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

Fifth International Congress of Chinese Mathematicians Part 2

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/051.2 AMS/IP Studies in Advanced Mathematics

Volume 51, Part 2

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

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)

c 2012 by the American Mathematical Society and International Press. All rights reserved. The American Mathematical Society and International Press retain all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ Visit the International Press home page at URL: http://www.intlpress.com/ 10987654321 171615141312 Contents

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 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 J¨urgen 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 Teichm¨uller 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 J¨urgen Jost, Wilfrid Kendall, Umberto Mosco, Michael R¨ockner, 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 develop- ments 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

978-0-8218-7587-2 (Part 2) 978-0-8218-7555-1 (Set)

AMSIP/51.2