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Fifth International Congress of Chinese Mathematicians Part 1
AMS/IP Studies in Advanced Mathematics S.-T. Yau, Series Editor Fifth International Congress of Chinese Mathematicians Part 1 Lizhen Ji Yat Sun Poon Lo Yang Shing-Tung Yau Editors American Mathematical Society • International Press Fifth International Congress of Chinese Mathematicians https://doi.org/10.1090/amsip/051.1 AMS/IP Studies in Advanced Mathematics Volume 51, Part 1 Fifth International Congress of Chinese Mathematicians Lizhen Ji Yat Sun Poon Lo Yang Shing-Tung Yau Editors American Mathematical Society • International Press Shing-Tung Yau, General Editor 2000 Mathematics Subject Classification. Primary 05–XX, 08–XX, 11–XX, 14–XX, 22–XX, 35–XX, 37–XX, 53–XX, 58–XX, 62–XX, 65–XX, 20–XX, 30–XX, 80–XX, 83–XX, 90–XX. All photographs courtesy of International Press. Library of Congress Cataloging-in-Publication Data International Congress of Chinese Mathematicians (5th : 2010 : Beijing, China) p. cm. (AMS/IP studies in advanced mathematics ; v. 51) Includes bibliographical references. ISBN 978-0-8218-7555-1 (set : alk. paper)—ISBN 978-0-8218-7586-5 (pt. 1 : alk. paper)— ISBN 978-0-8218-7587-2 (pt. 2 : alk. paper) 1. Mathematics—Congresses. I. Ji, Lizhen, 1964– II. Title. III. Title: 5th International Congress of Chinese Mathematicians. QA1.I746 2010 510—dc23 2011048032 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. -
Lie Groups and Automorphic Forms This Page Intentionally Left Blank AMS/IP Studies in Advanced Mathematics
Lie Groups and Automorphic Forms This page intentionally left blank https://doi.org/10.1090/amsip/037 AMS/IP Studies in Advanced Mathematics Volume 37 Lie Groups and Automorphic Forms Proceedings of the 2003 Summer Program Zhejiang University Center of Mathematical Sciences Hangzhou, China Lizhen Ji, Managing Editor Jian-Shu Li, H. W. Xu, and Shing-Tung Yau, Editors American Mathematical Society • International Press Shing-Tung Yau, General Editor 2000 Mathematics Subject Classification. Primary 20Gxx, 22Exx, 11-XX, 55Nxx. The paper, "On the cohomology of locally symmetric spaces and of their compactifi- cations," by Leslie Saper, starting on page 169, is used by permission. © 2002, Current Developments in Mathematics, International Press, Somerville, MA. The photograph of Armand Borel on page v is courtesy of Armand Borel's daughter, Dominique Borel. Library of Congress Cataloging-in-Publication Data Lie groups and automorphic forms / [Lizhen Ji, editor]. p. cm. — (Studies in advanced mathematics, ISSN 1089-3288 ; v. 37) Includes bibliographical references. ISBN-13: 978-0-8218-4198-3 (alk. paper) ISBN-10: 0-8218-4198-X (alk. paper) 1. Lie groups. 2. Automorphic forms. I. Ji, Lizhen, 1964- QA387.L52 2006 512'.482—dc22 2006048421 Copying and reprinting. Material in this book may be reproduced by any means for edu• cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg• ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. -
A Brief Overview of the Work of Shing-Tung Yau
A brief overview of the work of Shing-Tung Yau Lizhen Ji Dept of Math University of Michigan Ann Arbor, MI 48109 May 19, 2010 Abstract The purpose of this article is to give a brief overview of the work of Yau by compiling lists of his papers and books together with brief comments on some part of his work, and describing a sample of some open problems raised by him. It tries to complement other more detailed commentaries on particular aspects of his work by experts in this volume. We hope that such an overview will convey both the depth and width of his work. Contents 1 Introduction 2 2 A summary of some major works of Yau 5 3 Topics Yau has worked on 6 4 Basics on K¨ahler-Einsteinmetrics and Calabi conjectures 10 5 Some applications of K¨ahler-Einsteinmetrics and Calabi-Yau manifolds 11 6 Harmonic maps 13 7 Rigidity of K¨ahlermanifolds 15 8 Super-rigidity of spaces of nonpositive curvature 20 9 Survey papers by Yau 21 10 Open problems by Yau 23 11 Books written and co-written by Yau 27 12 Books edited and co-edited by Yau 30 13 Ph.D. students of Yau 40 1 14 Partial list of papers and books of Yau 43 15 Papers and books by others 67 1 Introduction Shing-Tung Yau has revolutionized the broad field of geometric analysis by combining partial differ- ential equations with differential geometry and applying geometric analysis to algebraic geometry. Besides his celebrated solution to the Calabi conjecture on K¨ahler-Einsteinmetrics and applications of the Calabi-Yau manifolds to the string theory and the mirror symmetry, he has also initiated highly original approaches to harmonic analysis on manifolds by using gradient estimates and Har- nack inequalities, topology of manifolds using minimal surfaces, rigidity of complex manifolds and algebraic varieties using harmonic maps, and the positive mass conjecture in general relativity us- ing minimal surfaces. -
Locally Symmetric Spaces
Locally symmetric spaces J¨urgen Rohlfs (Katholische Universit¨at Eichst¨att–Ingolstadt), Leslie Saper (Duke University), Birgit Speh (Cornell University) May 18 - 23, 2008 1 The setting A locally symmetric space is the quotient Γ\X of a globally symmetric space X = G/K , where G is a non compact reductive Lie Group with maximal compact subgroup K and where Γ is a discrete subgroup of G . Locally symmetric spaces are important in geometry, analysis, and as well as in number theory. We considered at the conference both arithmetic and non-arithmetic subgroups Γ as well as compact and non-compact locally symmetric spaces Γ\X . The topics covered at the conference belong to - compactifications, - differential geometry and topology, and - arithmetic. Techniques and results from all these areas were represented. 2 Background material Several developments in different fields over the past few years made this an opportune time for this work- shop. 2.1 Compactifications When X/Γ is non-compact, various compactifications (often singular) have been introduced to address dif- ferent problems in geometry and number theory. The construction and relationship between various compac- tifications has become more clear through the work of several mathematicians, particularly Borel, Goresky, Ji, and Zucker. Various types of cohomology associated to these compactifications are important in number theory. The relation between the L2-cohomology and the intersection cohomology of the Baily-Borel compactification is important for Langlands’s program and was resolved by Looijenga and Saper-Stern. The further relationship between this and the weighted cohomology of the (less singular) reductive Borel-Serre compactification was settled by Goresky, Harder, and MacPherson. -
Advanced Lectures in Mathematics
Advanced Lectures in Mathematics (ALM) ALM 1: Superstring Theory ALM 2: Asymptotic Theory in Probability and Statistics with Applications ALM 3: Computational Conformal Geometry ALM 4: Variational Principles for Discrete Surfaces ALM 6: Geometry, Analysis and Topology of Discrete Groups ALM 7: Handbook of Geometric Analysis, No. 1 ALM 8: Recent Developments in Algebra and Related Areas ALM 9: Automorphic Forms and the Langlands Program ALM 10: Trends in Partial Differential Equations ALM 11: Recent Advances in Geometric Analysis ALM 12: Cohomology of Groups and Algebraic K-theory ALM 13: Handbook of Geometric Analysis, No. 2 ALM 14: Handbook of Geometric Analysis, No. 3 ALM 15: An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices ALM 16: Transformation Groups and Moduli Spaces of Curves ALM 17: Geometry and Analysis, No. 1 ALM 18: Geometry and Analysis, No. 2 ALM 19: Arithmetic Geometry and Automorphic Forms ALM 20: Surveys in Geometric Analysis and Relativity ALM 21: Advances in Geometric Analysis ALM 22: Differential Geometry: Under the Influence of S.-S. Chern ALM 23: Recent Developments in Geometry and Analysis ALM 24: Handbook of Moduli, Volume I ALM 25: Handbook of Moduli, Volume II ALM 26: Handbook of Moduli, Volume III ALM 27: Number Theory and Related Areas ALM 28: Selected Expository Works of Shing-Tung Yau with Commentary, Volume I ALM 29: Selected Expository Works of Shing-Tung Yau with Commentary, Volume II ALM 30: Automorphic Forms and L-functions ALM 31: Handbook of Group Actions, -
An Introduction to Arithmetic Groups
An introduction to arithmetic groups Lizhen Ji CMS, Zhejiang University Hangzhou 310027, China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109 June 27, 2006 Plan. 1. Examples of arithmetic groups 2. Definition of arithmetic groups 3. Basic properties of arithmetic groups 4. Discrete (lattice) subgroups of Lie groups 5. Rigidities of lattices and locally symmetric spaces 6. Compactifications of locally symmetric spaces 7. Spectral theory of automorphic forms 8. Large scale geometry of discrete groups 9. Cohomology of arithmetic groups The most basic example of arithmetic groups is Z, the ring of integers. (Recall that by arithmetic, one means the study of integers). So it is natural that Z is an arithmetic group. When we consider Z as an arithmetic group, we consider it as a subgroup of R. Z ⊂ R. This embedding is important since we can for the quotient Z\R. The Poisson summation for- mula relates the spectral theory and the lengths of closed geodesics. It is fundamental in many applications. Z is an infinite cyclic group. A natural gen- n eralization is Z , the free abelian group on n- generators. n The Poisson formula for Z and other lattices n in R is crucial to recent work on lattice sphere packing. A non-abelian generalization SL(2, Z) ! a b SL(2, ) = { | a, b, c, d ∈ , ad − bc = 1}. Z c d Z This is often called the modular group. It is an arithmetic subgroup of the Lie group SL(2, R). A further generalization is SL(n, Z) ⊂ SL(n, R), n ≥ 2. -
Advanced Lectures in Mathematics (ALM)
Advanced Lectures in Mathematics (ALM) ALM 1: Superstring Theory ALM 2: Asymptotic Theory in Probability and Statistics with Applications ALM 3: Computational Conformal Geometry ALM 4: Variational Principles for Discrete Surfaces ALM 6: Geometry, Analysis and Topology of Discrete Groups ALM 7: Handbook of Geometric Analysis, No. 1 ALM 8: Recent Developments in Algebra and Related Areas ALM 9: Automorphic Forms and the Langlands Program ALM 10: Trends in Partial Differential Equations ALM 11: Recent Advances in Geometric Analysis ALM 12: Cohomology of Groups and Algebraic K-theory ALM 13: Handbook of Geometric Analysis, No. 2 ALM 14: Handbook of Geometric Analysis, No. 3 ALM 15: An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices ALM 16: Transformation Groups and Moduli Spaces of Curves ALM 17: Geometry and Analysis, No. 1 ALM 18: Geometry and Analysis, No. 2 ALM 19: Arithmetic Geometry and Automorphic Forms ALM 20: Surveys in Geometric Analysis and Relativity ALM 21: Advances in Geometric Analysis ALM 22: Differential Geometry: Under the Influence of S.-S. Chern ALM 23: Recent Developments in Geometry and Analysis Advanced Lectures in Mathematics Volume XXIII Recent Developments in Geometry and Analysis edited by Yuxin Dong · Jixiang Fu · Guozhen Lu Weimin Sheng · Xiaohua Zhu International Press 浧䷘㟨十⒉䓗䯍 www.intlpress.com HIGHER EDUCATION PRESS Advanced Lectures in Mathematics, Volume XXIII Recent Developments in Geometry and Analysis Volume Editors: Yuxin Dong (Fudan University) Jixiang Fu (Fudan University) Guozhen Lu (Wayne State University) Weimin Sheng (Zhejiang University) Xiaohua Zhu (Peking University) Copyright © 2012 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. -
Advanced Lectures in Mathematics (ALM)
Advanced Lectures in Mathematics (ALM) ALM 1: Superstring Theory ALM 2: Asymptotic Theory in Probability and Statistics with Applications ALM 3: Computational Conformal Geometry ALM 4: Variational Principles for Discrete Surfaces ALM 6: Geometry, Analysis and Topology of Discrete Groups ALM 7: Handbook of Geometric Analysis, No. 1 ALM 8: Recent Developments in Algebra and Related Areas ALM 9: Automorphic Forms and the Langlands Program ALM 10: Trends in Partial Differential Equations ALM 11: Recent Advances in Geometric Analysis ALM 12: Cohomology of Groups and Algebraic K-theory ALM 13: Handbook of Geometric Analysis, No. 2 ALM 14: Handbook of Geometric Analysis, No. 3 ALM 15: An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices ALM 16: Transformation Groups and Moduli Spaces of Curves ALM 17: Geometry and Analysis, No. 1 ALM 18: Geometry and Analysis, No. 2 ALM 19: Arithmetic Geometry and Automorphic Forms ALM 20: Surveys in Geometric Analysis and Relativity Advanced Lectures in Mathematics Volume IX Automorphic Forms and the Langlands Program edited by Lizhen Ji, Kefeng Liu, Shing-Tung Yau, and Zhu-Jun Zheng International Press 浧䷘㟨十⒉䓗䯍 www.intlpress.com HIGHER EDUCATION PRESS Advanced Lectures in Mathematics, Volume IX Automorphic Forms and the Langlands Program Volume Editors: Lizhen Ji (University of Michigan) Kefeng Liu (University of California at Los Angeles) Shing-Tung Yau (Harvard University) Zhu-Jun Zheng (Henan University) Copyright © 2010 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is published and sold in China exclusively by Higher Education Press of China. -
Honoring the Class of 2020
Honoring the Class of 2020 UNIVERSITY OF MICHIGAN Winter 2020 Graduates Honoring the Class of 2020 WINTER GRADUATION UNIVERSITY OF MICHIGAN 2020 This book includes a list of the candidates for degrees to be granted upon completion of formal requirements. Candidates for graduate degrees are recommended jointly by the Executive Board of the Horace H. Rackham School of Graduate Studies and the faculty of the school or college awarding the degree. Following the School of Graduate Studies, schools are listed in order of their founding. Candidates within those schools are listed by degree then by specialization, if applicable. Horace H. Rackham School of Graduate Studies .....................................................................................................15 College of Literature, Science, and the Arts ..............................................................................................................19 Medical School .........................................................................................................................................................25 Law School ..............................................................................................................................................................25 School of Dentistry ..................................................................................................................................................25 College of Pharmacy ................................................................................................................................................26 -
Fourth International Congress of Chinese Mathematicians
AMS/IP Studies in Advanced Mathematics S.-T. Yau, Series Editor Fourth International Congress of Chinese Mathematicians Lizhen Ji Kefeng Liu Lo Yang Shing-Tung Yau Editors American Mathematical Society • International Press amsip-48-ji2-cov.indd 1 7/6/10 8:56 AM Fourth International Congress of Chinese Mathematicians https://doi.org/10.1090/amsip/048 AMS/IP Studies in Advanced Mathematics Volume 48 Fourth International Congress of Chinese Mathematicians Lizhen Ji Kefeng Liu Lo Yang Shing-Tung Yau Editors American Mathematical Society • International Press Shing-Tung Yau, General Editor 2000 Mathematics Subject Classification. Primary 00Bxx, 05–XX, 08–XX, 11–XX, 14–XX, 22–XX, 35–XX, 37–XX, 80–XX. For additional information and updates on this book, visit www.ams.org/bookpages/amsip-48 Library of Congress Cataloging-in-Publication Data International Congress of Chinese Mathematicians (4th : 2007: Hangzhou, China) p. cm. (AMS/IP studies in advanced mathematics ; v. 48) Includes bibliographical references. ISBN 978-0-8218-5021-3 (alk. paper) 1. Mathematics—Congresses. I. Ji, Lizhen, 1964– II. Title. QA1.I746 2007 510—dc22 2010019768 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society and International Press. -
List of All Ph.D. Students and Post-Doctors of S.T. Yau
List of all Ph.D. students and post-doctors of S.T. Yau Part I. Ph.D. students: 1. John Nash, Stanford University, 1976, 2. Richard Schoen, Stanford University, 1977, [email protected], currently at Stanford University. 3. Thomas Parker, Stanford University, 1980, [email protected], cur- rently at Michigan State University. 4. Andrejs Treibergs, Stanford University, 1980, [email protected], currently at University of Utah. 5. Richard Klotz, Stanford University, 1982, 6. Shigetoshi Bando, Princeton Univ, 1983, [email protected], currently at Tohoku University, Japan. 7. Robert Bartnik, Princeton University, 1983, [email protected], currently at Monash University, Australia. 8. Leslie Saper, Princeton University, 1984, [email protected], cur- rently at Duke University. 9. Mark Stern, Princeton Univ, 1984, [email protected], currently at Duke University. 10. Huai-Dong Cao, Princeton University, 1986, [email protected], cur- rently at Lehigh University. 11. Bennett Chow, Princeton University, 1986, [email protected], currently at UC San Diego. 12. Alex Freire, Princeton University, 1988, [email protected], currently at University of Tennessee. 13. Gang Tian, Harvard University, 1988, [email protected], cur- rently at Princeton University. 14. Jun Li, Harvard University, 1989, [email protected], currently at Standford University. 1 15. Tzuemn-Renn Lin, University of California, San Diego, 1989. 16. Wen-Xiang Wang, Princeton University, 1989, [email protected], cur- rently at California State University, San Bernardino. 17. Fangyang Zheng, Harvard University, 1989, [email protected] currently at Ohio State University. 18. Rui-Tao Dong, University of California, San Diego, 1990, [email protected] 19. -
Cohomology of Groups and Algebraic K-Theory
Advanced Lectures in Mathematics Volume XII Cohomology of Groups and Algebraic K-theory Editors: Lizhen Ji, Kefeng Liu, and Shing-Tung Yau International Press 浧䷘㟨十⒉䓗䯍 www.intlpress.com HIGHER EDUCATION PRESS Lizhen Ji Kefeng Liu University of Michigan University of California at Los Angeles Shing-Tung Yau Harvard University Copyright © 2010 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is published and sold in China exclusively by Higher Education Press of China. No part of this work can be reproduced in any form, electronic or mechanical, recording, or by any information storage and data retrieval system, without prior approval from International Press. Requests for reproduction for scientific and/or educational purposes will normally be granted free of charge. In those cases where the author has retained copyright, requests for permission to use or reproduce any material should be addressed directly to the author. ISBN 978-1-57146-144-5 Typeset using the LaTeX system. Printed in the USA on acid-free paper. ADVANCED LECTURES IN MATHEMATICS Executive Editors Shing-Tung Yau Kefeng Liu Harvard University University of California at Los Angeles Zhejiang University Lizhen Ji Hangzhou, China University of Michigan, Ann Arbor Editorial Board Chongqing Cheng Tatsien Li Nanjing University Fudan University Nanjing, China Shanghai, China Zhong-Ci Shi Zhiying Wen Institute of Computational Mathematics Tsinghua University Chinese Academy of Sciences (CAS) Beijing, China Beijing, China Lo Yang Zhouping Xin Institute of Mathematics The Chinese University of Hong Kong Chinese Academy of Sciences (CAS) Hong Kong, China Beijing, China Weiping Zhang Xiangyu Zhou Nankai University Institute of Mathematics Tianjin, China Chinese Academy of Sciences (CAS) Beijing, China Xiping Zhu Zhongshan University Guangzhou, China Preface Cohomology of groups is a fundamental tool in many subjects in modern mathematics.