Third Internationa l Congress o f Chines e Mathematicians This page intentionally left blank AMS/IP https://doi.org/10.1090/amsip/042.2 Studies in Advanced Mathematics
Volume 42, Part 2
Third Internationa l Congress o f Chines e Mathematicians
Ka-Sing Lau, Zhou-Ping Xin, and Shing-Tung Yau, Editors
American Mathematical Society • International Press Shing-Tung Yau , Genera l Edito r
2000 Mathematics Subject Classification. Primar y 05-XX , 08-XX , 11-XX , 14-XX , 22-XX, 35-XX , 37-XX , 80-XX .
All photo s courtes y o f Shing-Tun g Ya u an d Internationa l Pres s
Library o f Congres s Cataloging-in-Publicatio n Dat a International Congres s o f Chines e Mathematician s (3r d : 2004 : Chines e Universit y o f Hon g Kong) Third Internationa l Congres s o f Chines e Mathematician s : proceeding s o f ICCM04 , Decem - ber 17-22 , 2004 , the Chines e Universit y o f Hong Kong , Hon g Kong , Chin a / Ka-Sin g Lau . Zhou - Ping Xin , Shing-Tun g Yau , editors . p. cm . — (AMS/I P studie s i n advance d mathematics , ISS N 1089-328 8 ; v. 42 ) Includes bibliographica l references . ISBN-13: 978-0-8218-4454- 0 (pt . 1 : alk . paper ) ISBN-13: 978-0-8218-4452- 6 (pt . 2 : alk . paper ) ISBN-13: 978-0-8218-4416- 8 (set : alk . paper ) 1. Mathematics—China—Congresses . 2 . Mathematicians—China—Congresses . I . Lau , Ka-Sing. II . Xin , Zhou-Ping , 1959 - III . Yau, Shing-Tung , 1949 - IV . Series . QA1.1746 200 4 510—dc22 200706057 6
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© 200 8 by the America n Mathematica l Societ y an d Internationa l Press . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / Visit th e Internationa l Pres s hom e pag e a t URL : http://www.intlpress.com / 10 9 8 7 6 5 4 3 2 1 1 3 1 2 1 1 1 0 09 0 8 Contents
Preface x i Welcome letters xii i A Chines e Essa y i n Tribute to Professo r Shiing-She n Cher n xvi i Committees o f ICCM 200 4 xi x Morningside Lifetim e Achievemen t Awar d i n Mathematics xxii i Morningside Meda l o f Mathematics xx v Morningside Meda l o f Mathematics - Selectio n Committe e xxvi i Morningside Awar d Medalist s xxi x Chern Priz e Recipient s l i ICCM International Cooperatio n Awar d Recipien t lii i Photographs l v List o f Speaker s lxvi i
Morningside Lecture s Shock Wave s and Cosmolog y J.A. SMOLLE R AN D J.B . TEMPL E 1
Plenary Lecture s Variational Constructio n o f Diffusio n Orbit s i n Conve x Hamiltonian System s with Multipl e Degree s o f Freedo m CHONG-QING CHEN G AN D JU N YA N 1 1 Saddlepoint Approximation s an d Boundar y Crossin g Probabilities fo r Rando m Field s an d Thei r Application s TZE LEUN G LA I 2 9 Recognizing Certai n Rationa l Homogeneou s Manifold s o f Picard Numbe r 1 from Thei r Varietie s o f Minima l Rational Tangent s NGAIMING MO K 4 1 CONTENTS
Discontinuous Galerki n Methods fo r Convectio n Dominate d Partial Differentia l Equation s CHI-WANG SH U 6 3 Singularity Behavio r o f the Mea n Curvatur e Flo w XU-JIA Wan g 7 5 Localization an d Dualit y JIAN ZHO U 8 3 Surgical Ricc i Flo w o n Four-Manifolds wit h Positiv e Isotropic Curvatur e BING-LONG CHE N AN D XI-PIN G ZH U 10 1 Special Subvarietie s o f Ag ECKART VlEHWE G AN D KANG ZU O 11 1
Invited Talks-Par t 1
Number Theor y Local Monodromy o f the Kloosterma n Shea f a t o o LEI F U AN D DAQIN G WA N 12 5 Hilbert Modula r Function s an d Thei r CM Value s TONGHAI YAN G 13 5
Algebraic an d Comple x Geometr y Geometric Invarian t Theor y an d Birationa l Geometr y Yi Hu 15 5 Remarks o n Gieseker' s Degeneratio n an d it s Normalizatio n XIAOTAO SU N 17 7 Bundle Rigidit y o f Complex Surface s WING-SUM CHEUN G AN D BU N WON G 19 3 CR Equivalenc e Proble m o f Strongly Pseudoconve x C R Manifold s STEPHEN S.-T . YA U 19 7 Vector Bundle s o n Non-Primary Hop f Manifold s wit h Abelia n Fundamental Grou p Liu WEIMIN G AN D ZHO U XIANGY U 20 9
Geometry Subelliptic PDE' s an d SubRiemannia n Geometr y DER-CHEN CHAN G AN D PETE R GREINE R 22 3 The Q- Curvature Flo w o n a Close d 3-Manifol d o f Positiv e Q-Curvature SHU-CHENG CHAN G 23 9 The Spac e o f Symplectic Structure s o n Close d 4-Manifold s TIAN-JUN L i 25 9 Ancient Solution s to Kahler-Ricc i Flo w LEI N I 27 9 CONTENTS
A Convergence Resul t o f the Lagrangia n Mea n Curvatur e Flo w MU-TAO WAN G 29 1 On Piecewis e Algebrai c Variet y REN-HONG WAN G 29 7
Topology An Introduction t o Chira l Equivarian t Cohomolog y BONG H . LIA N 30 7
Lie Group s an d Li e Algebra s Large Scal e Geometry, Compactification s an d th e Integra l Novikov Conjecture s fo r Arithmeti c Group s LIZHEN J i 31 7
Operator Algebra s an d Functiona l Analysi s Normal Dilation s M.-D. CHO I 34 5 On the Generato r Proble m o f von Neumann Algebra s LIMING G E AN D JUNHA O SHE N 35 7 A Survey o f Results o n the Groun d Stat e o f Semilinea r Elliptic Equation s MAN KA M KWON G 37 7 Separating an d Extendin g Subgroup s o f a Locall y Compact Grou p EBERHARD KANIUT H AN D ANTHON Y T . LA U 38 5 The Triangl e o f Operators, Topologies , Bornologie s NGAI-CHING WON G 39 5
Mathematical Physic s The Positiv e Mas s Theore m Nea r Nul l Infinit y XIAO ZHAN G 42 3
Invited Talks-Par t 2
Analysis Dirichlet Form s an d Marko v Semigroup s o n Non-Associativ e Vector Bundle s CHO-HO CH U AN D ZHONGMI N QIA N 43 3 Decomposition Principl e an d Rando m Cascade s Ai HU A FA N AN D JEA N PIERR E KAHAN E 44 7 Holomorphic Motion s an d Norma l Form s i n Comple x Analysi s YUNPING JIAN G 45 7 On Pseudo-Hermitian C R Manifold s SONG-YING L i 46 7 CONTENTS
Recent Progres s o n the Dirichle t Proble m i n Lipschit z Domain s ZHONGWEI SHE N 48 3 Refinable Function s wit h Non-Intege r Dilation s XIN-RONG DAI , DE-JU N FEN G AN D YAN G WAN G 49 3 On Whitney's Critica l Set s ZHI-YING WE N AN D LI-FEN G X I 51 3 Applications o f Nevanlinna Theor y t o Geometri c Problem s PIT-MANN WON G 52 3
Ordinary Differentia l Equation s an d Dynamica l System s Some Results o n Smale' s Mea n Valu e Conjectur e YUEFEI WAN G 59 5
Partial Differentia l Equation s Localized Non-Blowu p Condition s fo r the 3 D Incompressibl e Euler Equation s JIAN DENG , THOMA S Y . HOU , XINWE I Y U 60 3 Separation o f Bound Stat e Solution s o f Systems o f Nonlinear Schrodinge r Equation s TAI-CHIA LI N AN D YU-WEN HS U 61 3 The C a Regularit y o f a Clas s o f Ultraparabolic Equation s ZHANG LIQU N 61 9 Stability o f Basic Wave Patterns fo r Ga s Motion s TONG YAN G AN D HUI-JIAN G ZHA O 62 3
Probability an d Statistic s On Stron g Near-epoc h Dependenc e ZHENGYAN LI N 64 7 Multifractal Analysi s o f Branching Measur e o n a Galton-Watson Tre e PETER MORTER S AN D NARN-RUEI H SHIE H 65 5 Convex Duality Theor y fo r Optima l Investmen t JlANMING Xl A AN D JlA-AN YA N 66 3 Backward Stochasti c Volterr a Integral Equation s JlONGMIN YON G 67 9
Combinatorics Set Additio n an d Se t Multiplicatio n M.-C. CHAN G 70 7 Collineation Group s o f Translation Plane s CHAT YI N H O 72 1
Numerical Analysi s an d Scientifi c Computin g Intelligent an d Informativ e Scientifi c Computing , Trend s and Example s QIANG D u 73 1 CONTENTS
Scattered Dat a Interpolatio n b y Bo x Spline s ZOUWEI SHE N AN D SHAYN E WALDRO N 74 9 Piecewise Functio n Generate d b y the Solution s o f Linear Ordinar y Differentia l Equatio n ZONGMIN W U 76 9
Control Theor y an d Optimizatio n Step-Sizes fo r th e Gradien t Metho d YA-XIANG YUA N 78 5
Applications o f Mathematic s i n the Science s Number-Theoretic Method s i n Experimental Design s KAI-TAI FAN G AN D YUA N WAN G 79 7 Ear Modelin g an d Soun d Signa l Processin g JACK XI N 81 9 The Mathematica l Proble m o f Inertial Wave s i n Rapidl y Rotating Planet s an d Star s XlNHAO LlA O AN D KEKE ZHAN G 83 1
Mathematics Educatio n an d Popularizatio n o f Mathematics Mathematics, Mathematic s Educatio n an d th e Mous e Siu MA N KEUN G 86 1 This page intentionally left blank Preface
Shing-Tung Ya u President, International Congres s o f Chinese Mathematician s
In December o f 2004, the Third Internationa l Congres s o f Chinese Mathemati - cians wa s hel d a t th e Chines e Universit y o f Hon g Kong . Th e openin g ceremony , in the Exhibitio n Hal l o f Hong Kong , wa s attended b y a thousand people . A s th e president o f the Congress , I announce d tha t i t woul d dedicate d t o the memor y o f our teacher and our leader: Professo r Shiing-she n Chern , who recently passed away . Professor Cher n devoted his life to the advancement o f mathematics i n general, bu t especially i n China . H e was very muc h supportiv e o f the Congress , havin g offere d many importan t suggestion s an d contribute d a hundred thousan d yue n t o it . Un - fortunately, h e passed awa y just afte r h e had bough t hi s tickets to trave l t o Hon g Kong an d atten d th e Congress . W e mourn th e los s o f this grea t mathematician . We were honored to have President L u o f the Academy o f Science agree to serve as the Honorabl e Presiden t o f the Congress . H e presented th e Morningside Medal s to certai n Chines e mathematician s i n recognitio n o f thei r outstandin g work . Th e medals ar e awarde d b y th e Morningsid e Foundatio n t o mathematician s younge r than 4 5 years o f age. Th e Congres s sa w Morningside Medal s awarde d t o Professo r Zhou-Ping Xi n o f the Chines e Universit y o f Hon g Kong , an d t o Professo r Kefen g Liu o f the University o f California a t Lo s Angeles, who received the Gol d Medal fo r Pure Mathematics . Professo r To m Ho u o f the Californi a Institut e o f Technolog y and Professo r Chi-Lian g Yi n o f Columbi a Universit y share d th e Gol d Meda l fo r Applied Mathematics . Silve r medal s wer e awarde d t o Professo r Xi-Pin g Zh u o f Zhongshan Universit y o f China, to Jin-Yi Ca i o f Wisconsin University , an d to Aik o Liu o f the Universit y o f Californi a a t Berkeley . The Congress awarded Cher n medals to Professor Fang-Hu a Lin o f the Couran t Institute o f Mathematics, an d t o Professo r Yangl o o f the Morningsid e Institut e o f Mathematics. Moreover , i n recognitio n o f ou r oversea s friend s wh o hav e mad e great contribution s t o th e Mathematic s Communit y i n China , w e hav e institute d a ne w awar d calle d the International Cooperatio n Meda l fo r Mathematics . A t th e Congress, w e had the hono r o f presenting this special distinction to Professor Joh n Coates o f Cambridge Universit y i n England . The Congres s spanne d five day s an d covered man y area s o f mathematics, in - cluding pur e mathematics , applie d mathematics , statistics , an d compute r science . We were very please d a t th e euthusias m o f everyone participating .
xi xii PREFAC E
The purpos e o f th e Internationa l Congres s o f Chines e Mathematician s i s t o encourage exchang e o f idea s betwee n mathematician s al l ove r th e world , an d thi s was clearl y visibl e a t th e 200 4 Congress . I t i s my wis h tha t thi s comin g togethe r will strengthen an d promot e mathematic s i n China . Man y mathematician s no t o f Chinese descent cam e to participate, an d w e are very grateful fo r their support an d help. W e shoul d lik e ver y muc h t o invit e eve n mor e friend s t o com e fo r ou r nex t gathering o f the Congress , whic h i s to be hel d a t Zhejin g Universit y i n Hangzhou , China. Welcome Letter s
Shing-Tung Ya u Chairman, International Congres s o f Chinese Mathematician s
To My Fello w Mathematicians an d Distinguishe d Guests ,
As Chairman o f the International Congres s o f Chinese Mathematicians, I would like to welcom e yo u t o th e Thir d Internationa l Congres s o f Chines e Mathematicians . Since 1998, the Congress has grown from strength to strength. I am deeply gratifie d by the response and interest o f my fellow mathematicians in this triennial gathering. Over seven hundred mathematicians fro m around the world are expected to convene for si x days o f intensive discussions , talks, an d presentations . In th e past decade , mathematics ha s steadil y improve d i n China , Hon g Kong , and Taiwan . Th e dedicatio n an d visio n o f several leading Chines e mathematician s has emboldene d a ne w generatio n o f mathematicians t o pursu e career s i n mathe - matical researc h an d teaching . I n particular , I woul d lik e t o recogniz e Professo r Yongxiang Lu, the President o f the Chinese Academy o f Sciences, for his leadership and suppor t i n advancin g mathematics i n China . At the opening ceremony o f each International Congres s o f Chinese Mathemati- cians, the winners o f the Morningside Meda l o f Mathematics ar e honored i n a spe- cial ceremony fo r their outstanding achievement s in pure and applied mathematics . This year , th e secon d Cher n Priz e an d th e first ICC M Internationa l Cooperatio n Award wil l also be presented t o the three individual s fo r thei r contribution s t o th e development o f mathematics . The Thir d Internationa l Congres s o f Chines e Mathematician s woul d no t b e possible withou t th e har d wor k an d effort s o f man y individuals . I woul d lik e t o express m y sincer e appreciatio n t o Mr . Ronni e C . Chan , th e co-founde r o f TH E MORNINGSIDE GROUP , fo r hi s generous support. I would als o like to recogniz e Professors Ka-Sin g Lau and Zhouping Xin at The Chinese University o f Hong Kong for thei r leadershi p i n organizing an d executin g this congress . Th e members o f the various ICC M 200 4 committee s an d th e staf f a t Th e Institut e o f Mathematica l Sciences at The Chinese University o f Hong Kong have played a key role in makin g this even t a tremendous succes s a s well.
Xlll Ka-Sing La u Chairman, Departmen t o f Mathematic s The Chines e Universit y o f Hong Kon g
Zhouping Xi n Associate Director , Th e Institut e o f Mathematical Science s The Chines e Universit y o f Hong Kon g
Dear Fello w Mathematicians an d Guests ,
On behal f o f the Organizing Committee, w e would like to welcome you to the Thir d International Congres s o f Chinese Mathematicians (ICC M 2004) . Mathematics ha s experience d phenomena l growt h i n Hon g Kon g i n th e las t decade. Loca l universitie s hav e establishe d leadin g mathematic s institute s an d centers conducting majo r researc h on pure, applied, an d computational mathemat - ics, founded program s t o attract eminen t mathematician s an d promisin g students , and hoste d colloqui a on cutting-edge topics. A s a result, Hon g Kong has develope d a cor e grou p o f internationall y renowne d researchers . W e dre w o n thi s larg e poo l of talen t i n Hon g Kon g t o hos t ICC M 2004 . Ove r seve n hundre d distinguishe d mathematicians an d professor s fro m aroun d th e worl d wil l gather a t Th e Chines e University o f Hong Kong to discus s their researc h wor k an d t o speak o n the lates t developments i n mathematics . The presentatio n o f the Morningsid e Meda l o f Mathematics, th e Cher n Prize , and th e ICC M Internationa l Cooperatio n Awar d mark s th e beginnin g o f the con - gress. Thes e awards, whic h are the highest honor s bestowed a t ICCM , ar e given to mathematicians wh o hav e mad e significant contribution s t o thi s field, an d whos e work have had and wil l continue to impact the future developmen t o f mathematica l sciences. Held ever y three years , the International Congres s o f Chines e Mathematician s is a considerabl e undertakin g tha t require s th e suppor t o f severa l organizations . ICCM 200 4 i s mad e possibl e b y TH E MORNINGSID E GROU P an d th e organi - zational effort s o f The Institut e o f Mathematical Science s an d th e Departmen t o f Mathematics a t Th e Chines e Universit y o f Hong Kong . Suppor t fro m othe r spon - sors an d academi c institution s i n Hong Kon g an d Chin a als o played a n invaluabl e role i n ensuring the succes s o f this congress . We are pleased that yo u are joining us at the opening ceremony o f ICCM 2004 .
XIV Ronnie C . Cha n Co-Founder The Morningsid e Grou p
Distinguished Mathematicians , Guests , an d Friends ,
On behal f o f THE MORNINGSID E GROUP , I woul d lik e to welcom e yo u t o the 200 4 Morningsid e Meda l o f Mathematic s Award s Presentatio n an d Sympo - sium, whic h kick s of f the Third Internationa l Congres s o f Chinese Mathematician s (ICCM). In 1996 , the n Presiden t Jian g Zemi n aske d Field s Medalis t Professo r Shing - Tung Yau to help develop excellent Chinese mathematicians. Workin g together with Professor Yongxian g Lu o f the Chinese Academy o f Sciences (CAS) , the three o f us founded th e Morningside Cente r o f Mathematics o n the CA S campus i n Beijing. A building, which won international architectural acclaim, was also constructed wher e renowned mathematicians from around the world can interact with local talents and engage i n mathematics research . Professor Ya u an d I have als o se t u p th e Morningsid e Meda l o f Mathematics . It i s give n onc e ever y thre e year s a t th e ICC M t o youn g Chines e mathematician s anywhere i n th e worl d fo r thei r achievement s an d contribution s t o thi s field o f science. Th e Selectio n Committee i s composed o f top mathematicians fro m aroun d the globe , wit h th e onl y ethni c Chines e bein g Professo r Yau , wh o als o serve s a s the chair . Th e firs t medal s wer e presente d i n Beijin g i n 1998 , followe d b y Taipe i in 2001 . Thi s year , i t wil l be hel d i n Hon g Kong , an d toda y w e will celebrate th e work o f seven Morningside Medalist s i n pure and applie d mathematics . I am honored an d pleased to be associated with these meaningful endeavors . I t is heartening to see the rise o f a generation o f young Chinese mathematicians. Ma y they enric h th e worl d wit h thei r mathematica l revelations , an d b e a n inspiratio n for other s to follow . This page intentionally left blank A Chines e Essa y i n Tribut e t o Professo r Shiing-Shen Cher n
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Honorary President s Zhili Chen , Stat e Councilor , Stat e Counci l o f the People' s Republi c o f Chin a Shiing She n Chern , Nanka i Universit y Yongxiang Lu , Chines e Academ y o f Science s
Congress Chairma n Shing-Tung Yau , Th e Chines e Universit y o f Hong Kong/Harvard Universit y
Steering Committe e Shiu Yue n Cheng , Hon g Kon g University o f Scienc e an d Technolog y Kai La i Chung, Stanfor d Universit y Chaohao Gu , Fuda n Universit y Lei Guo , Chines e Academ y o f Science s Ka-Sing Lau , Th e Chines e Universit y o f Hong Kon g Zhong-Ci Shi , Chines e Academ y o f Science s Yuan Wang , Chines e Academ y o f Science s Wen-Tsun Wu , Chines e Academ y o f Science s Zhouping Xin , The Chines e Universit y o f Hong Kon g Lo Yang, Chines e Academ y o f Science s
Scientific Committe e Shing-Tung Yau , Harvar d Universit y an d Th e Chines e Universit y o f Hong Kon g Ngai-Hang Chan , Th e Chines e Universit y o f Hong Kon g Tony Chan , Universit y o f California, Lo s Angele s Shiu Yue n Cheng , Hon g Kong Universit y o f Science and Technolog y Man-Duen Choi , University o f Toront o Jiaxing Hong , Fudan Universit y Sze-Bi Hsu, National Tsin g Hu a Universit y Ming-Chang Kang , Nationa l Taiwa n Universit y Tsit-Yuen Lam , Universit y o f California , Berkele y Ka-Sing Lau , Th e Chines e Universit y o f Hong Kon g Jun Li , Stanford Universit y Bong Lian , Brandei s Universit y Chang-Shou Lin , National Tsin g Hu a Universit y
xix xx COMMITTEE S O F ICC M 200 4
Song Su n Lin , Nationa l Chia o Tun g Universit y Kefeng Liu , Zhejian g Universit y an d Universit y o f California , Lo s Angele s Yiming Long , Nankai Universit y Jiang-Hua Lu , The Universit y o f Hong Kon g Shengli Tan, Eas t Chin a Norma l Universit y Zhi Ying Wen , Tsinghu a Universit y Wing-Hung Wong , Harvard Universit y Sijue Wu , Universit y o f Marylan d Zhouping Xin , The Chines e Universit y o f Hong Kon g Lo Yang, Chines e Academ y o f Science s Andrew Chi-Chi h Yao , Princeton Universit y Horng-Tzer Yau , Ne w York Universit y Stephen S . T. Yau , Universit y o f Illinois at Chicag o Jing Yu , National Tsin g Hua Universit y Qiang Zhang , Th e Cit y Universit y o f Hong Kon g
Executive Committe e Alice S . Chang, Princeto n Universit y Louis H. Y. Chen , National Universit y o f Singapor e Guoshun Cheng , Nationa l Chiaotun g Universit y Xiaxi Din g , Chinese Academ y o f Science s Weinan E , Princeto n Universit y Jianqing Fan , Th e Chines e Universit y o f Hong Kon g Kai-tai Fang , Hon g Kong Baptist Universit y Sheng Gong , Chines e Academ y o f Science s Thomas Y . Hou , Californi a Institut e o f Technolog y Zixin Hou , Nanka i Universit y Daren Huang , Zhongsha n Universit y Daqian Li , Fudan Universit y Qikeng Lu , Chines e Academ y o f Science s Min-you Qi , Wuhan Universit y Kok-Lay Teo , Hong Kong Polytechni c Universit y Howell Tong , The Universit y o f Hong Kon g Renhong Wang , Dalia n Universit y o f Technolog y Yuefei Wang , Chines e Academ y o f Science s Roderick S . C . Wong, The Cit y Universit y o f Hong Kon g Jeff Wu , Georgi a Institute o f Technolog y Zhihong Xia , Northwester n Universit y Stephen S . T. Yau , Universit y o f Illinois a t Chicag o Yaxiang Yuan , Chines e Academ y o f Science s Tao Zhan , Shandon g Universit y Jiping Zhang , Pekin g Universit y Weiping Zhang , Nanka i Universit y Ya-Qin Zhang , President Microsof t Researc h Asi a Xiangyu Zhou , Chines e Academ y o f Science s
Local Organizin g Committe e Ka-Sing Lau , Chairman , Th e Chines e Universit y o f Hong Kon g Zhouping Xin , Chairman , Th e Chines e Universit y o f Hong Kon g LIST O F COMMITTEE S O F ICCM0 4
Raymond Chan , Vice-chairman , Th e Chines e Universit y o f Hong Kon g Tom Wan, Vice-chairman , Th e Chines e Universit y o f Hong Kon g Thomas Au , The Chines e Universit y o f Hong Kon g Wing-Sum Cheung , Th e Universit y o f Hong Kon g Edmund Chiang , Hon g Kon g Universit y o f Scienc e and Technolog y Fred J . Hickernell , Hon g Kon g Baptist Universit y Patrick, Tuen-Wa i Ng , The Universit y o f Hong Kon g Liqun Qi , Hong Kong Polytechni c Universit y Chong-Sze Tong , Hon g Kon g Baptist Universit y Xiaoping Wang , Hon g Kong University o f Scienc e and Technolog y Ngai-Ying Wong , The Chines e Universit y o f Hong Kon g Ding-Xuan Zhou , Th e Cit y Universit y o f Hong Kon g
Committee o n Basi c Educatio n Chairmen: Thoma s Au , Th e Chines e Universit y o f Hong Kon g Siu-Leung Ma , Director , Hon g Kon g Education Cit y Limite d
Pundraising Committe e Chairman: Banke e Kwan , Celestia l Asi a Securitie s Holding s Limite d Kin Chun g Lam , Stron g Ma n Investmen t Limite d This page intentionally left blank Morningside Lifetim e Achievemen t Award i n Mathematic s
Shiing-Shen Cher n
Professor Shiing-She n Cher n received th e Morningsid e Lifetim e Achievement Awar d i n 200 1 fo r hi s work o n developin g th e foundatio n o f Chinese mathematics , hi s epocha l con - tributions to research i n differential ge- ometry, an d hi s nurturin g o f leadin g mathematicians i n Chin a an d abroad . Professor Cher n i s considere d on e o f the greates t geometer s o f th e twenti - eth century and a pioneer i n differentia l geometry. Som e o f hi s majo r achieve - ments includ e th e Cher n characteris - tic classe s i n fiber space s an d hi s proo f of th e Gauss-Bonne t theorem . I n th e 1940s, differentia l geometr y wa s onl y starting t o b e understood an d studied . Today, i t i s a majo r subjec t are a i n mathematics largel y du e t o Professo r Chern's researc h wor k an d hi s trainin g of talented mathematician s i n this field. Professor Cher n wa s born i n 191 1 in Jiaxing , Zhejian g provinc e i n China . H e received his B.S. from Nanka i University i n 1930 , his M.Sc. fro m Tsinghu a Univer - sity in 1934 , and his Ph.D. from the University o f Hamburg in Germany. Fro m 194 3 to 1945 , he was a member o f the Institute fo r Advance d Stud y at Princeton Univer - sity. Fro m 194 6 to 1948 , he was the Acting Director o f the Institute o f Mathematics at Academi a Sinic a i n Nanjing. Fro m 194 9 to 1960 , he taught a t th e Universit y o f Chicago befor e joinin g th e facult y o f the Universit y o f California , Berkeley . Fro m 1982 to 1984 , he was the Founding Director o f the Mathematical Science s Researc h Institute (MSRI) , th e world' s premie r cente r fo r collaborativ e researc h acros s th e whole spectrum o f mathematical sciences . Fro m 198 5 to 1993 , he was the Foundin g Director o f the Nanka i Institute o f Mathematics i n Tianjin .
xxiii xxiv SHIING-SHE N CHER N
Professor Cher n was a member or foreign member o f many academies, includin g Academia Sinica, the U.S. National Academy o f Sciences, the Third World Academy of Sciences , th e Roya l Societ y o f London , L'Academi e d e Science s d e Paris , th e Chinese Academ y o f Sciences , an d th e Russia n Academ y o f Sciences . Hi s award s included th e U.S . Nationa l Meda l o f Scienc e i n 1975 , whic h i s th e highes t hono r bestowed b y th e U.S . President t o distinguishe d scientist s an d engineer s fo r thei r pioneering scientifi c research , th e Wol f Priz e i n 1983 , whic h i s on e o f th e mos t prestigious award s fo r scientists , an d th e Sha w Priz e i n Mathematica l Science s i n 2004. Professor Cher n passe d awa y o n Decembe r 3 , 2004 . H e wa s towerin g figur e in mathematic s an d ha d a caree r tha t spanne d continent s an d cultures . A s a teacher an d researcher , Professo r Cher n helpe d forg e link s wit h mathematician s across th e glob e whil e advancin g knowledg e o f differentia l geometry . H e founde d and serve d a s the foundin g directo r o f three to p mathematica l researc h institutes : the Institute o f Mathematics at the Chinese Academy o f Sciences, the Mathematica l Sciences Research Institute at the University o f California, Berkeley , and the Nankai Institute o f Mathematics i n Tianjin . Professo r Cher n wil l remain a n inspirationa l figure t o al l mathematicians . Morningside Meda l o f Mathematic s
The Morningside Meda l o f Mathematics i s awarded to outstanding mathemati - cians o f Chinese descent t o encourage them i n their pursui t o f mathematical truth . Gold an d silve r medal s ar e awarde d t o mathematician s unde r th e ag e o f forty-fiv e for exceptiona l achievement s i n pur e an d applie d mathematics . Th e Morningsid e Medals ar e awarde d ever y thre e year s a t th e Internationa l Congres s o f Chines e Mathematicians. A committee of internationally renowned mathematicians, chaire d by Professo r Shing-Tun g Yau , select s the medalists . The firs t medal s were awarded o n December 12 , 199 8 at the First Internationa l Congress o f Chinese Mathematicians (ICC M 1998 ) a t th e Grea t Hal l o f the Peopl e in Beijing . Th e secon d medal s wer e awarde d o n Decembe r 17 , 2001 at th e Gran d Hotel i n Taipe i a t th e Secon d Internationa l Congres s o f Chines e Mathematician s (ICCM 2001) . A special Morningside Lifetime Achievemen t Awar d i n Mathematic s and a cash award o f US$50,000 were also presented to Professor Shiing-She n Cher n at ICC M 2001 . Th e thir d medal s wer e awarde d o n Decembe r 17 , 200 4 a t th e The Hon g Kon g Conventio n an d Exhibitio n Centr e i n Hon g Kon g a t th e Thir d International Congres s o f Chinese Mathematician s (ICC M 2004) . Each medalis t receive s a certificate, medal , an d cas h awar d o f US$25,000 fo r a gold medal or US$10,000 fo r a silver medal. Beside s the name o f the medalist, eac h Morningside Meda l o f Mathematics ha s the imag e o f a Mobius ban d an d a funda - mental domain . The y wer e chose n no t onl y fo r th e simplicit y o f their image , bu t also becaus e o f their significant contribution s t o th e developmen t o f mathematic s in the twentiet h century . The Mobiu s band (als o known a s the Mobiu s strip ) wa s discovere d i n 185 8 by the Germa n mathematicia n an d astronome r Augus t Ferdinan d Mobius . Thi s curi - ous onesided surfac e doe s not hav e any orientation, ye t has a distinct topographica l character. I t wa s on e o f the mos t importan t discoverie s o f the twentiet h century , which ha s profoundl y influence d moder n physics , classica l physics , an d moder n mathematics, includin g geometry . A fundamental domai n i s related to the concep t o f a group, which can be use d to expres s symmetr y i n mathematics. Durin g the lat e nineteent h century , infinit e xxvi MORNINGSID E MEDA L O F MATHEMATIC S discrete grou p wa s studie d bu t onl y unti l th e twentiet h century , di d i t becom e a main subjec t are a i n mathematics . Thi s fiel d o f stud y i s no t onl y importan t i n geometry bu t als o i n numbe r theory . Practicall y al l th e famou s development s i n modern number theory ar e related to concepts o f fundamental domai n an d discret e group. Morningside Meda l o f Mathematics - Selectio n Committee
The Morningsid e Meda l o f Mathematic s Selectio n Committe e i s chaire d b y Professor Shing-Tun g Yau . A nomination committe e fo r the Morningside Meda l o f Mathematics, comprisin g a maximum o f fifty Chines e mathematician s worldwide , nominates individual s based o n their research , qualifications , an d curriculum vitae . The nominatio n committe e the n submit s th e name s o f the nominate d individuals , along wit h supportin g materials , t o th e selectio n committee . Afte r a thoroug h multi-step review , th e selectio n committe e make s a final decisio n regardin g th e medalists. Th e winners o f the Morningside Medal o f Mathematics are announced a t an awards presentation durin g the opening ceremony o f the International Congres s of Chinese Mathematicians . All members o f the selectio n committee , wit h th e exceptio n o f the committe e chair, ar e non-Chinese , thereb y ensurin g th e independenc e an d integrit y o f thei r decision. Th e te n member s o f the 200 4 Morningside Meda l o f Mathematics Selec - tion Committe e are :
• Pete r Bicke l (Professo r o f Statistic s a t th e Universit y o f California , Berkeley) • Joh n Coate s (Th e Sadleiria n Professo r o f Pur e Mathematic s a t Cambridge University ) • Ronal d R . Coifma n (Professo r o f Mathematics an d Compute r Scienc e at Yal e University ) • Davi d Gieseke r (Professo r o f Mathematic s a t th e Universit y o f California, Lo s Angeles ) • Stanle y Oshe r (Professo r o f Mathematics , th e Directo r fo r Specia l Projects a t th e Institute fo r Pur e an d Applie d Mathematics , an d th e Di - rector o f Applie d Mathematic s a t th e Universit y o f California , Los Angeles ) • Joe l Smolle r (Th e Lambert o Cesar i Chai r i n Mathematic s a t th e University o f Michigan, An n Arbor ) • Cumru n Vaf a (Th e Donner Professo r o f Scienc e at Harvar d University ) • Lesli e G . Valian t (Th e T . Jefferso n Coolidg e Professo r o f Compute r Science an d Applie d Mathematic s a t Harvar d University )
xxvii xxviii MORNINGSID E MEDA L O F MATHEMATIC S - SELECTIO N COMMITTE E
• Srinivas a S . R . Varadha n (Professo r o f Mathematic s a t th e Couran t Institute o f Mathematical Science s at Ne w York University ) • Shing-Tun g Ya u (Th e Willia m Caspe r Graustei n Professo r o f Mathe - matics a t Harvar d University , the Director o f The Institute o f Mathemat - ical Science s a t Th e Chines e Universit y o f Hon g Kong , an d th e Directo r of th e Morningsid e Cente r o f Mathematic s a t th e Chines e Academ y o f Sciences i n Beijing ) Morningside Awar d Medalist s
Morningside Gol d Meda l o f Mathematics : Recipients Kefeng Liu , Professo r o f Mathematics, Universit y o f California , Los Angele s Zhouping Xin , Willia m M . W. Mon g Professo r o f Mathematics, The Chines e Universit y o f Hong Kon g
Morningside Gol d Meda l o f Applie d Mathematics : Joint Recipient s Thomas Yizha o Hou , Charle s Le e Powell Professor o f Applied an d Computational Mathematics , Californi a Institut e o f Technolog y Zhiliang Ying , Professo r o f Statistics, Columbi a Universit y
Morningside Silve r Meda l o f Mathematics : Recipients Jin-Yi Cai , Professo r o f Computer Science , University o f Wisconsin , Madison Ai-Ko Liu , Assistan t Professo r o f Mathematics, Universit y o f California, Berkele y Xi-Ping Zhu , Professo r o f Mathematics, Zhongsha n Universit y
XXIX This page intentionally left blank The Wor k o f Kefun g Li u
Jim L i
Dr. Li u is one o f the leading young Chinese mathematicians. H e has worked o n a wide range o f mathematical subjects , an d ha s shaped man y aspect s o f the topic s he touched . His first spectacular work is on the rigidity and vanishing of elliptic genus. Ellip - tic genu s was discovered throug h th e joint effor t o f mathematicians an d physicists . It i s the inde x theor y o f loo p spaces . Th e rigidit y conjecture s o f ellipti c genu s b y Witten wer e mentioned a s on e o f the thre e mai n contribution s fo r Witten' s Field s medal. Taube s gave a proof to a rigidity theorem, whic h is very complicated. Bott - Taubes simplified i t to some extent. I n his work, Li u realized the role o f modularity to the rigidit y conjectur e an d provide d a n elegan t proof t o the know n rigidit y the - orem. H e als o discovere d man y ne w result s i n thi s field. Hi s proo f reshape d ou r understanding o f the rigidit y theorem . His wor k o n usin g hea t kerne l t o stud y modul i space s o f vecto r bundle s ove r smooth curve s wa s just a s profound . B y pullin g bac k th e hea t kerne l an d the n applying localization , h e gav e a ne w an d simpl e proo f t o the Witten's nonabelia n localization; i t influence d th e late r wor k i n this field. His joint wor k with B.Lian and S.T . Ya u on the Mirror symmetry conjectur e o f quintic threefold s an d it s generalization dre w eve n a wider applause . Thi s subjec t began wit h Candelas ' amazin g generatin g functio n o n th e "numbers " o f rationa l curves in a general quintic CY , derived i n early 9 0 via mirror symmetry conjecture . By using localization technique, Liu and his collaborators derived a set o f quadratic relations i n th e equivarian t Cho w rin g o f th e linea r E-model . The y the n showe d that i n many cases, which include many hypersurfaces i n toric varieties and balloo n manifolds, th e "virtual " numbe r o f rationa l curve s coul d b e reconstructe d fro m these relations . Th e Candelas ' formul a i s a consequenc e o f their work . Thi s wor k is on e o f th e mos t importan t i n thi s subject . I t i s als o th e mos t powerfu l one , covering man y example s i n loca l Mirro r symmetry , hypersurface s i n tori c varietie s and balloo n varieties . Most recently , togethe r wit h M . Li u an d J . Zhou , Li u ha s prove d a conjectur e of Marino-Vafa o n Hodge integrals o n the modul i o f stable curves. Thi s conjectur e was based on relating open string Gromov-Witten invariant s with the Cheng-Simo n invariants o f three manifolds. It s nove l origin ensured it s far reac h consequence bu t also came with difficult y i n proving this. I n their proof , they applie d th e techniqu e of localization to the moduli spac e o f relative stable maps; it i s a beautiful proof ; i t
xxx i xxxii J . L I is als o a n importan t theore m tha t establishe d man y previousl y unknow n identitie s among Hodg e integrals . His work with X.-F.Su n an d S.-T . Ya u o n the metric geometr y o f the moduli o f Riemann surface s ha s reshape d thi s researc h subject . Th e question s t o ho w thes e metrics, includin g th e Peter-Wilso n metric , th e hyperboli c metric , an d man y othe r metrics, puzzle d man y mathematician s fo r decades . I n tryin g t o understan d thei r relations, mor e metric s wer e introduced . I t wa s onl y afte r th e wor k o f Li u an d hi s collaborators tha t a complet e pictur e immerged . Seeing the lis t o f publications o f Liu, i t i s impossible not to have been impresse d by th e breat h an d th e dept h o f hi s mathematics . H e ha s worke d i n area s rangin g from differentia l geometry , topology , algebrai c geometr y t o mathematica l physics . On eac h o f the are a he has touched, h e lef t a permanent mar k fo r the later to follow .
DEPARTMENT O F MATHEMATICS , STANFOR D UNIVERSITY , 45 0 SERR A MALL , BLDG . 380, STANFORD , C A 94305-212 5 E-mail addresss: [email protected] u Summary o f Main Researche s o f Zhoupin g Xi n
by Joe l Smolle r
The focu s o f researches o f Xi n ha s bee n the field o f theoretical an d numerica l analysis an d computation s o f nonlinea r partia l differentia l equation s arisin g fro m continuum mechanics , such as Euler an d Navier-Stoke s systems fo r both compress - ible and incompressible, and nonlinear Boltzmann type equations, etc. H e has made substantial contribution s t o man y problem s i n various subareas o f this field. Som e of the highlight s o f his works include :
Theory o f Multi-dimensiona l Shoc k Wave s Compared wit h th e recen t advance s i n th e one-dimensiona l theor y o f shoc k waves, the theor y fo r th e multi-dimensio n i s fa r fro m bein g develope d du e t o th e great complexit y an d lac k o f understanding . However , suc h a multi-dimensiona l theory i s o f fundamenta l importanc e i n bot h mathematica l theor y an d physica l applications. I t seem s necessar y t o star t wit h som e basi c problem s whic h ar e fun - damental i n fluid dynamics an d there ar e a lot physica l experiments an d numerica l data ar e available , suc h a s transonic flows with shocks , supersoni c pas t soli d bod - ies, and shoc k reflection problems , etc. Recently , Xi n has achieved som e significant results alon g these lines : 1. Studie s o n Transonic Shoc k Wave s Phenomena involvin g transonic flows and transonic flows with shock s is a fundamental subjec t i n fluid dynamics, especially gas dynamics. A generic transonic flow pattern involve s shock s i n genera l sinc e smoot h transoni c flows are unstabl e i n th e presenc e o f physica l boundaries . I n [1] , jointly with H. Yin, Xin has studied the problem o f existence and uniqueness o f a transonic shoc k to the stead y flow through a general nozzl e with variabl e sections i n both 2- D an d 3- D cases . Thei r approac h seem s quite powerfu l and robus t i n th e sens e tha t othe r problem s involvin g transoni c shock s can be treated similarly . I n particular, this approach ca n be used to prove the existence and uniqueness o f strong transonic shock fo r supersonic flow past a 3-dimensiona l wedge . 2. Studie s o n Globa l Shock s fo r Supersoni c Flo w Past A Pointed Bod y In [2] , wit h hi s collaborators , Xi n establishe d th e globa l existenc e an d structural stabilit y towar d self-simila r shoc k wave s o f a shoc k wav e solu - tion t o th e stead y supersoni c ga s flows past a n infinit e curve d an d ax - isymmetric cone . Fo r a steady supersoni c flow hitting o n a sharp curve d xxxiv J . SMOLLE R
body whic h i s a smal l perturbatio n o f a circula r cone , i t i s show n i n [2 ] that a n entropy wea k solution to this problem exist s globally in the whol e space wit h a pointe d shoc k attache d a t th e ti p o f the con e an d tend s t o a self-simila r solutio n correspondin g unperturbe d circular . Furthermore , the solution i s smooth everywher e except a t the global shock and i s struc- turally stable. Thi s settles a fundamental proble m in gas dynamics, whic h is a well-know n difficul t proble m an d ha s bee n attacke d b y man y peopl e without success . Th e approac h i n [2 ] i s base d o n lookin g fo r multiplier s for an enlarged linearized system, and hence truly multi-dimensional. Th e analysis shoul d she d light s on man y othe r multi-dimensiona l problems .
Boundary Laye r Theor y an d Prandtl' s Syste m In the presence o f physical boundaries, the asymptotic behavior o f viscous flo w for larg e Reynolds numbers i s of great importanc e physicall y an d i s extremely diffi - cult to study mathematically du e to the appearance o f the boundary laye r phenom - ena. I n 1904 , Prandtl propose d a theor y whic h show s that th e invisci d idea l flo w can approximat e th e rea l flui d onl y i n th e regio n awa y fro m th e physica l bound - aries and i n a thin region near the boundary the fluid velocit y changes dramaticall y and i s governed b y the Prandtl's equations . Thi s i s now called Prandtl's boundar y layer theor y an d ha s becom e a n essentia l par t o f th e flui d dynamics . I n collabo - ration wit h Yanagisawa , Xi n ha s justifie d th e Prandtl' s theor y fo r th e linearize d Navier-Stokes flow s an d obtaine d detaile d asymptoti c structur e an d optima l rat e of convergence . Th e nonlinea r proble m i s muc h mor e difficult . On e o f th e mai n difficulties i s th e well-posednes s o f th e initia l o r initia l boundar y valu e problem s for th e Prandtl' s equation s i n a Sobole v space . Despit e th e tremendou s effort s by man y people , th e bes t availabl e result s alon g thi s lin e unti l recentl y ar e du e to OLeini k wh o establishe d i n a serie s o f wor k th e LOCA L well-posednes s i n th e class o f monoton e data . I t i s a longstandin g ope n proble m t o achiev e th e globa l well-posedness o f the solutio n t o the Prandtl' s syste m i n the Oleinik' s clas s i n th e case tha t th e pressur e i s favorable . Recently , i n [3] , joint wit h hi s collaborators , Xin ha s solve d thi s ope n proble m completely . First , Xi n an d Zhan g establishe d the globa l existenc e o f wea k solutio n i n th e spac e o f bounde d tota l variation s b y introducing a splitting metho d an d a n elaborate d Nash-Mose r iteratio n schem e i n [3]. Mor e recently , Xin , together wit h Zhan g an d Zhao , has show n that suc h a B V solution depend s o n the dat a continuousl y i n Li-norm, an d henc e i s unique. Thi s is achieved b y studying the structure o f BV solutions carefull y base d o n geometri c measure theory . Furthermore , the y hav e show n suc h wea k solution s ar e i n fac t classical b y developin g a genera l regularit y theor y fo r degenerat e ultra-paraboli c equations base d o n the Heisenbur g grou p structur e o f the linearize d equation s [3] . This i s a result lon g sought b y the expert s i n the field .
Blow-up o f Smoot h Solutio n t o th e Compressibl e N- S Equation s Vacuum states play an essential role in the well-posedness theory o f solutions to the viscou s compressibl e Navier-Stoke s system , a s pointed ou t b y many importan t studies du e t o Hoff-Smoller , P.L.ions , etc . I n [4] , Xin obtain s th e firs t an d rathe r significant result s o n the finit e tim e blow-u p o f smooth solution s t o th e compress - ible N- S equation s wit h initia l densit y o f compac t support . Th e essentia l ide a i s SUMMARY O F MAI N RESEARCHE S O F ZHOUPIN G XI N xxx v a ver y elegan t analysi s o f the deca y o f total pressur e base d o n estimat e o f meno - ments fo r the Navier-Stoke s equations . Thi s i s the firs t dispersiv e estimat e fo r th e total pressur e o f the compressibl e N- S system . A s a consequence , h e showe d tha t there are no global (i n time) bounde d smoot h solutions to the general compressibl e Navier-Stokes equation s n o matte r ho w smal l th e initia l dat a ar e a s lon g a s th e initial densit y ha s compac t support .
Relaxation Approximatio n an d Relaxatio n Scheme s Instead o f the usua l parabolic regularizing , Xi n and Ji n firs t propose d th e ne w idea o f relaxatio n approximation s t o a clas s o f quasi-linea r firs t orde r system s o f PDE's suc h a s system s o f conservatio n law s i n an y spac e dimensions , Hamilton - Jacobi equations, an d equation s fo r curvature-dependen t front s motions , etc . Thi s was firs t propose d b y Xi n an d Ji n i n [5 ] fo r system s o f conservation s i n arbitrar y dimensions, whic h i s no w bein g calle d Jin-Xi n relaxatio n models . Th e mai n ide a is that fo r a give n syste m o f conservation laws , on e construct s a n enlarge d hyper - bolic system (calle d relaxation system ) wit h linear constant convectio n an d specia l nonlinear sourc e term s whic h ar e designe d s o that th e solutio n t o th e relaxatio n system relaxe s to th e solutio n o f the give n syste m o f conservation law s fas t i n th e limit o f smal l relaxation . Du e t o th e specia l structur e o f th e ne w relaxatio n sys - tems, the y hav e bee n abl e t o construc t man y hig h resolutio n numerica l scheme s (called relaxation schemes) .
Vortex Sheet s Motio n an d Vorte x Method s One o f the mos t importan t 2- D incompressibl e invisci d flow s ar e evolutio n o f vortex sheet s i n whic h th e vorticit y i s a finit e Rado n measur e concentrate d o n a curve and exhibits the classical Kelvin-Helmholtz instability . Th e existence o f clas- sical weak solutions to the 2- D incompressibl e Eule r equation s wit h genera l vorte x sheets i s o f fundamental importanc e bot h mathematicall y an d physically . An d th e vortex methods (includin g both the vortex blob methods and pointed vortex meth - ods) ar e Lagrangian method s whic h ar e especiall y effectiv e t o comput e suc h flow s and ar e mostly use d i n practical engineerin g computations . Ver y interesting struc - ture wer e revealed afte r singularit y i n the vortex roll-u p i n many computation s b y Kransy b y th e vorte x methods . Analysi s o f suc h problem s i s extremel y difficul t since th e solutio n i s ver y wild . I n [7] , Xin an d J . Li u gav e th e firs t resul t o n th e convergence o f the vortex method fo r suc h "wild " flo w consisting o f a signed Rado n measure globally in time (beyon d the conjectured tim e o f the singularity formatio n in th e vorte x sheet) , an d the y prove d th e globa l convergenc e t o a classica l wea k solution o f approximation solution s generate d b y the vorte x blo b methods fo r th e vortex sheet s initia l dat a a s lon g a s the initia l vorticit y i s o f one-sign . On e o f th e main difficultie s her e i s the wea k consistency . Thi s analysi s wa s late r generalize d to the pointed-vorte x method s b y Xin an d J . Liu .
Dissipative Effect s t o System s o f Hyperbolic Conservatio n Law s One o f the mos t importan t problem s an d th e main driving forc e i n mathemat - ical theory o f flui d dynamic s i s understanding th e relatio n betwee n wea k solution s to the invisci d hyperboli c system s (suc h a s the compressibl e Eule r equations ) an d the solutions to the corresponding suitable viscous systems (suc h as the well-know n compressible Navier-Stoke s equations , whic h ar e paraboli c system s i n general ) i n xxxvi J . SMOLLE R the limi t o f smal l viscosity . Thi s i s s o partly du e t o th e non-uniquenes s o f wea k solutions t o th e hyperboli c systems , an d th e physica l significanc e o f th e slightl y viscous fluids i n reality . However , thi s pose s a lon g standin g challengin g mathe - matical proble m du e t o th e highl y singula r natur e o f the limi t i n th e presenc e o f shock discontinuities an d boundaries. Xi n has made many important contribution s to the subjec t whic h includes : 1. Larg e time asymptoti c stabilit y o f linear an d nonlinea r viscou s wave s The complete theory was made by Xin and Szeppes y in [9] . In this theory, they discovere d th e surprisin g phenomen a tha t a generi c perturbation o f a give n shoc k profil e produce s no t onl y phas e shif t i n th e shoc k profil e itself an d diffusiv e wave s i n th e transversa l wav e directions , bu t als o in - troduces resonan t diffusio n wave s i n the shoc k wave s regio n du e to wav e interactions. 2. Vanishin g viscosity limit fo r piecewise smooth solutions to systems o f con- servation law s In [11] , Xin and Goodma n prove d that fo r genera l systems, an y piecewis e smooth solutions with finitely many shoc k discontinuities whic h satisfyin g Lax-entropy conditions are in fact the limits of solutions of the correspond- ing viscous systems .
Fluid-Dynamic Limi t fo r th e Broadwel l Mode l o f Th e Nonlinea r Boltzmann Equatio n i n Th e Presenc e o f Shock s Fluid-dynamic limi t o f the kineti c equation s suc h a s the Boltzman n equatio n is muc h mor e singula r tha n th e viscou s system . Th e genera l Boltzman n equatio n of kineti c theor y give s a statistica l descriptio n o f a ga s o f interactin g particles . An importan t propert y o f this equatio n i s its asymptoti c equivalenc e t o the Eule r and Navier-Stoke s system s o f compressibl e fluid dynamic s i n th e limi t o f smal l mean fre e path . Al l the previou s work s (du e to Grad , Nishida , Caflisch , Caflisch - Papanicoulou, etc. ) dea l wit h macroscopi c smoot h fluid flows (i n othe r words , before shoc k formations , (thi s i s true fo r mos t o f the wor k o n the statistica l larg e particle syste m limit)) . I n [12] , Xin proved th e first theore m o n the validit y o f the fluid-dynamic limi t to the piecewis e smooth fluid flows for the Broadwel l equation .
References [1] Transoni c shock s i n a nozzle, I , 2- D case, (wit h H . Yin), Comm. Pure Appl . Math . 5 8 (2005) , no. 8 , 999-1050 , on-lin e publicatio n sinc e April , 2004 ; and , Multi-dimensiona l Transoni c shocks i n a nozzl e II , 3- D case , (wit h H . Yin), IM S Researc h Repor t IMS-2005-1 1 (135) . [2] Globa l shock waves for the supersonic flow past a perturbed cone , (with S . Chen an d H . Yin), Commun. Math . Phys . Vol . 228 , 47-84 (2002) . [3] O n th e globa l existenc e o f solution s o f Prandtl' s system , (wit h L . Q . Zhang) , Advance s i n Math., 18 1 (2004), 88-133, and, O n the continuou s dependenc e and regularit y o f the solutio n (Part II) , (wit h L . Zhan g an d J . Zhao ) wil l appear . [4] O n the blow-up o f smooth solutions to the compressible Navier-Stokes equations with compac t density, Comm . Pur e Appl . Math. , Vol . 5 1 (1998) , 229-240 . [5] Th e relaxation schemes for systems o f conservation laws in arbitrary dimensions , (wit h S . Jin), Comm. Pur e Appl . Math. , Vol . 48 (1995) , 61-105. [6] Relaxatio n scheme s for the curvature-dependent fron t propagations , (wit h S . Jin and M . Kat - soulakis), Comm . Pur e Appl . Math. , Vol . 5 2 (1999) , 1587-1615 . SUMMARY O F MAI N RESEARCHE S O F ZHOUPIN G XI N xxxvi i
[7] Convergenc e o f vorte x method s fo r wea k solution s t o th e 2- D Eule r equation s wit h vorte x sheets data , (wit h J . Liu) , Comm . Pur e Appl . Math. , Vo l 4 8 (1995) , 611-628 . [8] Existenc e o f vortex sheet s with reflectio n symmetr y i n two space dimensions , (wit h H . Lope s and M . Lopes) , Arch . Rat . Mech . Anal. , Vol . 158(2001) , 235-257 . [9] Nonlinea r Stabilit y o f viscou s shoc k waves , (wit h A . Szepessy) , Arch . Rat . Mech . Anal. , Vol. 12 2 (1993) , 217-256 . [10] Pointwis e deca y t o contac t discontinuitie s fo r system s o f viscou s conservatio n laws , (with T . P . Liu) , Asia n J . Math. , Vol . 1 (1997), 34-84 . [11] Viscou s limi t t o piecewis e smoot h solution s o f system s o f conservatio n laws , (wit h J. Goodman) , Arch . Rat . Mech . Anal. , Vol . 12 1 (1992) , 235-265. [12] Th e fluid-dynamic limi t o f the Broadwel l mode l o f the nonlinea r Boltzman n equatio n i n th e presence o f shocks, Comm . Pur e Appl . Math. , Vol.4 4 (1991) , 679-713. This page intentionally left blank Research Accomplishment s o f Thomas Yizha o Ho u
Stanley Oshe r
1. Introductio n Thomas Yizha o Ho u i s the Charle s Le e Powel l professor o f applie d an d com - putational mathematic s a t Caltech , an d i s on e o f th e leadin g expert s i n applie d and numerica l analysi s fo r vorte x dynamic s an d multiscal e problems . Hi s researc h interests ar e centere d aroun d developin g analytica l tool s an d effectiv e numerica l methods fo r vorte x dynamics , interfacia l flows, an d multiscal e problems . H e re - ceived hi s B.S . fro m Sout h Chin a Universit y o f Technolog y i n 198 2 and hi s Ph.D. from UCL A in 1987 . Upo n graduating fro m UCLA, he joined the Courant Institut e as a postdoc an d then becam e a faculty membe r i n 1989 . H e moved to the applie d math departmen t a t Caltec h i n 1993 , and serve d a s the chai r o f the departmen t o f applied an d computationa l mathematic s fro m 200 0 to 2006 . Dr . Ho u has receive d a number o f honors and awards, including the Computational an d Applied Science s Award fro m Unite d State s Associatio n o f Computationa l Mechanic s i n 2005 , th e SI AM Wilkinson Prize in Numerical Analysis and Scientifi c Computing in 2001, the Francois N . Frenkiel Awar d fro m th e Divisio n o f Fluid Mechanic s o f APS i n 1998 , the Fen g Kan g Priz e i n Scientifi c Computin g i n 1997 , a Sloa n fello w fro m 199 0 to 1992. H e was an invited plenary speaker at the International Congres s o f Industrial and Applied Mathematics i n 2003, and an invited speaker o f the International Con - gress o f Mathematicians i n 1998 . H e has been als o the foundin g Editor-in-Chie f o f the S I AM Journal o n Multiscal e Modelin g an d Simulatio n sinc e 2002. Dr. Ho u was awarded the first Morningside Gol d Medal Prize in Applied Math - ematics during the Third International Congres s o f Chinese Mathematicians whic h was hel d i n Hon g Kon g fro m Decembe r 1 7 to 22 , 2004 . Accordin g t o th e cita - tion o f the Morningsid e Gol d Meda l Prize , Dr . Ho u wa s honore d "fo r hi s semina l research o n applie d partia l differentia l equations , scientifi c computatio n an d nu - merical analysis . Thi s researc h include s convergenc e o f the poin t vorte x method , accurate numerica l method s fo r fluid interface s wit h surfac e tension , analysi s o f three dimensiona l vorte x sheets , an d singularit y criteri a fo r th e thre e dimensiona l Euler equation. " Belo w I will give a brief revie w o f some o f his work cited above .
2. Localize d non-blowu p criteri a fo r th e 3 D Eule r equation s 2.1. A brie f review . On e o f Dr . Hou' s mai n contribution s i s t o develo p localized non-blowu p criteri a fo r th e 3 D incompressibl e Eule r equation s an d t o apply the m t o stud y th e dynami c depletio n o f th e vorte x stretchin g i n th e 3 D xxxi x xl S. OSHE R
Euler equations . Th e questio n o f whether th e 3 D incompressibl e Eule r equation s with smoot h initia l dat a ca n develo p a finit e tim e singularit y i s on e o f th e mos t challenging ope n question s i n applie d mathematics . Th e understandin g o f thi s fundamental questio n woul d enhanc e ou r understandin g o f flui d dynami c stabilit y and she d usefu l ligh t i n our understanding o f the onse t o f turbulence. T o illustrat e some key points, w e consider the 3D incompressible Eule r equations in the vorticit y stream functio n formulatio n [14 ]
(1) uo t - f u - Vuo = Vu - UJ, where u i s velocit y an d to = V x u i s vorticity . Th e initia l conditio n i s smoot h and decay s rapidly a t infinity . Velocit y can be recovered fro m vorticit y through th e stream function , i/;, by solvin g
-Atp = a; , u = V x ij). The ter m o n th e righ t han d sid e o f (1 ) i s calle d th e vorte x stretchin g term . Thi s term i s absen t fo r th e two-dimensiona l problem . Formally , th e vorte x stretchin g term has a scaling which i s quadratic i n vorticity. Du e to the presence o f the vorte x stretching term, there is currently n o global regularity resul t fo r the 3D Euler equa - tions [14] . The best known result i s Beale-Kato-Majda's blowu p criterion [2] , which states tha t a necessar y an d sufficien t conditio n fo r th e solutio n t o develo p a finit e time singularity a t T i s that J Q \\u\\L
2.2. A localized non-blowup criterio n of Deng-Hou-Yu. I n [4 , 5], Deng, Hou, an d Y u develope d a shar p localize d non-blowu p criterio n fo r th e 3 D Eule r equations. Thi s i s one o f the important development s i n recent years. The y showe d that th e geometri c regularit y o f vorte x lines , eve n i n a n extremel y localize d re - gion containin g th e maximu m vorticity , ca n lea d t o depletio n o f nonlinea r vorte x RESEARCH ACCOMPLISHMENT S O F THOMA S YIZHA O HO U xl i stretching, thu s avoidin g finite tim e singularit y formatio n o f th e 3 D Eule r equa - tions. Specifically , assum e that a t each time t there exists some vortex line segment Lt o n whic h th e loca l maximu m vorticit y i s comparabl e t o th e globa l maximu m vorticity. Further , denot e L(t) a s the arclengt h o f L t an d K, a s the mea n curvatur e of L t. Deng-Hou-Y u showe d tha t i f (i ) th e velocit y field alon g L t i s bounde d b y a Cu(T - t)~ fo r som e a < 1 , (ii ) C L(T - tf < L(t) < C 0/maxLt(|«|, | V • £|), for som e (3 < 1 — a, the n th e solutio n o f the 3 D Eule r equation s remain s regula r up t o T. Kerr' s computation s violat e th e non-blowu p condition s o f Constantin - Fefferman-Majda theor y [3] , but fal l i n the critica l cas e o f the non-blowu p theor y of Deng-Hou-Yu [4 , 5], which corresponds to a = / ? = 1/2 . T o get a definite answer , one need s t o chec k whethe r th e scalin g constants , Cj/ , Co , an d CL i n condition s (i)-(ii) satisf y a n algebrai c inequality [5] . However , such scaling constants wer e not available i n [13].
2.3. Dynami c depletio n of vortex stretching. T o validate Deng-Hou-Yu' s non-blowup theory , Ho u an d L i [12 ] repeate d Kerr' s computation s usin g a well - resolved pseudo-spectra l metho d wit h resolutio n u p t o 153 6 x 102 4 x 3072 . Thi s is a very challengin g computation . Thei r numerica l result s demonstrate d tha t th e maximum vorticit y di d no t gro w faste r tha n doubl e exponentia l i n time , u p t o T = 19 , beyon d th e singularit y tim e predicte d b y Kerr' s computation s [13] . Th e velocity field was shown to be bounded throughout the computations. Wit h velocit y field bein g bounded , th e localize d non-blowu p criteri a o f Deng-Hou-Y u [4 ] can b e applied with a = 0 , (3 = 1/2 , whic h implies the non-blowup o f the 3D incompressible Euler equation s u p to T = 19 . T o gain a n insigh t t o the dynami c depletio n o f th e vortex stretchin g term , the y examine d th e degre e o f nonlinearit y i n th e vorte x stretching term . A n 0(T — t)~ l blowu p rat e i n th e maximu m vorticit y woul d imply that th e nonlinearity i n the vortex stretching term i s quadratic a s a functio n of the maximum vorticity . However , their numerica l results showed that th e vorte x stretching term , whe n projecte d t o th e uni t vorticit y vector , £ , i s bounde d b y cll^Hoo logdl^Hoo). Thi s implies that ther e i s tremendous dynami c depletion i n th e vortex stretchin g term . Wit h suc h uppe r boun d o n th e vorte x stretchin g term , one ca n easil y sho w tha t th e maximu m vorticit y canno t gro w faste r tha n doubl e exponential i n time. Thi s wa s indeed confirme d b y their numerica l results . The Deng-Hou-Y u localize d non-blowu p criteri a an d thei r numerica l valida - tion represen t a significan t progres s i n thi s field i n recen t years . I t make s peopl e re-evaluate th e whol e blowu p scenari o o f th e 3 D Eule r equation s an d pa y mor e attention t o the dynami c depletio n mechanism .
3. Singularit y formatio n i n 3 D vorte x sheet s The Kelvin-Helmholtz instabilit y i s a fundamental instabilit y o f incompressible fluid flow a t hig h Reynold s numbers . Th e idealizatio n o f a shea r layere d flow a s a vorte x shee t separatin g tw o region s o f potentia l flow ha s ofte n bee n use d a s a mode l t o stud y mixin g properties , boundar y layer s an d coheren t structure s o f fluids. Althoug h singularit y formatio n ha s bee n wel l studied fo r 2 D vortex sheet s using comple x analysis , the techniques use d fo r 2 D vortex sheets d o not generaliz e naturally t o 3 D vorte x sheets . A s a result , ther e ha s bee n no t muc h progres s i n studying singularit y formatio n o f 3 D vortex sheets . xlii S. OSHE R
In [10] , Hou, Hu , an d Zhan g studie d th e singularit y o f 3 D vortex sheet s usin g a ne w approach . First , the y derive d a leadin g orde r approximatio n t o the bound - ary integra l equatio n governin g th e 3 D vorte x sheet . Thi s leadin g orde r equatio n captures th e mos t singula r contributio n o f the integra l equation . Moreover , afte r applying a nonloca l integra l transformatio n t o th e physica l variables , the y foun d that this leading order 3D vortex sheet equation de-generates into a two-dimensional vortex shee t equatio n i n the directio n o f the tangential velocit y jump. Thi s rathe r surprising resul t showe d tha t th e drivin g mechanis m fo r singularit y formatio n o f 3D vortex sheets is the same a s that o f 2D vortex sheets. B y analyzing the reduce d leading order system, they showed that the singularity type o f the three-dimensiona l problem is the same as that o f the two-dimensional problem. Moreover , they derived a ne w 3 D vorte x shee t mode l whic h capture s th e leadin g orde r singula r behavio r of th e 3 D vorte x shee t equatio n an d ca n b e compute d efficientl y an d accuratel y using the Fast Fourie r Transform . Thei r extensiv e numerical studies confirme d th e analytic results , an d reveale d th e generi c for m o f the 3 D vortex shee t singularity . SI AM News featured a long article describing their results in March, 2002 . Th e article concludes that "Th e new results o f Hou and his colleagues have brought 'th e ever distan t goal ' o f understandin g turbulenc e an d hydrodynami c instabilit y a t least a fe w steps closer" . Dr . Hou' s Ph.D. students, Gan g Hu an d Xinwe i Y u were awarded th e SIA M Charle s DiPrim a Outstandin g Dissertatio n Priz e i n Applie d Mathematics i n 200 2 and 200 6 respectively .
4. Convergenc e o f the poin t vorte x metho d The idea o f representing an incompressible, invisci d flow by a collection o f mov- ing Lagrangia n poin t vortice s i s physicall y appealin g an d ha s bee n use d b y man y physicists. However , the particle velocity induced fro m othe r particles may becom e unbounded a s neighboring particle s approac h on e another. Fo r a lon g time, man y leading experts i n this field had widel y believed that suc h method wa s numericall y unstable, and the vortex blob method has been introduced to alleviate this difficulty . In hi s joint wor k wit h Goodma n an d Lowengrub , Dr . Ho u prove d a ver y sur - prising result [6 , 7] : the point vorte x method i s stable and convergen t wit h secon d order accuracy fo r two and three dimensional Euler equations. Th e key observation is that tw o neighboring particle s ar e i n fact separate d b y a distance o f order 0(h), where h is the mes h siz e o f the initia l grid . The y showe d that th e 0(h) separatio n is just enoug h t o obtai n stabilit y o f the poin t vorte x method , b y usin g a classica l result du e to C alder on-Zygmund. Anothe r important contributio n o f their analysi s was t o establis h a n asymptoti c erro r expansio n i n th e eve n power s o f h. Base d on these observations, they proved nonlinea r stabilit y an d convergenc e o f the poin t vortex method. Thei r convergence result has changed the landscape o f the field and has le d to a number o f subsequent theoretica l development s an d man y interestin g physical applications , especiall y fo r fluid interfac e problems .
5. Stabl e an d efficien t numerica l method s fo r interfacia l flow s Many physically interesting problems involv e propagation o f free surfaces . Wa - ter waves , boundarie s betwee n immiscibl e fluids, vorte x sheets , Hele-Sha w cells , thin-film growth , crysta l growt h an d solidificatio n ar e som e o f th e bette r know n examples. Numerica l simulation s fo r interfacia l flows pla y a n increasingl y impor - tant rol e i n understandin g th e comple x interfacia l dynamics , patter n formations , RESEARCH ACCOMPLISHMENT S O F THOMA S YIZHA O HO U xlii i and interfacia l instabilities . Du e to the underlyin g physica l instabilities , numerica l schemes are known to be very sensitive to numerical instabilities. Fo r many years, it was not clea r whether th e observed instabilitie s i n numerical calculations wer e du e to physica l o r numerica l instability . On e o f the mai n contribution s o f Dr . Hou' s work i s t o identif y th e sourc e o f numerica l instability , t o propos e a ne w clas s o f computational method s an d to analyz e the convergenc e o f these methods . In [1] , Beale Hou, and Lowengru b analyze d the stability o f a class o f boundary integral method s fo r interfacia l flow s wit h o r without surfac e tension . The y foun d that ther e i s a compatibilit y conditio n betwee n th e choic e o f quadrature rul e fo r the singula r velocit y integra l an d th e choic e o f a spatia l derivative . Man y exist - ing boundar y integra l method s violate d thi s compatibilit y condition , resultin g i n unstable discretizations . Thei r analysi s clarifie d som e confusio n i n th e literatur e regarding the stabilizing effect o f surface tension. Base d on these observations, the y proved the convergenc e o f a class o f spectrally accurat e boundary integra l method s for water waves, and applied these methods to study the stabilizing effec t o f surfac e tension fo r interfacia l flows. Dr. Hou' s anothe r contributio n fo r interfacia l flows i s to introduc e a clas s o f efficient an d stable boundary integral methods fo r interfacial flows with surface ten - sion. Du e t o th e hig h orde r regularizin g effec t an d th e Lagrangia n discretization , boundary integra l method s suffe r fro m a sever e time-ste p stabilit y constrain t fo r interfacial flows with surfac e tension . I n [8] , Hou, Lowengrub, and Shelle y success - fully remove d this stiffness constrain t b y using an efficien t implici t schem e based o n a reformulatio n o f the problem . Thi s reformulatio n use s the arclengt h metri c an d tangent angl e a s new dynamical variables . The y then develope d a n effectiv e Smal l Scale Decomposition techniqu e to extract th e leading order contribution o f the stif f terms. Thi s lead s to a n efficien t implici t discretizatio n a t th e sam e cos t a s a n ex - plicit method . Thi s metho d offer s a hug e facto r (thousand s o r more ) o f speed-u p over the conventiona l explici t methods . Man y application s whic h wer e previousl y unattainable no w becam e possibl e usin g thei r metho d an d ne w phenomen a wer e discovered. Thi s metho d ha s bee n widel y use d b y man y peopl e i n severa l disci - plines ranging fro m fluid dynamics , to materials science , chemistry, an d biology . I t has made a big impact i n these application areas . Usin g this method, the y foun d a new type o f topological singularitie s fo r interfacia l flows with surfac e tension , an d some surprisin g regularizin g effec t o f surfac e tension . Thei r 199 7 Physics o f Flui d paper [9 ] gav e a n in-dept h stud y o f this ne w type o f topological singularitie s an d was awarde d th e Frenkie l Priz e b y th e Flui d Dynamic s Divisio n o f the America n Physical Societ y i n 1998 .
References [1] J . T . Beale , T . Y . Hou , an d J . Lowengrub , Convergence of a Boundary Integral Method for Water Waves, SIA M J . Numer . Anal. , 3 3 (1996) , pp. 1797-1843 . [2] J . T . Beale , T . Kato , an d A . Majda , Remarks on the Breakdown of Smooth Solutions of the 3-D Euler Equations, Comm . Math . Phys. , 9 6 (1984) , pp. 61-66 . [3] P . Constantin , C . Fefferman , an d A . Majda , Geometric Constraints on Potentially Singular Solutions for the 3-D Euler Equation, Commun . PDEs , 2 1 (1996) , pp. 559-571 . [4] J . Deng , T . Y . Hou , an d X . Yu , Geometric Properties and the Non-Blowup of the Three- Dimensional Euler Equation, Commun . PDEs , 3 0 (2005) , pp. 225-243 . [5] J . Deng , T . Y . Hou , an d X . Yu , Improved Geometric Conditions for Non-blowup of the 3D Incompressible Euler Equation, Commun . PDEs , 3 1 (2006) , pp. 293-306 . xliv S. OSHE R
[6] J . Goodman, T . Y. Hou, an d J. Lowengrub, The Convergence of the Point Vortex Method for the 2D Euler Equations, Comm . Pur e an d Appl . Math. , 43 (1990), pp. 415-430 . [7] T . Y. Hou an d J . Lowengrub, The Convergence of a Point Vortex Method for the 3D Euler Equations, Comm . Pur e and Appl. Math. , 43 (1990), pp. 965-981 . [8] T . Y. Hou, J. Lowengrub, and M . Shelley , Removing the Stiffness from Interfacial Flows with Surface Tension, J . Comput. Phys , 114 (1994) , pp. 312-338 . [9] T . Y. Hou, J . Lowengrub , an d M. Shelley, The Long-Time Motion of Vortex Sheets with Surface Tension, Phys . o f Fluid, A , 9 (1997), pp. 1933-1954 . [10] T . Y. Hou, G. Hu, and P . Zhang, Singularity Formation in 3D Vortex Sheets. Phys . o f Fluids, 15 (2003) , pp. 147-172 . [11] T . Y. Hou and C.M . Li , Dynamic Stability of the 3D Axisymmetric Navier-Stokes Equations with Swirl, preprint , submitte d t o Comm. Pur e Appl . Math. , 2006 . [12] T . Y. Hou and R. Li, Dynamic depletion of vortex stretching and non-blowup of the 3D incompressible Euler equations, J . Nonlinear Science , 1 6 (2006), pp. 639-664 . [13] R . M . Kerr, Evidence for a Singularity of the Three Dimensional, Incompressible Euler Equations, 5 (1993), pp. 1725-1746 . [14] A . J. Majda an d A . L. Bertozzi, Vorticity an d Incompressibl e Flow , Cambridg e Univer - sity Press , 2002 .
DEPARTMENT O F MATHEMATICS, UCLA , Lo s ANGELES , C A 90095-1555 . E-mail addresss: [email protected] . The Researc h o f Jin-Yi Ca i
Andrew Chi-Chi h Ya o
Ladies an d Gentlemen , It i s my grea t pleasur e t o presen t th e wor k o f Professor Jin-Y i Cai , whic h ha s won hi m a prestigious Morningsid e Silve r Medal . Cai i s a distinguished leadin g researche r i n theoretica l compute r science , wh o has mad e significan t an d lastin g contribution s ove r a wid e spectru m o f topic s i n theory o f computation. I discuss below in some detail one o f his sensational accom - plishments, i n whic h h e resolve d tw o famou s conjecture s o f Juri s Hartmani s tha t had bee n ope n fo r ove r fifteen years . A "spars e set " i s a set wit h polynomiall y bounde d density . I n 197 8 Hartmanis conjectured, i n connection wit h th e P versu s NP question , that n o sparse set s ca n be P-complete unles s P equal s LOGSPACE. Thi s came to be known a s Hartmanis' Conjecture fo r P-hard sparse sets, and regarded as a fundamental questio n in theory of computation. I n 1995 Ogihara made a breakthrough on this seemingly intractable problem b y showin g tha t n o spars e set s ca n b e P-complet e unles s P i s containe d in squar e logspace . Buildin g o n Ogihara' s work , Ca i an d Sivakuma r obtaine d i n their 199 5 paper "Spars e hard set s fo r P" a complete resolutio n o f this conjecture . Later on , Ca i an d Sivakuma r resolve d a relate d conjectur e b y Hartmani s fo r NL - hard spars e sets . Th e proo f o f Ca i an d Sivakuma r wa s a n amazin g masterpiece , requiring several truly ingenious ideas beyond the innovation contained in Ogihara's work. Cai ha s mad e outstandin g contribution s o n man y othe r subjects , includin g boolean circuits , grap h isomorphism , average-cas e complexit y theory , an d th e re - lation betwee n unifor m an d non-unifor m complexit y classes . Fo r instance , i n 198 6 while still a graduate student , Ca i proved that subexponential-siz e bounded-dept h circuits cannot compute the parity function eve n approximately. Thi s has great im - plications on Turing machine complexity theory. I n 2001 Cai obtained the stronges t extension todate o f the classic Karp-Lipton theorem which relates uniform an d non- uniform complexit y classes . In summary, Jin-Y i Cai has been one of the deepest an d most productive schol- ars o f his generation i n the theory o f computation. Th e awarding o f a Morningsid e
xlv xlvi A . C.-C . YA O
Silver Meda l i s a fitting recognitio n o f his grea t scientifi c accomplishments .
CENTER FO R ADVANCE D STUDY , TSINGHU A UNIVERSITY , BEIJING 100084 , P.R . CHINA .
E-mail addresss: [email protected] m Ai-Ko Liu' s Wor k
Tian-Jun L i
Ai-Ko Li u work s o n 4-manifold s an d Mirro r symmetry . Hi s accomplishment s are mainl y i n th e followin g thre e areas : 1 . Seiberg-Witte n invariant s o f smoot h 4-manifolds; 2 . geometr y an d topolog y o f symplectic 4-manifolds ; 3 . enumeratio n of nodal curve s i n algebrai c surface s an d Calabi-Ya u 3-folds . The Seiberg-Wit t en invariant s ar e smoot h invariant s o f 4-manifolds . Fo r a smooth, closed, oriented 4-manifold M , onc e a spinc structure is chosen, its Seiberg- Witten invariant s ar e describe d b y on e Z—value d functio n SW o n H 2(M;Z) i f 6+(M) > 2 , an d b y tw o Z-value d functio n SW+ an d SW~ i f 6+(M ) = 1 . I n a join t wor k wit h T-J . Li , h e foun d th e s o calle d genera l wal l crossin g formul a which calculate s th e differenc e SW + — SW~ . Thi s i s a fundamenta l formul a i n the Seiberg-Witte n theory . I n addition , Li u systematicall y develope d th e famil y Seiberg-Witten theor y fo r a fibr e bundl e o f smoot h 4-manifolds . Especially , h e established severa l basi c formula s includin g the famil y wal l crossing formul a (wit h T-J. Li) , the famil y blo w up formula , an d th e famil y switchin g formula . A symplectic structure o n a smooth manifol d M i s a closed non-degenerat e 2 - form u. Th e most basi c invariant o f a symplectic manifol d (M , u) i s the symplecti c canonical class K^, whic h is an element in H2(M; Z) . Whe n combined with Taubes' symplectic Seiberg-Witte n theory , th e genera l wal l crossin g formul a ha s strikin g applications t o symplectic 4-manifolds . Th e mos t noteworth y i s Liu's proo f o f the Gompf conjecture , whic h say s tha t a minima l symplecti c 4-manifol d (M , u) wit h K% < 0 mus t b e rationa l o r ruled . Her e {M,UJ) i s sai d t o b e rationa l i f M i s diffeomorphic t o CP 2 o r S 2 x S2, an d ruled i f M i s diffeomorphic t o an S' 2—bundle. In addition , h e classifie d symplecti c 4-manifold s wit h metric s o f positiv e scala r curvature. I n join t work s wit h T-J . Li , h e prove d th e uniquenes s o f symplecti c structures u p t o deformatio n o n rationa l o r rule d manifolds , th e uniquenes s o f symplectic canonica l clas s o n manifold s wit h 6 + = 1 , an d determine d completel y the symplecti c con e when 6 + = 1 . Finally, Li u applie d th e famil y Seiberg-Witte n theor y t o th e enumeratio n o f nodal curves in algebraic surfaces an d Calabi-Ya u 3-fold s wit h spectacular success . In particular, he confirmed a beautiful conjectur e o f Gottsche expressing the number of nodal curve s i n a sufficientl y ampl e linea r syste m L o n a n algebrai c surfac e M in terms o f universal polynomial s i n 2 2 L , L-a(M), Cl (M) , c 2(M).
xlvii xlviii T.-J. L I
This formul a i s a non-linea r analogu e o f th e Riemann-Roc h formul a an d include s as a specia l cas e th e famou s Yau-Zaslo w conjectur e prove d b y Brya n an d Leung . Recently Li u als o confirme d th e Harvey-Moor e conjectur e fo r noda l curve s i n K3- fibred Calabi-Ya u 3-folds .
SCHOOL O F MATHEMATICS , UNIVERSITY OF MINNESOTA , MINNEAPOLIS , MN 55455 , E-mail addresss: [email protected] u The Wor k o f Xi-Ping Zh u
Huai-Dong Ca o
Xi-Ping Zhu' s research i n past a fe w years has been focuse d o n geometric flows and it s applications . I n particular , h e ha s mad e importan t contribution s t o th e study o f Ricc i flow o n Kahle r manifold s an d th e understandin g o f geometr y o f complete noncompac t Kahle r manifold s wit h positiv e holomorphi c bisectiona l cur - vature. Belo w I shall describ e som e o f his important work s o n the subject .
1. Geometr y o f complet e Kahle r manifold s wit h positiv e bisectiona l curvature
Joint wit h B.-L . Chen , Xi-Pin g Zh u foun d som e dee p hidde n connection s be - tween th e sig n o f (positive ) curvatur e an d th e volum e growt h rat e a s wel l a s cur - vature deca y rate. The y prove d
n THEOREM 1 (2002) . Any complete noncompact Kahler manifold X with pos- itive bisectional curvature has the following properties: (a) Volume growth of order at least n: V(xo,r) > Cr n for 1 < r < oo. (b) Average curvature (on geodesic balls) decays at least linearly:
/ R(x)dx < —— — V(x0,r)JB(xQir) ~ 1+r " Here, V(xo,r) denote s the volum e o f the geodesi c bal l B(xo,r) centere d a t XQ and o f radius r an d R denote s the scala r curvature .
2. Uniformizatio n o f complet e Kahle r surface s In 1980's , Yau mad e the followin g importan t conjecture : A complete noncom - pact Kahle r manifol d X n o f positiv e holomorphi c bisectiona l curvatur e i s biholo - morphic t o C n. Join t wit h Bing-Lon g Che n an d Siu-Hun g Tang , Zh u combine d
xlix 1 H.-D. CA O techniques fro m Ricc i flo w an d algebrai c geometr y t o prov e the followin g result :
2 THEOREM 2 (2002) . Let X be a complete noncompact Kahler surface with bounded and positive (or nonnegative) bisectional curvature. Assume X has Eu- clidean volume growth: V(xo,r) > Cr 4. Then X 2 is biholomorphic to C 2.
This improve s a previous resul t b y Mo k and give s a partial affirmativ e answe r to the abov e Yau's conjectur e i n comple x dimensio n two .
3. Shar p Dimensio n estimate s o f holomorphi c function s
To understand spac e o f holomorphic function s o n a complete Kahle r manifol d is an important problem . I n 2003 , i n a joint wor k with hi s three students, Xi-Pin g Zhu gav e a complet e resolutio n o f Yau' s conjectur e o n dimensio n estimat e o f th e space o f holomorphic functions . The y proved :
n THEOREM 3 (2003) . Let X be a complete noncompact Kahler manifold with nonnegative bisectional curvature. Let Pd(X n) denote the space of holomorphic functions of polynomial growth of order at most d on X n. Then, for all d > 0,
n dim(Pd(X )) < dim(P [d](C")) with equality holds for some positive integer d if and only if X = C n.
This improve s a previous resul t o f Le i Ni, who firs t prove d the abov e theore m under th e additiona l assumptio n tha t X n ha s Euclidea n volum e growth .
4. Th e Ricc i flo w o n compac t Kahle r manifold s wit h positiv e bisectional curvatur e Combining the non-collapsing result o f Perelman, the Li-Yau-Hamilton inequal - ity o f Ca o fo r Kahler-Ricc i flow , Xi-Pin g Zh u (togethe r wit h H.-D . Ca o an d B.-L . Chen) obtaine d th e followin g
THEOREM 4 (2003) . Any solution g{i) to the Kahler-Ricci flow on a compact Kahler manifold M n with positive bisectional curvature is necessarily nonsingular, i.e., the curvature of g(t) is uniformly bounded independent of t for all t > 0 . Moreover, there exists a subsequence {tj} such that g(tj) converges to a Kahler- Ricci soliton.
This answer s a n important questio n raise d b y Hamilton an d Ya u i n mid 80's .
DEPARTMENT O F MATHEMATIC S LEHIG H UNIVERSIT Y BETHLEHEM , P A 18015 , E-mail addresss: [email protected] Chern Priz e Recipient s
Chern Priz e
The Cher n Priz e i n mathematics wa s established i n 200 1 in honor o f Professo r Shiing-Shen Chern , on e o f the greates t geometer s an d Chines e mathematician s o f the twentieth century. Thes e awards are presented ever y three years to mathemati - cians o f Chinese descent wh o have made exceptional contributions to mathematica l research o r to publi c servic e activitie s i n support o f mathematics .
2004 Cher n Priz e Recipient s
Lo Yan g Professor L o Yan g i s awarde d th e 200 4 Cher n Priz e fo r hi s dedicatio n t o th e development o f mathematic s i n Chin a an d fo r hi s rol e i n promotin g mathemati - cal activitie s an d endeavor s ove r th e pas t twent y years . Professo r Yan g i s a dis - tinguished mathematicia n wh o ha s mad e severa l importan t contribution s t o th e modular an d angula r distributio n o f entire and meromorphic function s an d norma l families. Wit h Professo r Zhan g Guanghou , the y establishe d th e close d relatio n between the number o f deficient value s and the Borel directions o f entire and mero- morphic functions . H e als o settle d som e problem s pose d b y Davi d Drasi n an d W. K . Hayman ; i n collaboratio n wit h Professo r Hayman , the y solve d a conjec - ture b y J . E . Littlewood . Currently , Professo r Yan g i s the Deput y Directo r o f th e Morningside Cente r o f Mathematics . H e wa s a forme r presiden t o f th e Chines e Mathematical Society , a forme r directo r o f the Institut e o f Mathematics , an d th e founding directo r o f the Academy o f Mathematics an d System s Sciences . Professo r Yang i s a 196 2 graduate o f Pekin g Universit y an d complete d graduat e studie s a t the Institut e o f Mathematics. H e has bee n a membe r o f the Chines e Academ y o f Sciences sice 1980 . Hi s honors include the Hua Lookeng Award, the Ho Lin and H o Li Award , th e Ta n Ka h Ke e Prize , an d th e Nationa l Natura l Scienc e Foundatio n Award.
Fang Hu a Li n Professor Fan g Hu a Li n i s awarded th e 200 4 Cher n Priz e fo r hi s fundamenta l contributions t o th e theor y o f liqui d crystal , harmoni c maps , Ginzburg-Landa u
li Hi CHERN PRIZ E RECIPIENT S equations, the stati c a s well a s dynamic theor y o f topological defects , Skyrm e an d Faddeev models , an d th e Navier-Stoke s equations . Professo r Li n i s a pionee r i n the stud y o f liqui d crystal s an d publishe d a serie s o f papers tha t establishe d tha t the limitin g phenomeno n i n Ginzburg-Landa u equation s i s governe d b y a finite dimensional system associated to the BBH renormalization energy . Professo r Li n is the Silve r Professo r o f Mathematics a t Ne w Yor k University , wher e h e als o serve s on th e facult y o f th e Couran t Institut e o f Mathematica l Sciences . I n 2004 , h e was elected to the America n Academ y o f Arts an d Sciences . Hi s other award s an d honors include the Bocher Prize o f the American Mathematical Societ y in 2002, the Outstanding Researc h Award fro m NSF-China i n 1998 , and the Presidential Youn g Investigator Awar d an d a Sloa n Fellowshi p i n 1989 . Professo r Li n ha s hel d serva l visiting professorships, includin g the Ordwa y Chai r a t th e University o f Minnesot a in 199 9 and the Cheun g Kong Professorshi p a t Zhejian g Universit y i n 200 0 as well as at Fuda n Universit y i n 1999 . H e received hi s B.S. from Zhejian g Universit y an d his Ph.D. fro m th e Universit y o f Minnesota . ICCM Internationa l Cooperatio n Awar d Recipien t
ICCM Internationa l Cooperatio n Awar d
The ICCM International Cooperatio n Awar d i s presented to an individual wh o has promoted th e developmen t o f mathematics i n China , Hon g Kong , an d Taiwa n through collaboration , teaching , and support o f Chinese mathematicians. Th e firs t award i s presented a t the Third International Congres s o f Chinese Mathematicians .
Recipient
John Coate s Professor Joh n Coate s i s awarde d th e firs t ICC M Internationa l Cooperatio n Award. Fo r th e pas t twent y years , h e ha s generousl y give n hi s knowledg e an d time t o Chines e mathematician s whic h ha s enable d the m t o mak e grea t stride s in al l area s o f mathematics . A s th e Sadleiria n Professo r o f Pure Mathematic s a t Cambridge University , Professo r Coate s ha s traine d man y Chines e student s an d post-doctoral fellows , an d ha s bee n a tremendou s influenc e o n thei r growt h a s a mathematician. H e has been a member o f the Monrningside Meda l o f Mathematics Selection Committe e sinc e 1998 , an d wa s th e forme r Chairma n o f th e Oversea s Committee fo r Th e Institut e o f Mathematical Science s a t Th e Chines e Universit y of Hon g Kong . Professo r Coate s als o organize d severa l importan t workshop s i n Beijing and Hangzhou, where he devoted himself to helping Chinese mathematicians understand moder n mathematics . I n 2004 , he wa s made a n Honorary Professo r a t Zhejiang Universit y an d th e Universit y o f Scienc e and Technolog y o f China .
liii This page intentionally left blank Photographs
The Thir d Internationa l Congres s o f Chines e Mathematicians, 200 4
Opening Ceremon y a t th e Hon g Kong Conventio n an d Exhibitio n Centr e
lv PHOTOGRAPHS
On th e stag e (fro m lef t t o right) : Lesli e G . Valiant , Mrs . Ma y Chu (th e daughte r o f Professo r Shiing-She n Chern) , Lawrenc e J . Lau (Vice-Chancellor , CUHK) , Yongxian g Lu , Shing-Tun g Yau , Ronnie Cha n an d Joh n Coate s
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Ronnie C . Chan, Co-Founde r o f THE MORNINGSID E GROUP, an d Chairma n o f the Hang Lung Grou p PHOTOGRAPHS lvii
Shing-Tung Yau , Chairma n o f the Internationa l Congres s o f Chi - nese Mathematician s
Yongxiang Lu , Vice-Chairma n o f the Standin g Committe e o f th e 10th Nationa l People' s Congress , People's Republi c o f Chin a lviii PHOTOGRAPHS
Mr. Yongxian g L u presentin g th e Morningsid e Gol d Meda l o f Mathematics t o Kefun g Li u
Mr. Yongxian g L u presentin g th e Morningsid e Gol d Meda l o f Mathematics t o Zhoupin g Xi n PHOTOGRAPHS lix
Prof. Shing-Tun g Ya u presenting th e Morningside Gol d Meda l o f Applied Mathematic s to Thomas Yizha o Hou
Prof. Shing-Tun g Ya u presenting th e Morningside Gol d Meda l o f Applied Mathematic s to Zhiliang Yin g PHOTOGRAPHS
Prof. Lesli e G . Valiant , selectio n committe e o f the Morningsid e Medal o f Mathematics, presentin g the Silver Meda l to Jin-Yi Cai
Prof. Joh n Coates , selectio n committee o f the Morningside Meda l of Mathematics, presentin g the Silver Meda l to Ai-Ko Liu PHOTOGRAPHS lxi
Prof. Lawrenc e J . Lau , th e Vice-Chancello r o f Th e Chines e Uni - versity o f Hong Kong, presentin g the Silve r Medal to Xi-Ping Zh u
INM ^
2004 Chern Priz e recipients, Fan g Hua Li n (left ) an d L o Yang lxii PHOTOGRAPHS
Mr. Ronni e Chan presenting the ICCM International Cooperatio n Award t o Joh n Coate s
ft y\* t, % The Morningsid e Award s Recipient s PHOTOGRAPHS lxiii Prof. Ju n Li presenting the research o f Kefeng Liu !%iim^ici' •••. Prof. Stanle y Oshe r presentin g the research o f Thomas Yizha o Hou PHOTOGRAPHS Prof. Jia-A n Ya n presenting the research o f Zhiliang Yin g Prof. Andre w Ya o presenting the research o f Jin-Yi Cai PHOTOGRAPHS Prof. Tian-Ju n L i presenting th e research o f Ai-Ko Liu ;;.'- v<„p, - " \ ,- s •W Prof. Huai-Don g Ca o presenting the research o f Xi-Ping Zhu This page intentionally left blank List o f Speaker s Morningside Lecture s Shock Wave s an d Cosmolog y JOEL A . SMOLLE R (CONJOIN T WIT H BLAK E TEMPLE ) Plenary Lecture s Variational Constructio n o f Diffusio n Orbit s i n Conve x Hamiltonia n Systems wit h Multipl e Degree s o f Freedo m CHONG-QING CHEN G (CONJOIN T WIT H JU N YAN ) Saddlepoint Approximation s an d Boundar y Crossin g Probabilitie s fo r Random Field s an d Thei r Application s TZE LEUN G LA I Recognizing Certai n Rationa l Homogeneou s Manifold s o f Picard Numbe r 1 from Thei r Varietie s o f Minimal Rationa l Tangent s NGAIMING MO K Discontinuous Galerki n Method s fo r Convectio n Dominate d Partia l Differential Equation s Cm-WANG SH U Singularity Behavio r o f the Mea n Curvatur e Flo w XU-JIA WAN G Localization an d Dualit y JIAN ZHO U Surgical Ricc i Flo w o n Four-Manifold s wit h Positiv e Isotropi c Curvatur e XI-PING ZH U (CONJOIN T WIT H BING-LON G CHEN ) Special Subvarietie s o f A g KANG ZU O (CONJOIN T WIT H ECKAR T VIEHWEG ) 45-Minute Invite d Talk s Subelliptic PDE's an d SubRiemannia n Geometr y DER-CHEN CHAN G (CONJOIN T WIT H PETE R GREINER ) lxvii lxviii LIS T O F SPEAKER S Set Additio n an d Se t Multiplicatio n MEI-CHU CHAN G The Q-Curvatur e Flo w o n a Close d 3-Manifol d o f Positive Q-Curvatur e SHU-CHENG CHAN G Normal Dilation s MAN-DUEN CHO I Dirichlet Form s an d Marko v Semigroup s o n Non-Associative Vecto r Bundle s CHO-HO CH U (CONJOIN T WIT H ZHONGMI N QIAN ) Intelligent an d Informativ e Scientifi c Computing , Trend s an d Example s QIANG D u Decomposition Principl e an d Rando m Cascade s Ai HU A FA N (CONJOIN T WIT H JEA N PIERR E KAHANE ) Number-Theoretic Method s i n Experimental Design s KAI-TAI FAN G (CONJOIN T WIT H YUA N WANG ) Local Monodrom y o f the Kloosterma n Shea f a t o o LEI F U (CONJOIN T WIT H DAQIN G WAN ) On the Generato r Proble m o f von Neumann Algebra s LIMING G E (CONJOIN T WIT H JUNHA O SHEN ) Collineation Group s o f Translation Plane s CHAT YI N H O Localized Non-Blowu p Condition s fo r the 3 D Incompressibl e Euler Equation s THOMAS YIZHA O HO U (CONJOIN T WIT H JIA N DEN G AN D XINWE I YU ) Geometric Invarian t Theor y an d Birationa l Geometr y Yi H u Large Scal e Geometry, Compactification s an d th e Integral Noviko v Conjectures fo r Arithmeti c Group s LIZHEN J i Holomorphic Motion s an d Norma l Form s i n Comple x Analysi s YUNPING JIAN G A Surve y o f Results o n the Groun d Stat e o f Semilinear Ellipti c Equation s MAN KA M KWON G Separating an d Extendin g Subgroup s o f a Locall y Compac t Grou p ANTHONY TO-MIN G LA U (CONJOIN T WIT H EBERHAR D KANIUTH ) On Pseudo-Hermitia n C R Manifold s SONG-YING L i The Spac e o f Symplectic Structure s o n Close d 4-Manifold s TIAN-JUN L i An Introductio n t o Chira l Equivariant Cohomolog y BONG H . LIA N LIST O F SPEAKER S Separation o f Bound Stat e Solution s o f Systems o f Nonlinea r Schrodinger Equation s TAI-CHIA LI N (CONJOIN T WIT H YU-WE N HSU ) On Stron g Near-epoc h Dependenc e ZHENGYAN LI N Ancient Solution s to Kahler-Ricc i Flo w LEI N I Recent Progres s o n the Drichle t Proble m i n Lipschitz Domain s ZHONGWEI SHE N Scattered Dat a Interpolatio n b y Bo x Spline s ZUOWEI SHE N (CONJOIN T WIT H SHAYN E WALDRON ) Multifractal Analysi s o f Branching Measur e o n a Galton-Watson Tre e NARN-RUEIH SHIE H (CONJOIN T WIT H PETE R MORTERS ) Mathematics, Mathematic s Educatio n an d th e Mous e Siu MA N KEUN G Remarks o n Gieseker' s Degeneratio n an d it s Normalizatio n XIAOTAO SU N A Convergenc e Resul t o f the Lagrangian Mea n Curvatur e Flo w MU-TAO WAN G On Piecewis e Algebrai c Variet y REN-HONG WAN G Refinable Function s wit h Non-Intege r Dilation s YANG WAN G (CONJOIN T WIT H XIN-RON G DA I AN D DE-JU N FENG ) Some Results o n Smale' s Mea n Valu e Conjectur e YUEFEI WAN G On Whitney's Critica l Set s ZHI-YING WE N (CONJOIN T WIT H LI-FEN G XI ) Bundle Rigidit y o f Complex Surface s BUN WON G (CONJOIN T WIT H WING-SU M CHEUNG ) The Triangl e o f Operators, Topologies , Bornologie s NGAI-CHING WON G Applications o f Nevanlinna Theor y t o Geometri c Problem s PIT-MANN WON G Piecewise Functio n Generate d b y the Solution s o f Linear Ordinar y Differential Equatio n ZONGMIN W U Ear Modelin g an d Soun d Signa l Processin g JACK XI N Convex Dualit y Theor y fo r Optima l Investmen t JIA-AN YA N (CONJOIN T WIT H JIANMIN G XIA ) lxx LIS T O F SPEAKER S Stability o f Basic Wave Patterns fo r Ga s Motion s TONG YAN G (CONJOIN T WIT H HUI-JIAN G ZHAO ) Hilbert Modula r Function s an d Thei r CM Value s TONGHAI YAN G CR Equivalenc e Proble m o f Strongly Pseudoconve x C R Manifold s STEPHEN S.-T . YA U Backward Stochasti c Volterr a Integra l Equation s JlONGMIN YON G Step-Sizes fo r the Gradien t Metho d YA-XIANG YUA N The Mathematical Proble m o f Inertial Wave s in Rapidl y Rotating Planet s and Star s KEKE ZHANG M (CONJOIN T WIT H XINHA O LIAO ) The C a Regularit y o f a Clas s o f Ultraparabolic Equation s ZHANG LIQU N The Positiv e Mas s Theore m Nea r Nul l Infinit y XIAO ZHAN G Vector Bundle s o n Non-Primary Hop f Manifold s wit h Abelia n Fundamental Grou p XIANGYU ZHO U (CONJOIN T WIT H WEIMIN G LIU )