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A Brief Summary of My Researches
A Brief Summary of My Researches Kefeng Liu My research interest is to solve problems from mathematics and physics with simple and clean ideas, and along the way to learn and to ¯nd new math- ematical structures and theories. Localization techniques in various settings, combining with various subjects of mathematics, have played central roles in my researches. See my ICM lecture [38] for more detailed discussions. I divide my past researches into the following several parts: (1). The applications of modular invariance, the in¯nite dimensional sym- metry in Kac-Moody algebras and mathematical physics, to geometry and topology. From modular invariance we derived several quite general rigid- ity, divisibility and vanishing results about certain topological invariants of a manifold. (2). The use of the explicit expressions of heat kernels on Lie groups to study the geometry and topology of moduli spaces of flat connections on Riemann surfaces, this proves certain conjectural formulas of Witten and gives several new results including general vanishing formulas of intersection numbers. (3). The understanding of mirror symmetry and counting curves on pro- jective manifolds. My joint work with Lian and Yau solves some very general conjectures about the relationship between the numbers of algebraic curves in projective manifolds and hypergeometric series. (4). The proof of the Marino-Vafa conjecture on Hodge integrals on mod- uli spaces of Deligne-Mumford curves, which expresses the generating series of three term Hodge integrals over all genera and any number of marked points in terms of a ¯nite expression of representation of symmetric groups, or Chern-Simons knot invariants. -
VITA Sheldon Katz Urbana, IL 61801 Phone
VITA Sheldon Katz Urbana, IL 61801 Phone: (217) 265-6258 Education S.B. 1976 Massachusetts Institute of Technology (Mathematics) Ph.D. 1980 Princeton University (Mathematics) Professional Experience University of Illinois Professor 2001{ Dean's Special Advisor, College of LAS 2018{ Interim Chair, Department of Mathematics Fall 2017 Chair, Department of Mathematics 2006{2011 Oklahoma State University Regents Professor 1999{2002 Southwestern Bell Professor 1997{1999 Professor 1994{2002 Associate Professor 1989{1994 Assistant Professor 1987{1989 University of Oklahoma Assistant Professor (tenure 1987) 1984{1987 University of Utah Instructor 1980{1984 Visiting Positions MSRI Simons Visiting Professor 2018 Simons Center 2012 Mittag-Leffler Institute, Sweden Visiting Professor 1997 Duke University Visiting Professor 1991{1992 University of Bayreuth, West Germany Visiting Professor 1989 Institute for Advanced Study Member 1982{1983 Publications 1. Degenerations of quintic threefolds and their lines. Duke Math. Jour. 50 (1983), 1127{1135. 2. Lines on complete intersection threefolds with K=0. Math. Z. 191 (1986), 297{302. 3. Tangents to a multiple plane curve. Pac. Jour. Math. 124 (1986), 321{331. 4. On the finiteness of rational curves on quintic threefolds. Comp. Math. 60 (1986), 151{162. 5. Hodge numbers of linked surfaces in P4. Duke Math. Jour. 55 (1987), 89{95. 6. The cubo-cubic transformation of P3 is very special. Math. Z. 195 (1987), 255{257. 7. Iteration of Multiple point formulas and applications to conics. Proceed- ings of the Algebraic Geometry Conference, Sundance, UT 1986, SLN 1311, 147{155. 8. Varieties cut out by quadrics: Scheme theoretic versus homogeneous gen- eration of ideals (with L. -
Mirror Symmetry Is a Phenomenon Arising in String Theory in Which Two Very Different Manifolds Give Rise to Equivalent Physics
Clay Mathematics Monographs 1 Volume 1 Mirror symmetry is a phenomenon arising in string theory in which two very different manifolds give rise to equivalent physics. Such a correspondence has Mirror Symmetry Mirror significant mathematical consequences, the most familiar of which involves the enumeration of holomorphic curves inside complex manifolds by solving differ- ential equations obtained from a “mirror” geometry. The inclusion of D-brane states in the equivalence has led to further conjectures involving calibrated submanifolds of the mirror pairs and new (conjectural) invariants of complex manifolds: the Gopakumar Vafa invariants. This book aims to give a single, cohesive treatment of mirror symmetry from both the mathematical and physical viewpoint. Parts 1 and 2 develop the neces- sary mathematical and physical background “from scratch,” and are intended for readers trying to learn across disciplines. The treatment is focussed, developing only the material most necessary for the task. In Parts 3 and 4 the physical and mathematical proofs of mirror symmetry are given. From the physics side, this means demonstrating that two different physical theories give isomorphic physics. Each physical theory can be described geometrically, and thus mirror symmetry gives rise to a “pairing” of geometries. The proof involves applying R ↔ 1/R circle duality to the phases of the fields in the gauged linear sigma model. The mathematics proof develops Gromov-Witten theory in the algebraic MIRROR SYMMETRY setting, beginning with the moduli spaces of curves and maps, and uses localiza- tion techniques to show that certain hypergeometric functions encode the Gromov-Witten invariants in genus zero, as is predicted by mirror symmetry. -
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. -
Download Enumerative Geometry and String Theory Free Ebook
ENUMERATIVE GEOMETRY AND STRING THEORY DOWNLOAD FREE BOOK Sheldon Katz | 206 pages | 31 May 2006 | American Mathematical Society | 9780821836873 | English | Providence, United States Enumerative Geometry and String Theory Chapter 2. The most accessible portal into very exciting recent material. Topological field theory, primitive forms and related topics : — Nuclear Physics B. Bibcode : PThPh. Bibcode : hep. As an example, consider the torus described above. Enumerative Geometry and String Theory standard analogy for this is to consider a multidimensional object such as a garden hose. An example is the red circle in the figure. This problem was solved by the nineteenth-century German mathematician Hermann Schubertwho found that there are Enumerative Geometry and String Theory 2, such lines. Once these topics are in place, the connection between physics and enumerative geometry is made with the introduction of topological quantum field theory and quantum cohomology. Enumerative Geometry and String Theory mirror symmetry relationship is a particular example of what physicists call a duality. Print Price 1: The book contains a lot of extra material that was not included Enumerative Geometry and String Theory the original fifteen lectures. Increasing the dimension from two to four real dimensions, the Calabi—Yau becomes a K3 surface. This problem asks for the number and construction of circles that are tangent to three given circles, points or lines. Topological Quantum Field Theory. Online Price 1: Online ISBN There are infinitely many circles like it on a torus; in fact, the entire surface is a union of such circles. For other uses, see Mirror symmetry. As an example, count the conic sections tangent to five given lines in the projective plane. -
Awards of ICCM 2013 by the Editors
Awards of ICCM 2013 by the Editors academies of France, Sweden and the United States. He is a recipient of the Fields Medal (1986), the Crafoord Prize Morningside Medal of Mathematics in Mathematics (1994), the King Faisal International Prize Selection Committee for Science (2006), and the Shaw Prize in Mathematical The Morningside Medal of Mathematics Selection Sciences (2009). Committee comprises a panel of world renowned mathematicians and is chaired by Professor Shing-Tung Björn Engquist Yau. A nomination committee of around 50 mathemati- Professor Engquist is the Computational and Applied cians from around the world nominates candidates based Mathematics Chair Professor at the University of Texas at on their research, qualifications, and curriculum vitae. Austin. His recent work includes homogenization theory, The Selection Committee reviews these nominations and multi-scale methods, and fast algorithms for wave recommends up to two recipients for the Morningside propagation. He is a member of the Royal Swedish Gold Medal of Mathematics, up to two recipients for the Morningside Gold Medal of Applied Mathematics, and up to four recipients for the Morningside Silver Medal of Mathematics. The Selection Committee members, with the exception of the committee chair, are all non-Chinese to ensure the independence, impartiality and integrity of the awards decision. Members of the 2013 Morningside Medal of Mathe- matics Selection Committee are: Richard E. Borcherds Professor Borcherds is Professor of Mathematics at the University of California at Berkeley. His research in- terests include Lie algebras, vertex algebras, and auto- morphic forms. He is best known for his work connecting the theory of finite groups with other areas in mathe- matics. -
String-Math 2012
Volume 90 String-Math 2012 July 16–21, 2012 Universitat¨ Bonn, Bonn, Germany Ron Donagi Sheldon Katz Albrecht Klemm David R. Morrison Editors Volume 90 String-Math 2012 July 16–21, 2012 Universitat¨ Bonn, Bonn, Germany Ron Donagi Sheldon Katz Albrecht Klemm David R. Morrison Editors Volume 90 String-Math 2012 July 16–21, 2012 Universitat¨ Bonn, Bonn, Germany Ron Donagi Sheldon Katz Albrecht Klemm David R. Morrison Editors 2010 Mathematics Subject Classification. Primary 11G55, 14D21, 14F05, 14J28, 14M30, 32G15, 53D18, 57M27, 81T40. 83E30. Library of Congress Cataloging-in-Publication Data String-Math (Conference) (2012 : Bonn, Germany) String-Math 2012 : July 16-21, 2012, Universit¨at Bonn, Bonn, Germany/Ron Donagi, Sheldon Katz, Albrecht Klemm, David R. Morrison, editors. pages cm. – (Proceedings of symposia in pure mathematics; volume 90) Includes bibliographical references. ISBN 978-0-8218-9495-8 (alk. paper) 1. Geometry, Algebraic–Congresses. 2. Quantum theory– Mathematics–Congresses. I. Donagi, Ron, editor. II. Katz, Sheldon, 1956- editor. III. Klemm, Albrecht, 1960- editor. IV. Morrison, David R., 1955- editor. V. Title. QA564.S77 2012 516.35–dc23 2015017523 DOI: http://dx.doi.org/10.1090/pspum/090 Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. 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 select pages 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. -
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 Advanced Lectures in Mathematics Volume XVIII Geometry and Analysis No. 2 Editor: Lizhen Ji International Press 浧䷘㟨十⒉䓗䯍 www.intlpress.com HIGHER EDUCATION PRESS Advanced Lectures in Mathematics, Volume XVIII Geometry and Analysis, No. 2 Volume Editor: Lizhen Ji, University of Michigan, Ann Arbor 2010 Mathematics Subject Classification. 58-06. Copyright © 2011 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. All rights reserved. Individual readers of this publication, and non-profit 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. -
View This Volume's Front and Back Matter
AMS/IP Studies in Advanced Mathematics S.-T. Yau, Series Editor Fifth International Congress of Chinese Mathematicians Part 2 Lizhen Ji Yat Sun Poon Lo Yang Shing-Tung Yau Editors American Mathematical Society • International Press Fifth International Congress of Chinese Mathematicians https://doi.org/10.1090/amsip/051.2 AMS/IP Studies in Advanced Mathematics Volume 51, Part 2 Fifth International Congress of Chinese Mathematicians Lizhen Ji Yat Sun Poon Lo Yang Shing-Tung Yau Editors American Mathematical Society • International Press Shing-Tung Yau, General Editor 2000 Mathematics Subject Classification. Primary 05–XX, 08–XX, 11–XX, 14–XX, 22–XX, 35–XX, 37–XX, 53–XX, 58–XX, 62–XX, 65–XX, 20–XX, 30–XX, 80–XX, 83–XX, 90–XX. All photographs courtesy of International Press. Library of Congress Cataloging-in-Publication Data International Congress of Chinese Mathematicians (5th : 2010 : Beijing, China) p. cm. (AMS/IP studies in advanced mathematics ; v. 51) Includes bibliographical references. ISBN 978-0-8218-7555-1 (set : alk. paper)—ISBN 978-0-8218-7586-5 (pt. 1 : alk. paper)— ISBN 978-0-8218-7587-2 (pt. 2 : alk. paper) 1. Mathematics—Congresses. I. Ji, Lizhen, 1964– II. Title. III. Title: 5th International Congress of Chinese Mathematicians. QA1.I746 2010 510—dc23 2011048032 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. -
Morningside Medals, the Chern Prizes, and the International Cooperation Awards Excerpts from the 2010 ICCM Booklet
Morningside Medals, the Chern Prizes, and the International Cooperation Awards excerpts from the 2010 ICCM booklet A fundamental domain is related to the concept of a group, which can be used to express symmetry in Morningside medals of mathematics and mathematics. During the late 19th century, infinite dis- applied mathematics crete group was studied but only until the 20th century, The Morningside Medal of Mathematics is awarded to did it become a main subject area in mathematics. This outstanding mathematicians of Chinese descent to en- field of study is not only important in geometry but also courage them in their pursuit of mathematical truth. Up in number theory. Practically all the famous develop- to six medals, two gold and four silver, are awarded every ments in modern number theory are related to concepts three years at the International Congress of Chinese of fundamental domain and discrete group. Morningside Medals of Mathematics Selection Committee The selection committee for the Morningside Medal of Mathematics is chaired by Professor Shing-Tung Yau. A The Morningside Gold Medal awarded to Kefeng Liu in 2004 Mathematicians. The Morningside Medals also carry a cash award of US $25,000 for each of the gold medalists; and US $ 10,000 for each of the silver medalists. A com- mittee of internationally renowned mathematicians, chaired by Professor Shing-Tung Yau, selects the medal- ists. In addition to the name of medalists, each gold and silver medal has the image of a Möbius band and a fun- damental domain. They were chosen not only for the The Morningside Silver Medals awarded to Daqing Wan and simplicity of the image, but also because of their signifi- Chin-Lung Wang in 2001 cant contributions to the development of mathematics in 20th century. -
New Publications Offered by The
New Publications Offered by the AMS To subscribe to email notification of new AMS publications, please go to http://www.ams.org/bookstore-email. Algebra and Algebraic Analysis Geometry Differentiable Quantum Bounded Measures and the Symmetric Domains Malliavin Calculus Leonid L. Vaksman Vladimir I. Bogachev, Moscow Translated by Olga Bershtein and State University, Russia Sergey D. Sinel’shchikov This book provides the reader with the principal concepts and results related This book provides exposition of the to differential properties of measures basic theory of quantum bounded on infinite dimensional spaces. In the symmetric domains. The area became finite dimensional case such properties are described in terms of active in the late 1990s at a junction of densities of measures with respect to Lebesgue measure. In the noncommutative complex analysis and infinite dimensional case new phenomena arise. For the first time a extensively developing theory of quantum groups. It is well detailed account is given of the theory of differentiable measures, known that the classical bounded symmetric domains involve a initiated by S. V. Fomin in the 1960s; since then the method large number of nice constructions and results of the theory has found many various important applications. Differentiable of C∗-algebras, theory of functions and functional analysis, properties are described for diverse concrete classes of measures representation theory of real reductive Lie groups, harmonic arising in applications, for example, Gaussian, convex, stable, analysis, and special functions. In a surprising advance of the theory Gibbsian, and for distributions of random processes. Sobolev of quantum bounded symmetric domains, it turned out that many classes for measures on finite and infinite dimensional spaces are classical problems admit elegant quantum analogs. -
Superstring Theory
Advanced Lectures in Mathematics Volume I Superstring Theory Editors: Kefeng Liu, Shing-Tung Yau & Chongyuan Zhu International Press 高等教育出版社 www.intlpress.com HIGHER EDUCATION PRESS Kefeng Liu Chongyuan Zhu University of California at Los Chinese Academy of Sciences, Beijing Angeles Shing-Tung Yau Harvard University 2000 Mathematics Subject Classification. 83E30. Copyright 2008 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 978-1-57146-131-5 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, Los Angeles Lizhen Ji University of Michigan, Ann Arbor Editorial Board Chongqing Cheng Xiping Zhu Nanjing University Zhongshan University Nanjing, China Gangzhou, China Zhong-Ci Shi Tatsien Li Institute of Computational Fudan University Mathematics Shanghai, China Chinese Academy of Sciences Beijing, China Zhiying Wen Tsinghua University Zhouping Xin Beijing, China The Chinese University of Hong Kong Lo Yang Hong Kong, China Institute of Mathematics Chinese Academy of Sciences Weiping Zhang Beijing, China Nankai University Tianjin, China Preface In celebration of the opening of the Center of Mathematical Sci- ences at Zhejiang University, we held the 2002 String Theory Interna- tional Conference from August 12th to August 15th at the new Center.