Perspectives in Representation Theory
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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 Profile 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 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 -
Contemporary Mathematics 442
CONTEMPORARY MATHEMATICS 442 Lie Algebras, Vertex Operator Algebras and Their Applications International Conference in Honor of James Lepowsky and Robert Wilson on Their Sixtieth Birthdays May 17-21, 2005 North Carolina State University Raleigh, North Carolina Yi-Zhi Huang Kailash C. Misra Editors http://dx.doi.org/10.1090/conm/442 Lie Algebras, Vertex Operator Algebras and Their Applications In honor of James Lepowsky and Robert Wilson on their sixtieth birthdays CoNTEMPORARY MATHEMATICS 442 Lie Algebras, Vertex Operator Algebras and Their Applications International Conference in Honor of James Lepowsky and Robert Wilson on Their Sixtieth Birthdays May 17-21, 2005 North Carolina State University Raleigh, North Carolina Yi-Zhi Huang Kailash C. Misra Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Andreas Blass Abel Klein 2000 Mathematics Subject Classification. Primary 17810, 17837, 17850, 17865, 17867, 17868, 17869, 81T40, 82823. Photograph of James Lepowsky and Robert Wilson is courtesy of Yi-Zhi Huang. Library of Congress Cataloging-in-Publication Data Lie algebras, vertex operator algebras and their applications : an international conference in honor of James Lepowsky and Robert L. Wilson on their sixtieth birthdays, May 17-21, 2005, North Carolina State University, Raleigh, North Carolina / Yi-Zhi Huang, Kailash Misra, editors. p. em. ~(Contemporary mathematics, ISSN 0271-4132: v. 442) Includes bibliographical references. ISBN-13: 978-0-8218-3986-7 (alk. paper) ISBN-10: 0-8218-3986-1 (alk. paper) 1. Lie algebras~Congresses. 2. Vertex operator algebras. 3. Representations of algebras~ Congresses. I. Leposwky, J. (James). II. Wilson, Robert L., 1946- III. Huang, Yi-Zhi, 1959- IV. -
Conference Poster (.Pdf)
MMMAAATTTHHHEEEMMMAAATTTIIICCCSSS DDDEEEPPPAAARRRTTTMMMEEENNNTTT NNNooorrrttthhh CCCaaarrrooollliiinnnaaa SSStttaaattteee UUUnnniiivvveeerrrsssiiitttyyy LLLIIIEEE AAALLLGGGEEEBBBRRRAAASSS,,, VVVEEERRRTTTEEEXXX OOOPPPEEERRRAAATTTOOORRR AAALLLGGGEEEBBBRRRAAASSS AAANNNDDD TTTHHHEEEIIIRRR AAAPPPPPPLLLIIICCCAAATTTIIIOOONNNSSS TTTHHH SSSTTT MMMAAAYYY 111777 ––– 222111 ,,, 222000000555 Conference iin honor of James Lepowsky and Robert L. Wiillson Invited speakers include: • Special lecture by Bertram Kostant, MIT • Georgia Benkart, University of Wisconsin, Madison • Stephen Berman, University of Saskatchewan, Canada • Richard Block, University of California, Riverside • Vyjayanthi Chari, University of California, Riverside SSSpppooonnnsssooorrreeeddd bbbyyy::: • Chongying Dong, University of California, Santa Cruz TTThhheee NNNaaatttiiiooonnnaaalll SSSccciiieeennnccceee FFFooouuunnndddaaatttiiiooonnn • Rolf Farnsteiner, University of Bielefeld, Germany NNNCCCSSSUUU MMMaaattthhheeemmmaaatttiiicccsss DDDeeepppaaarrrtttmmmeeennnttt • Igor Frenkel, Yale University • Jurgen Fuchs, Institutionen for Fysik, Karlstads Universitet, Sweden • Howard Garland, Yale University Organized by: OOrrggaanniizzeedd bbyy:: • Seok-Jin Kang, Seoul National University, Korea KKKaaaiiilllaaassshhh CCC... MMMiiisssrrraaa • Rinat Kedem, University of Illinois, Urbana-Champaign mmmiiisssrrraaa@@@mmmaaattthhh...nnncccsssuuu...eeeddduuu • Alexander Kirillov, Jr. State University of New York, Stony Brook YYYiii---ZZZhhhiii HHHuuuaaannnggg • Haisheng Li, Rutgers University, -
Arxiv:2007.07270V1 [Hep-Th]
The Cosmological Constant: A Categorical View Fen Zuo∗ Abstract Some theoretic results related to the cosmological constant, obtained in the 80’s and 90’s of last century, are reviewed. These results exhibit some interesting underlying pattern when viewed from a category- theoretic perspective. In doing this, we illustrate how Baez and Dolan’s Periodic Table of k-tuply monoidal n-categories can serve as a basic framework of a quantum spacetime structure. Explicitly, we show how Einstein gravity (with the cosmological constant turned on) emerges when we de-categorify certain 2-tuply monoidal 2-categories properly, as Crane proposed a long time ago. In particular, we find that some recent work on Vertex Operator Algebras and 4-manifolds fits nicely into this framework. arXiv:2007.07270v1 [hep-th] 13 Jul 2020 ∗ Email: [email protected] 1 Contents I. Introduction 3 A. The notation 4 II. 3D 4 A. Euclidean case 4 B. Lorentzian case 7 III. 2D 8 A. Euclidean case 8 B. Lorentzian case 9 IV. 4D 10 A. Euclidean case 11 1. Barrett-Crane Model 11 2. Anyon-Condensation Picture 12 3. Gravity From Condensation? 15 4. Cosmological Constant 17 B. Lorentzian case 18 1. Identifying “Time” 18 2. Emergence of 4D Lorentz Symmetry 20 3. Cosmological Constant 20 V. Categorification 21 A. Motivation 21 B. General framework 24 1. Vertex Operator Algebras 24 2. Theory X and 4D topological invariants 24 3. VOA[M4] and 2-Tannaka-Krein duality 27 4. Gravity and Cosmological Constant 28 VI. Discussion 30 Note added: 31 2 Acknowledgments 32 References 32 I. -
Aaron D. Lauda
USCDEPARTMENTOFMATHEMATICS • 3620 S. VERMONT AVE, KAP 108 • LOS ANGELES, CA 90089-2532 E - M A I L:[email protected] • W E B: HTTP://SITES.GOOGLE.COM/VIEW/LAUDA-HOME A A R O N D. L A U D A September 16, 2021 RESEARCH INTERESTS • Low-dimensional topology and topological quantum field theory • Topological quantum computation • Categorification and its applications • Representation theory of Lie algebras and quantum groups EDUCATION 2021 Cambridge University Doctor of Science 2006 Cambridge University Ph.D. Mathematics Thesis: Open-closed topological field theory and Khovanov homology, Advisor: Martin Hyland 2003 University of California, Riverside M.S. in Physics Advisor: John Baez 2002 University of California, Riverside B.S in Pure Mathematics, Summa Cum Laude EMPLOYMENT February 2019 – present University of Southern California, Los Angeles, CA. Dean’s Leadership Fellow for the Natural Sciences and Mathematics March 2014 – present University of Southern California, Los Angeles, CA. Professor November 2013 - March 2014 University of Southern California, Los Angeles, CA. Associate Professor July 2011 - November 2013 University of Southern California, Los Angeles, CA. Assistant Professor Fall 2006– Spring 2011 Columbia University, New York, NY. Joseph Fels Ritt Assistant Professor Fall 2007 Fields Institute, Toronto, Canada. Special Program on ‘Geometric Aspects of Homotopy Theory’ Jerrold E. Marsden Postdoctoral Fellow 1 HONORS AND AWARDS 2021 Doctor of Science, Cambridge University 2019 Fellow of the American Mathematics Society 2016 Mathematical and Physical Sciences Simons Fellow 2014 Section Chair for Seoul ICM Section - Lie Theory and Generalizations 2013 NSF CAREER Award 2011 Alfred P. Sloan Fellowship 2005 Rayleigh-Knight prize, Cambridge University PUBLICATIONS Following the convention in mathematics, authors are listed alphabetically. -
Evgeny Gorskiy
Curriculum vitae Evgeny Gorskiy University of California at Davis Phone: (530) 752-0827 Department of Mathematics One Shields Avenue, Davis CA 95616 http://www.math.ucdavis.edu/∼egorskiy Email: [email protected] Personal Data Date of Birth: December 12, 1984; Moscow, Russia. Languages: English, Russian. Employment 2018{present: Associate Professor, Department of Mathematics. UC Davis, USA 2014{2018: Assistant Professor, Department of Mathematics. UC Davis, USA (On leave for 2014{2015 academic year). 2013{2015: Ritt Assistant Professor, Department of Mathematics. Columbia University, USA. 2010{2013: James H. Simons Instructor, Department of Mathematics. Stony Brook University, USA. 2014{Present: Researcher, International Laboratory for Representation Theory and Mathematical Physics, National Research University { Higher School of Economics. Moscow, Russia. Research interests Theory of singularities, algebraic geometry, motivic integration, low-dimensional topology, knot homology, representation theory, double affine Hecke algebras. Education 2006{2009: PhD in Mathematics { Department of Higher Geometry and Topology, Faculty of Mathematics and Mechanics, Moscow State University Thesis title: "Motivic integration and invariants of algebraic knots". Thesis advisor: Professor Sabir M. Gusein-Zade. 2001{2006: M. Sc. in Mathematics { Department of Higher Geometry and Topology, Faculty of Mathematics and Mechanics, Moscow State University Diploma title: "Motivic integrals and functional equations". 2001{2006: M. Sc. in Mathematics { Independent University of Moscow Grade per average: 5.0 (out of 5) at both universities. Fellowships and awards 2016: Hellman fellowship 2013: Prize of the Moscow Mathematical Society 2009: Dynasty Foundation contest fellowship 2008: Honor diploma of the Euler contest in the "PhD students" nomination 2008: Fellowship of the Government of the Russian Federation 2007: 2nd prize of the Moebius contest in the "PhD students" nomination 2006: 1st prize of the Moebius contest in the "Students" nomination . -
On the Work of Igor Frenkel Introduction by Pavel Etingof Igor Frenkel Is One of the Leading Representation Theorists and Mathe- Matical Physicists of Our Time
On the work of Igor Frenkel Introduction by Pavel Etingof Igor Frenkel is one of the leading representation theorists and mathe- matical physicists of our time. Inspired by the mathematical philosophy of Herman Weyl, who recognized the central role of representation the- ory in mathematics and its relevance to quantum physics, Frenkel made a number of foundational contributions at the juncture of these fields. A quintessential mathematical visionary and romantic, he has rarely fol- lowed the present day fashion. Instead, he has striven to get aheadof time and get a glimpse into the mathematics of the future – at least a decade, no less. In this, he has followed the example of I. M. Gelfand, whose approach to mathematics has always inspired him. He would often write several foundational papers in a subject, and then leave it for the future generations to be developed further. His ideas have sometimes been so bold and ambitious and so much ahead of their time that they would not be fully appreciated even by his students at the time of their formulation, and would produce a storm of activity only a few years later. And, of course, as a result, many of his ideas are still waiting for their time to go off. This text is a modest attempt by Igor’s students and colleagues of vari- ous generations to review his work, and to highlight how it has influenced in each case the development of the corresponding field in subsequent years. 1 Contributions of Igor Frenkel to the rep- resentation theory of affine Lie algebras by Alistair Savage and Anthony Licata Among infinite-dimensional Lie algebras, it is the theory of affine Lie algebras that is the richest and most well understood. -
Evgeny Gorskiy
Curriculum vitae Evgeny Gorskiy University of California at Davis Phone: Department of Mathematics One Shields Avenue, Davis CA 95616 http://www.math.ucdavis.edu/∼egorskiy Email: [email protected] Personal Data Date of Birth: December 12, 1984; Moscow, Russia. Languages: English, Russian. Employment 2014{present: Assistant Professor, Department of Mathematics. UC Davis, USA (On leave for 2014{2015 academic year). 2013{2015: Ritt Assistant Professor, Department of Mathematics. Columbia University, USA. 2010{2013: James H. Simons Instructor, Department of Mathematics. Stony Brook University, USA. 2014{Present: Researcher, International Laboratory for Representation Theory and Mathematical Physics, National Research University { Higher School of Economics. Moscow, Russia. Research interests Theory of singularities, algebraic geometry, motivic integration, low-dimensional topology, knot homology, representation theory, double affine Hecke algebras. Education 2006{2009: PhD in Mathematics { Department of Higher Geometry and Topology, Faculty of Mathematics and Mechanics, Moscow State University Thesis title: "Motivic integration and invariants of algebraic knots". Thesis advisor: Professor Sabir M. Gusein-Zade. 2001{2006: M. Sc. in Mathematics { Department of Higher Geometry and Topology, Faculty of Mathematics and Mechanics, Moscow State University Diploma title: "Motivic integrals and functional equations". 2001{2006: M. Sc. in Mathematics { Independent University of Moscow Grade per average: 5.0 (of 5) at both institutes. Fellowships and awards 2013: Prize of the Moscow Mathematical Society 2009: Dynasty Foundation contest fellowship 2008: Honor diploma of the Euler contest in the "PhD students" nomination 2008: Fellowship of the Government of the Russian Federation 2007: 2nd prize of the Moebius contest in the "PhD students" nomination 2006: 1st prize of the Moebius contest in the "Students" nomination . -
Anton M. Zeitlin
ANTON M. ZEITLIN Contact Department of Mathematics Cell: (917) 7178815 Information Louisiana State University Fax: (225) 5784276 303 Lockett Hall E-mail: [email protected] Baton Rouge LA 70803 USA WWW: http://math.lsu.edu/~zeitlin Citizenship Russian, US Permanent Resident Current position Assistant Professor (tenure-track), Louisiana State University, August 21, 2017{ Previous J.F. Ritt Assistant Professor, Employment Department of Mathematics, Columbia University: July 1, 2012{July 1, 2017 On leave 2013-2014 academic year as EPDI fellow: September 1, 2013 - February 28, 2014, Max Planck Institute for Mathematics, Bonn, March 1, 2014 - August 31, 2014, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette. Research Representation theory with applications to geometry, topology and mathematical physics. Interests In particular: Enumerative geometry, Higher Teichm¨ullertheory, Supergeometry, Quantum groups, Kac-Moody agebras, Loop groups, Homotopy algebras, Conformal field theory, Integrable systems, Algebraic and Geometric structures of Quantum field theory and Gravity Education Yale University, Ph.D. in Mathematics, 2006-2012, St. Petersburg State University, St. Petersburg, Russia, M.S., High Energy Physics, 2005 Grants and AMS Simons Travel Grant: 2016-2018 Awards EPDI laureate (IHES): 2012-2014 Yale University: Sterling stipend, 2006-2008 Dynasty Foundation Stipend: 2003-2006 CRDF: Grant No. RUM1-2622-ST-04 (as a member of Group), 2005-2006 RFBR: grant 05-01-00922 (as a member of Group), 2005-2006 Teaching Louisiana State University, -
Lecture II-1: Symmetry Breaking
Lecture II-1: Symmetry breaking Edward Witten Notes by Pavel Etingof and David Kazhdan In this semester we will continue the discussion of quantum field theory, but now mostly dynamics rather than perturbation theory and purely formal things. The topics we will discuss are much more difficult to deal with rigorously than the formal theory we studied in the fall semester. We will try to do it when possible, but it will not always be possible. In general, we will try to form an intuitive picture of what is going on, and illustrate it by considering concrete examples. 1.0. Theories and realizations. [This section is explanatory and was written by the preparers of these notes in order to create a false sense of security i.e. an unsubstantiated feeling that we understand what we are talking about in the rest of the lecture. Unfortunately, this is not the case, at least if “understand” means what it usually means among mathematicians.] For the purposes of the present and forthcoming lectures, it will be important for us to distinguish theories and their realizations. So let us explain what we mean by a theory and what we mean by its realization. This explaination is not a mathematical definition (in fact, it is hard to give a definition which is both rigorous and useful), but we hope that it will make clear what we are talking about. Recall that a classical physical system Σ is usually described by defining the space of states X of Σ (a symplectic manifold, maybe infinite-dimensional) and a 1-parameter group gt of time translations which preserves symplectic structure. -
On the Work of Igor Frenkel
Contemporary Mathematics Volume 610, 2014 http://dx.doi.org/10.1090/conm/610/12155 On the work of Igor Frenkel John Duncan, Pavel Etingof, Ivan Ip, Mikhail Khovanov, Matvei Libine, Anthony Licata, Alistair Savage, and Michael Schlosser Introduction by Pavel Etingof Igor Frenkel is one of the leading representation theorists and mathematical physicists of our time. Inspired by the mathematical philosophy of Herman Weyl, who recognized the central role of representation theory in mathematics and its relevance to quantum physics, Frenkel made a number of foundational contribu- tions at the juncture of these fields. A quintessential mathematical visionary and romantic, he has rarely followed the present day fashion. Instead, he has striven to get ahead of time and get a glimpse into the mathematics of the future – at least a decade, no less. In this, he has followed the example of I. M. Gelfand, whose approach to mathematics has always inspired him. He would often write several foundational papers in a subject, and then leave it for the future generations to be developed further. His ideas have sometimes been so bold and ambitious and so much ahead of their time that they would not be fully appreciated even by his students at the time of their formulation, and would produce a storm of activity only a few years later. And, of course, as a result, many of his ideas are still waiting for their time to go off. This text is a modest attempt by Igor’s students and colleagues of various generations to review his work, and to highlight how it has influenced in each case the development of the corresponding field in subsequent years. -
Anton M. Zeitlin
ANTON M. ZEITLIN Contact Department of Mathematics Information Room 509, MC 4406 Cell: (917) 7178815 2990 Broadway Fax: (212) 8548962 Columbia University E-mail: [email protected] New York NY 10027 USA WWW: http://math.columbia.edu/~zeitlin Citizenship Russian, US Permanent Resident Current position J.F. Ritt Assistant Professor, Department of Mathematics, Columbia University: July 1, 2012{July 1, 2017 On leave 2013-2014 academic year as EPDI fellow: September 1, 2013 - February 28, 2014, Max Planck Institute for Mathematics, Bonn, March 1, 2014 - August 31, 2014, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette. Research Representation theory with applications to geometry, topology and mathematical physics. Interests In particular: Quantum equivariant K-theory, Higher Teichm¨ullertheory, Supergeometry, Quantum groups, Kac-Moody agebras, Loop groups, Homotopy algebras, Conformal field theory, Integrable systems, Algebraic structures of Quantum field theory and Gravity Education Yale University, New Haven, Connecticut USA Ph.D. in Mathematics, 2006-2012 • Dissertation Topic: “Semi-infinite Cohomology in Noncommutative Geometry and Conformal Field Theory" Advisor: Igor B. Frenkel St. Petersburg State University, St. Petersburg, Russia M.S., Mathematical Physics, 2005 • Thesis Topic: "Quantum Supersymmetric Integrable Models of KdV Type and Superconformal Field Theory" Grants and AMS Simons Travel Grant: 2016-2018 Awards EPDI laureate (IHES): 2012-2014 Yale University: Sterling stipend, 2006-2008 Dynasty Foundation Stipend: