehner on new forms, and a 1976 sequel of these two fundamental papers with Atkin. The trio had a lasting impact on the arithmetic of modular forms. Her subsequent work covered a wide range of topics in number theory, including Eisenstein series, Rankin-Selberg L-functions, local constants, Jacquet-Langlands corre- spondence, the converse theorem, class groups, and exponential sums. Her contribution extends to graph theory and coding theory, including the construction of Ramanujan graphs by arithmetic methods and the distribution of eigenvalues for regular graphs. In recent years Dr. Li revitalized the area of the arithmetic of non-congruence arithmetic subgroups and lead a group of young researchers in an adventure in an area that has been dormant for many years. Through her courses and lectures in the last twenty five years she has inspired gen- erations of Chinese mathematicians. Since May 2009 Dr. Li has been the Director of the National Center of Theoretical Sciences in Hsinchu, succeeding Dr. Jing Yu as the Direc- tor of the NCTS as well as the head of the Mathematical Division at the NCTS. The NCTS continues to flourish under her energetic leadership, as an institution with high esteem in the international mathematical community. The contribution of the NCTS to the development of mathematics in Taiwan is universally acknowledged.

ICCM International Cooperation Award

Richard M. Schoen Professor Schoen is awarded the third ICCM International Cooperation Award for his steadfast contribution to the training of Chinese mathematicians. Professor Schoen has cultivated close relations with Chinese mathematicians throughout his career. Professor Schoen’s own doctoral supervisor at Stanford University was Professor Shing-Tung Yau and now, as the Bass Professor of Humanities and Sciences at Stanford University, more than one third of Professor Schoen’s 34 Ph.D. students and many of his post doctoral researchers hail from mainland China, Hong Kong, and Taiwan. Professor Schoen is also recognized for his efforts in continually fostering the Chinese mathematics community, not only in the United States, but also worldwide. This year, Professor Schoen joined the Selection Committee of the Morningside Medal of Mathematics.

About the Medals and Awards of ICCM

At each International Congress of Chinese Mathematicians (ICCM), the winners of several prestigious awards are announced at the opening ceremony. These awards include the Morningside Medal of Mathematics, the Chern Prize in Mathematics, and the ICCM International Cooperation Award. THE COMMITTEES OF THE 2010 CONGRESS xix

Morningside Medal of Mathematics The Morningside Medal of Mathematics is awarded to exceptional mathematicians of Chinese descent under the age of forty-five for their seminal achievements in mathematics and applied mathematics. The winners of the Morningside Medal of Mathematics are traditionally announced at the opening ceremony of the triennial International Congress of Chinese Mathematicians. The inaugural medals were presented in 1998. Each Morn- ingside Medalist receives a certificate, a medal, and cash award of US$25,000 for a gold medal, or US$10,000 for a silver medal. Chern Prize in Mathematics The Chern Prize in Mathematics was established in 2001 in honor of Professor Shing-Shen Chern, one of the greatest geometers and Chinese mathematicians of the twentieth century. The Chern Prize is presented every three years at the International Congress of Chinese Mathematicians to mathematicians of Chinese descent who have made exceptional contributions to mathematical research or to public service activities in support of mathematics. ICCM International Cooperation Award The ICCM International Cooperation Award is presented to an individual who has promoted the development of mathematics in the Chinese mainland, Hong Kong, and Taiwan through collaboration, teaching, and support of Chinese mathematicians. The inaugural award was presented at the Third International Congress of Chinese Mathemati- cians in 2004.

The Committees of the 2010 Congress

Congress Chairman Shing-Tung Yau (Ϭ៤Ḥ )Harvard University 2010 Morningside Medal of Mathematics Selection Committee John H. Coates. Sadleirian Professor of Pure Mathematics at the University of Cambridge and a Fellow of Emmanuel College, Cambridge. Simon K. Donaldson. Royal Society Research Professor in Pure Mathematics at Imperial College London. Gerd Faltings. Director of the Max Planck Institute for Mathematics in Germany. Jean-Marc Fontaine. Professor Emeritus of Mathematics at the University of Paris-Sud. Benedict H. Gross. George Vasmer Leverett Professor of Mathematics at Harvard University. Nigel Hitchin. Savilian Professor of Geometry at Oxford University. Stanley J. Osher. Professor of Mathematics, Computer Science, Electrical Engineering at the University of California at Los Angeles. Richard M. Schoen. Bass Professor of Humanities and Sciences at Stanford University. Joel A. Smoller. Lamberto Cesari Chair of Mathematics at the University of Michigan, Ann Arbor. xx THE COMMITTEES OF THE 2010 CONGRESS

Grace Wahba. I.J. Schoenberg-Hilldale Professor of Statistics at the University of Wisconsin at Madison. Shing-Tung Yau. William Casper Graustein Professor of Mathematics at Harvard University. Scientific Committee Zhi-Ming Chen (䰜ᖫᯢ), Chinese Academy of Sciences Shiu-Yuen Cheng (䚥㒡䖰), Hong Kong University of Science and Technology Fan Chung Graham (䞥㢇㪝), University of California at San Diego Jiaxing Hong (⋾ᆊ݈), Fudan University Tze-Leung Lai (咢ᄤ㡃), Stanford University Jun Li (ᴢ偣), Stanford University Peter Li (ᴢӳܝ), University of California, Irvine Bong Lian (䖲᭛䈾 Brandeis University Chang-Shou Lin (ᵫ䭓ᇓ), National Chung Cheng University Kefeng Liu (߬ܟዄ), Zhejiang University and University of California at Los Angeles Ngaiming Mok (㥿↙ᯢ), The University of Hong Kong Yum-Tong Siu (㙪㤿ූ), Harvard University Shengli Tan (䇜㚰߽), East China Normal University Chuu-lian Terng (⒩Ἦ㦆), University of California at Irvine Xujia Wang (∾ᕤᆊ), National University of Australia Jie Xiao (㙪ᵄ), Tsinghua University Ling Xiao (㙪⦆), Chinese Academy of Science Zhouping Xin (䕯਼ᑇ), The Chinese University of Hong Kong Lo Yang (ᴼФ), Chinese Academy of Sciences Andrew Chi-Chih Yao (ྮᳳᱎ), Tsinghua University Shouwu Zhang (ᓴᇓ℺), Columbia University Weiping Zhang (ᓴӳᑇ), Nankai University Xiping Zhu (ᴅ➍ᑇ), Sun Yat-Sen University Executive Committee Huai-Dong Cao (᳍ᗔϰ), Lehigh University Gerard J. Chang (ᓴ䬛ढ), National Taiwan University Chiun-Chuan Chen (䰜ܼ֞), National Taiwan University Louis Chen (䰜ᰧѥ), National University of Singapore Chongqing Cheng (⿟ዛᑚ), Nanjing University Jih-Hsin, Cheng (䚥᮹ᮄ), Institute of Mathematics, Academia Sinica, Taipei Xiaqi Ding (ϕ໣⬺), Chinese Academy of Sciences Weinan E (䛖㓈फ), Princeton University Ka-Sing Lau (߬ᆊ៤), The Chinese University of Hong Kong Daqian Li (ᴢ໻┰), Fudan University Qikeng Lu (䰚ਃ䫓), Chinese Academy of Sciences Zhongci Shi (⷇䩳᜜), Chinese Academy of Sciences SPONSORS, DONORS AND CONTRIBUTORS xxi

Chiwang Shu (㟦݊ᳯ), Brown University Zhexian Wan (ϛ૆ܜ), Chinese Academy of Sciences Yuan Wang (⥟ܗ), Chinese Academy of Sciences Jeff Wu (ਈᓎ⽣), Georgia Institute of Technology Jia-An Yan (Ϲࡴᅝ), Chinese Academy of Sciences Horng-Tzer Yau (ྮ吓⋑), New York University Stephen S. T. Yau (Ϭ៤ᷟ), University of Illinois at Chicago Jing Yu (Ѣ䴪), National Tsing Hua University Liqun Zhang (ᓴゟ㕸), Chinese Academy of Sciences Organizing Committee Keqin Feng (ރܟࢸ), Tsinghua University Qi Feng (ރ⧺), Chinese Academy of Sciences Kefeng Liu (߬ܟዄ), University of California at Los Angeles Yat Sun Poon (┬᮹ᮄ), University of California at Riverside Yuefei Wang (⥟䎗亲), Chinese Academy of Sciences Jie Xiao (㙪ᵄ), Tsinghua University Jian Zhou (਼മ), Tsinghua University

Sponsors, Donors and Contributors

Sponsors Morningside Group Academy of Mathematics and Systems Science, Chinese Academy of Sciences Tsinghua University Co-Sponsors Department of Mathematics, Chinese University of Hong Kong School of Mathematical Sciences, Fudan University Department of Mathematics, Hong Kong University of Science & Technology Department of Mathematics, National Central University Department of Mathematics, National Taiwan University National Center for Theoretical Sciences, National Tsing Hua University Department of Mathematical Sciences, Renmin University of China Department of Mathematics, Capital Normal University Department of Mathematical Sciences, Beijing Institute of Technology Department of Mathematics, Beijing Normal University College of Applied Science, Beijing University of Technology School of Mathematics and Computational Science, Sun Yat-Sen University Department of Mathematics, Tongji University Department of Mathematics, University of Science and Technology of China Center of Mathematical Sciences, Zhejiang University National Natural Science Foundation of China Photographs PHOTOGRAPHS xxiii

The Opening Ceremony of ICCM 2010

Awardees and members of the selection committee at the opening ceremony of ICCM 2010

Prof. S.-T. Yau (Chairman of ICCM 2010) delivers the opening address. xxiv PHOTOGRAPHS

Mu-Tao Wang (center) is awarded the 2010 Morningside Gold Medal of Mathematics. With S.-T. Yau (left) and Yongxiang Lu (right, President of the Chinese Academy of Sciences).

Sijue Wu (center) is awarded the 2010 Morningside Gold Medal of Mathematics. With S.-T. Yau (left) and Ping Hao (right, Vice Minister of the Ministry of Education of the P. R. of China).

Jun S. Liu (center) is awarded the 2010 Morningside Gold Medal of Applied Mathematics. With S.-T. Yau (left) and Binglin Gu (right, President of Tsinghua University). PHOTOGRAPHS xxv

Jungkai Alfred Chen (second from right) and Meng Chen (second from left) are jointly awarded the 2010 Morningside Silver Medal of Mathematics. With S.-T. Yau (left) and Benedict H. Gross (right, Harvard University).

Jixiang Fu (center) is awarded the 2010 Morningside Silver Medal of Mathematics. With S.-T. Yau (left) and Stanislaw Janeczko (right, Polish Academy of Sciences, Warsaw Univ. of Technology).

Juncheng Wei (center) is awarded the 2010 Morningside Silver Medal of Mathematics. With S.-T. Yau (left) and Ari Laptev (right, Imperial College London). xxvi PHOTOGRAPHS

Wen-Ching Winnie Li (center) is awarded the 2010 Chern Prize. With S.-T. Yau (left) and Lo Yang (right, member of the Chinese Academy of Conan Nai-Chung Leung (left) is awarded the Sciences). 2010 Chern Prize. With S.-T. Yau (right).

Jiaxing Hong (center) is awarded the 2010 Chern Prize. With S.-T. Yau (left) and John Coates (right, University of Cambridge).

Richard M. Schoen (center) is awarded the 2010 ICCM International Cooperation Award. With S.-T. Yau (left) and Ronnie C. Chan (right, President of Hang Lung Properties). Titles in This Series

51.2 Lizhen Ji, Yat Sun Poon, Lo Yang, and Shing-Tung Yau, Editors, Fifth International Congress of Chinese Mathematicians, 2012 51.1 Lizhen Ji, Yat Sun Poon, Lo Yang, and Shing-Tung Yau, Editors, Fifth International Congress of Chinese Mathematicians, 2012 50 Jørgen E. Andersen, Hans U. Boden, Atle Hahn, and Benjamin Himpel, Chern-Simons Gauge Theory: 20 Years After, 2011 49 Matthias Plaue, Alan Rendall, and Mike Scherfner, Advances in Lorentzian Geometry, 2011 48 Lizhen Ji, Kefeng Liu, Lo Yang, and Shing-Tung Yau, Fourth International Congress of Chinese Mathematicians, 2010 47 Alexander Grigor’yan, Heat Kernel and Analysis on Manifolds, 2009 46.2 Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Editors, Lagrangian Intersection Floer Theory, 2009 46.1 Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian Intersection Floer Theory, 2009 45 Lydia Bieri and Nina Zipser, Extensions of the Stability Theorem of the Minkowski Space in General Relativity, 2009 44 Eric Sharpe and Arthur Greenspoon, Editors, Advances in String Theory, 2008 43 Lizhen Ji, Editor, Arithmetic Groups and Their Generalizations, 2008 42.1 Ka-Sing Lau, Zhou-Ping Xin, and Shing-Tung Yau, Editors, Third International Congress of Chinese Mathematicians, 2008 41 Wen-Ching Winnie Li, Editor, Recent Trends in Coding Theory and its Applications, 2007 40 Ovidiu Caliin, Der-Chen Chang, and Peter Greiner, Editors, Geometric Analysis on the Heisenberg Group and Its Generalizations, 2007 39 Zhijie Chen, Sheng-Li Tan, Jianpan Wang, and Stephen S.-T. Yau, Editors, Proceedings of the International Conference on Complex Geometry and Related Fields, 2007 38 Noriko Yui, Shing-Tung Yau, and James D. Lewis, Editors, Mirror Symmetry V, 2006 37 Lizhen Ji, Jian-Shu Li, H. W. Xu, and Shing-Tung Yau, Editors, Lie Groups and Automorphic Forms, 2006 36 Chuu-Lian Terng, Editor, Integrable Systems, Geometry, and Topology, 2006 35 Felix Finster, The Principle of the Fermionic Projector, 2006 34 Ren-Hong Wong, Editor, Computational Geometry, 2003 33 Eric D’Hoker, Duong Phong, and Shing-Tung Yau, Mirror Symmetry IV, 2002 32 Xi-Ping Zhu, Lectures on Mean Curvature Flows, 2002 31 Kiyoshi Igusa, Higher Franz-Reidemeister Torsion, 2002 30 Weiman Han and Mircea Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, 2002 29 S. T. Yau and Shuxing Chen, Editors, Geometry and Nonlinear Partial Diﬀerential Equations, 2002 28 Valentin Afraimovich and Sze-Bi Hsu, Lectures on Chaotic dynamical Systems, 2002 27 M. Ram Murty, Introduction to p-adic Analytic Number Theory, 2002 26 Raymond Chan, Yue-Kuen Kwok, David Yao, and Qiang Zhang, Editors, Applied Probability, 2002 25 Donggao Deng, Daren Huang, Rong-Qing Jia, Wei Lin, and Jian Zhong Wong, Editors, Wavelet Analysis and Applications, 2002 24 Jane Gilman, William W. Menasco, and Xiao-Song Lin, Editors, Knots, Braids, and Mapping Class Groups—Papers Dedicated to Joan S. Birman, 2001 23 Cumrun Vafa and S.-T. Yau, Editors, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, 2001 TITLES IN THIS SERIES

22 Carlos Berenstein, Der-Chen Chang, and Jingzhi Tie, Laguerre Calculus and Its Applications on the Heisenberg Group, 2001 21 Jürgen Jost, Bosonic Strings: A Mathematical Treatment, 2001 20 Lo Yang and S.-T. Yau, Editors, First International Congress of Chinese Mathematicians, 2001 19 So-Chin Chen and Mei-Chi Shaw, Partial Differential Equations in Several Complex Variables, 2001 18 Fangyang Zheng, Complex Differential Geometry, 2000 17 Lei Guo and Stephen S.-T. Yau, Editors, Lectures on Systems, Control, and Information, 2000 16 Rudi Weikard and Gilbert Weinstein, Editors, Differential Equations and Mathematical Physics, 2000 15 Ling Hsiao and Zhouping Xin, Editors, Some Current Topics on Nonlinear Conservation Laws, 2000 14 Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, 2000 13 Vasilios Alexiades and George Siopsis, Editors, Trends in Mathematical Physics, 1999 12 Sheng Gong, The Bieberbach Conjecture, 1999 11 Shinichi Mochizuki, Foundations of p-adic Teichmüller Theory, 1999 10 Duong H. Phong, Luc Vinet, and Shing-Tung Yau, Editors, Mirror Symmetry III, 1999 9 Shing-Tung Yau, Editor, Mirror Symmetry I, 1998 8 Jürgen Jost, Wilfrid Kendall, Umberto Mosco, Michael Röckner, and Karl-Theodor Sturm, New Directions in Dirichlet Forms, 1998 7 D. A. Buell and J. T. Teitelbaum, Editors, Computational Perspectives on Number Theory, 1998 6 Harold Levine, Partial Differential Equations, 1997 5 Qi-keng Lu, Stephen S.-T. Yau, and Anatoly Libgober, Editors, Singularities and Complex Geometry, 1997 4 Vyjayanthi Chari and Ivan B. Penkov, Editors, Modular Interfaces: Modular Lie Algebras, Quantum Groups, and Lie Superalgebras, 1997 3 Xia-Xi Ding and Tai-Ping Liu, Editors, Nonlinear Evolutionary Partial Differential Equations, 1997 2.2 William H. Kazez, Editor, Geometric Topology, 1997 2.1 William H. Kazez, Editor, Geometric Topology, 1997 1 B. Greene and S.-T. Yau, Editors, Mirror Symmetry II, 1997 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. American Mathematical Society www.ams.org

International Press www.intlpress.com

978-0-8218-7586-5 (Part 1) 978-0-8218-7555-1 (Set)

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