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Clay Mathematics Institute 2005 James A Contents Clay Mathematics Institute 2005 James A. Carlson Letter from the President 2 The Prize Problems The Millennium Prize Problems 3 Recognizing Achievement 2005 Clay Research Awards 4 CMI Researchers Summary of 2005 6 Workshops & Conferences CMI Programs & Research Activities Student Programs Collected Works James Arthur Archive 9 Raoul Bott Library CMI Profle Interview with Research Fellow 10 Maria Chudnovsky Feature Article Can Biology Lead to New Theorems? 13 by Bernd Sturmfels CMI Summer Schools Summary 14 Ricci Flow, 3–Manifolds, and Geometry 15 at MSRI Program Overview CMI Senior Scholars Program 17 Institute News Euclid and His Heritage Meeting 18 Appointments & Honors 20 CMI Publications Selected Articles by Research Fellows 27 Books & Videos 28 About CMI Profile of Bow Street Staff 30 CMI Activities 2006 Institute Calendar 32 2005 1 Euclid: www.claymath.org/euclid James Arthur Collected Works: www.claymath.org/cw/arthur Hanoi Institute of Mathematics: www.math.ac.vn Ramanujan Society: www.ramanujanmathsociety.org $.* $MBZ.BUIFNBUJDT*OTUJUVUF ".4 "NFSJDBO.BUIFNBUJDBM4PDJFUZ In addition to major,0O"VHVTU BUUIFTFDPOE*OUFSOBUJPOBM$POHSFTTPG.BUIFNBUJDJBOT ongoing activities such as JO1BSJT %BWJE)JMCFSUEFMJWFSFEIJTGBNPVTMFDUVSFJOXIJDIIFEFTDSJCFE the summer schools,UXFOUZUISFFQSPCMFNTUIBUXFSFUPQMBZBOJOnVFOUJBMSPMFJONBUIFNBUJDBM the Institute undertakes a 5IF.JMMFOOJVN1SJ[F1SPCMFNT SFTFBSDI"DFOUVSZMBUFS PO.BZ BUBNFFUJOHBUUIF$PMMÒHFEF number of smaller'SBODF UIF$MBZ.BUIFNBUJDT*OTUJUVUF $.* BOOPVODFEUIFDSFBUJPOPGB special projects each year. Four examples demonstrate64NJMMJPOQSJ[FGVOEGPSUIFTPMVUJPOPGTFWFOJNQPSUBOUDMBTTJDQSPCMFNT the breadth of the Institute’s XIJDIIBWFSFTJTUFETPMVUJPO5IFQSJ[FGVOEJTEJWJEFEFRVBMMZBNPOHUIF work. First is the TFWFOQSPCMFNT5IFSFJTOPUJNFMJNJUGPSUIFJSTPMVUJPOdigitization of the oldest surviving manuscript of Euclid’s5IF.JMMFOOJVN1SJ[F1SPCMFNTXFSFTFMFDUFECZUIFGPVOEJOH4DJFOUJmD Elements, dated to 888 "EWJTPSZ#PBSEPG$.*"MBJO$POOFT "SUIVS+BGGF "OESFX8JMFT BOE 5IF.JMMFOOJVN AD, when it was&EXBSE8JUUFOBGUFSDPOTVMUJOHXJUIPUIFSMFBEJOHNBUIFNBUJDJBOT copied from a predecessor in 5IFJSBJNXBTTPNFXIBUEJGGFSFOUUIBOUIBUPG)JMCFSUOPUUPEFmOFOFX Constantinople. DIBMMFOHFT CVUUPSFDPSETPNFPGUIFNPTUEJGmDVMUJTTVFTXJUIXIJDIFor the frst time, a facsimile of NBUIFNBUJDJBOTXFSFTUSVHHMJOHBUUIFUVSOPGUIFTFDPOENJMMFOOJVNUP 1SJ[F1SPCMFNT the founding documentSFDPHOJ[FBDIJFWFNFOUJONBUIFNBUJDTPGIJTUPSJDBMEJNFOTJPOUPFMFWBUF of mathematics is widely JOUIFDPOTDJPVTOFTTPGUIFHFOFSBMQVCMJDUIFGBDUUIBUJONBUIFNBUJDT UIF available. SecondGSPOUJFSJTTUJMMPQFOBOEBCPVOETJOJNQPSUBOUVOTPMWFEQSPCMFNTBOEUP is the James Arthur Archive on +$BSMTPO "+BGGF BOE"8JMFT &EJUPST the CMI website.FNQIBTJ[FUIFJNQPSUBODFPGXPSLJOHUPXBSETBTPMVUJPOPGUIFEFFQFTU This is an online library of the Letter from the president NPTUEJGmDVMUQSPCMFNT collected works 5IFQSFTFOUWPMVNFTFUTGPSUIUIFPGmDJBMEFTDSJQUJPOPGFBDIPGUIFTFWFOof Toronto mathematician James QSPCMFNTBOEUIFSVMFTHPWFSOJOHUIFQSJ[FT*UBMTPDPOUBJOTBOFTTBZCZ Arthur, noted for +FSFNZ(SBZPOUIFIJTUPSZPGQSJ[FQSPCMFNTJONBUIFNBUJDThis achievements in automorphic forms and representation theory. James Arthur will +$BSMTPO "+BGGF BOE"8JMFT &EJUPST be adding further material to the archive, including his own comments. The Clay Mathematics Institute ")JTUPSZPG1SJ[FTJO.BUIFNBUJDT +FSFNZ(SBZ hopes that there will be further efforts of this kind, #JSDIBOE4XJOOFSUPO%ZFS$POKFDUVSF "OESFX8JMFT both at CMI and elsewhere. The third example is the )PEHF$POKFDUVSF 1JFSSF%FMJHOF /BWJFSo4UPLFT&RVBUJPO $IBSMFT-'FGGFSNBO Institute’s support of George Csicsery’s documentary 1PJODBSÏ$POKFDUVSF +PIO.JMOPS flm on the life and work of Julia Robinson, noted 1WFSTVT/11SPCMFN 4UFQIFO$PPL XXXDMBZNBUIPSH 3JFNBOO)ZQPUIFTJT &OSJDP#PNCJFSJ for her key role in the solution XXXBNTPSHof Hilbert’s tenth problem. Robinson’s inspiring story will be of 2VBOUVN:BOHo.JMMT5IFPSZ "SUIVS+BGGFBOE&EXBSE8JUUFO James Carlson interest to both mathematicians and the general public. Finally, the Institute is supporting two small .13*;& Dear Friends of Mathematics ventures in Vietnam and India. The frst, part of Bao- Châu Ngô’s research award in 2004, is a donation of This past year marked the sixth anniversary of books to the Mathematical Institute in Hanoi. The the Clay Mathematics Institute’s program of second, part of Manjul Bhargava’s research award summer schools. The frst, held at Pine Manor in 2005, is two years of support for the Ramanujan College, was dedicated to mirror symmetry. The Mathematical Society (RMS) in Mumbai. The RMS last, held at MSRI in Berkeley, was on Ricci fow, will offer one or two Clay–Bhargava fellowships of 3-manifolds, and geometry. It was of special interest 50,000 Rupees to promising graduate students. because of Perelman’s proposed solution to the Poincaré conjecture and to Thurston’s geometrization The programs and projects mentioned above conjecture. It was also the culmination of Hamilton’s illustrate a guiding principle of the Clay Mathematics Ricci fow program and the deep work of many Institute: support a broad spectrum of initiatives in mathematicians who have dedicated their life’s work mathematics, large and small, in order to foster the to geometry and analysis. The aim of the summer creation of new knowledge. schools is to provide participants – graduate students and recent PhD’s within fve years of their degree – with the background needed to work successfully Sincerely, in an active and important area of mathematics. The next two schools, one on arithmetic geometry, the other on homogenous fows, dynamics, and number theory, will be held in Göttingen and Pisa, respectively. James A. Carlson 2 CMI ANNUAL REPORT The expected publication date is July 23, 2006. Copies of the book will be available at the ICM in Madrid, August 22–30, 2006, and also The prize problems sold at www.ams.org/bookstore The Millennium Prize Problems $.* $MBZ.BUIFNBUJDT*OTUJUVUF ".4 "NFSJDBO.BUIFNBUJDBM4PDJFUZ 0O"VHVTU BUUIFTFDPOE*OUFSOBUJPOBM$POHSFTTPG.BUIFNBUJDJBOT mathematics of historical dimension; to elevate in JO1BSJT %BWJE)JMCFSUEFMJWFSFEIJTGBNPVTMFDUVSFJOXIJDIIFEFTDSJCFE UXFOUZUISFFQSPCMFNTUIBUXFSFUPQMBZBOJOnVFOUJBMSPMFJONBUIFNBUJDBM 5IF.JMMFOOJVN1SJ[F1SPCMFNT the consciousness of the general public the fact that SFTFBSDI"DFOUVSZMBUFS PO.BZ BUBNFFUJOHBUUIF$PMMÒHFEF 'SBODF UIF$MBZ.BUIFNBUJDT*OTUJUVUF $.* BOOPVODFEUIFDSFBUJPOPGB in mathematics the frontier is still open and abounds 64NJMMJPOQSJ[FGVOEGPSUIFTPMVUJPOPGTFWFOJNQPSUBOUDMBTTJDQSPCMFNT in important unsolved problems; and to emphasize XIJDIIBWFSFTJTUFETPMVUJPO5IFQSJ[FGVOEJTEJWJEFEFRVBMMZBNPOHUIF TFWFOQSPCMFNT5IFSFJTOPUJNFMJNJUGPSUIFJSTPMVUJPO the importance of working toward a solution of the 5IF.JMMFOOJVN1SJ[F1SPCMFNTXFSFTFMFDUFECZUIFGPVOEJOH4DJFOUJmD deepest, most diffcult problems. "EWJTPSZ#PBSEPG$.*"MBJO$POOFT "SUIVS+BGGF "OESFX8JMFT BOE 5IF.JMMFOOJVN &EXBSE8JUUFOBGUFSDPOTVMUJOHXJUIPUIFSMFBEJOHNBUIFNBUJDJBOT 5IFJSBJNXBTTPNFXIBUEJGGFSFOUUIBOUIBUPG)JMCFSUOPUUPEFmOFOFX DIBMMFOHFT CVUUPSFDPSETPNFPGUIFNPTUEJGmDVMUJTTVFTXJUIXIJDI The Millennium Prize Problems gives the offcial NBUIFNBUJDJBOTXFSFTUSVHHMJOHBUUIFUVSOPGUIFTFDPOENJMMFOOJVNUP 1SJ[F1SPCMFNT SFDPHOJ[FBDIJFWFNFOUJONBUIFNBUJDTPGIJTUPSJDBMEJNFOTJPOUPFMFWBUF description of each of the seven problems and the JOUIFDPOTDJPVTOFTTPGUIFHFOFSBMQVCMJDUIFGBDUUIBUJONBUIFNBUJDT UIF GSPOUJFSJTTUJMMPQFOBOEBCPVOETJOJNQPSUBOUVOTPMWFEQSPCMFNTBOEUP +$BSMTPO "+BGGF BOE"8JMFT &EJUPST rules governing the prizes. It also contains an essay FNQIBTJ[FUIFJNQPSUBODFPGXPSLJOHUPXBSETBTPMVUJPOPGUIFEFFQFTU NPTUEJGmDVMUQSPCMFNT by Jeremy Gray on the history of prize problems in 5IFQSFTFOUWPMVNFTFUTGPSUIUIFPGmDJBMEFTDSJQUJPOPGFBDIPGUIFTFWFO mathematics. QSPCMFNTBOEUIFSVMFTHPWFSOJOHUIFQSJ[FT*UBMTPDPOUBJOTBOFTTBZCZ +FSFNZ(SBZPOUIFIJTUPSZPGQSJ[FQSPCMFNTJONBUIFNBUJDT +$BSMTPO "+BGGF BOE"8JMFT &EJUPST The seven Millennium Prize Problems range from the oldest, the Riemann Hypothesis, a problem in number ")JTUPSZPG1SJ[FTJO.BUIFNBUJDT +FSFNZ(SBZ theory stated in 1859, to the youngest, the P versus NP #JSDIBOE4XJOOFSUPO%ZFS$POKFDUVSF "OESFX8JMFT problem, a problem in theoretical computer science stated )PEHF$POKFDUVSF 1JFSSF%FMJHOF in 1971. Informal descriptions are given below. See the /BWJFSo4UPLFT&RVBUJPO $IBSMFT-'FGGFSNBO individual articles in the book for offcial statements 1PJODBSÏ$POKFDUVSF +PIO.JMOPS and background material. See also www.claymath.org/ 1WFSTVT/11SPCMFN 4UFQIFO$PPL XXXDMBZNBUIPSH millennium 3JFNBOO)ZQPUIFTJT &OSJDP#PNCJFSJ XXXBNTPSH 2VBOUVN:BOHo.JMMT5IFPSZ "SUIVS+BGGFBOE&EXBSE8JUUFO The Birch and Swinnerton-Dyer Conjecture: Let C be an elliptic curve over the rational numbers. The order to which its L function vanishes at s = 1 is the rank of the .13*;& group of rational points. The Hodge Conjecture: A rational cohomology class of type (p, p) on a projective algebraic manifold is represented by a rational sum of algebraic subvarieties. On August 8, 1900, at the second International The Navier–Stokes Equation. Show that the Navier– Congress of Mathematicians in Paris, David Hilbert Stokes equations on Euclidean 3-space have a unique, delivered the famous lecture in which he described smooth, fnite energy solution for all time greater than twenty-three problems that were to play an or equal to zero, given smooth, divergence-free, initial infuential role in future mathematical research. A conditions which “decay rapidly at large distances.” century later, on May 24, 2000, at a meeting at the Alternatively, show that there is no such solution.
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