Monopoles and Fueter Sections on Three-Manifolds
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Arxiv:1912.05718V1 [Math.DG] 12 Dec 2019
JNR MONOPOLES MICHAEL K. MURRAY AND PAUL NORBURY 1 Abstract. We review the theory of JNR, mass 2 hyperbolic monopoles in particular their spectral curves and rational maps. These are used to establish conditions for a spectral curve to be the spectral curve of a JNR monopole and to show that that rational map of a JNR monopole monopole arises by scattering using results of Atiyah. We show that for JNR monopoles the holomorphic sphere has a remarkably simple form and show that this can be used to give a formula for the energy density at infinity. In conclusion we illustrate some examples of the energy-density at infinity of JNR monopoles. Dedicated to the memory of Sir Michael Atiyah Contents 1. Introduction 2 2. Hyperbolic monopoles 4 3. JNR monopoles 5 3.1. JNR spectral curve 5 3.2. Scattering 7 3.3. JNR rational map 7 4. A criterion to be a JNR spectral curve 8 4.1. (N; N) spectral curves 8 arXiv:1912.05718v1 [math.DG] 12 Dec 2019 4.2. Grids 9 4.3. JNR monopoles and grids 9 5. Holomorphic spheres 10 2010 Mathematics Subject Classification. 81T13, 53C07, 14D21. MKM acknowledges support under the Australian Research Council's Discovery Projects funding scheme (project number DP180100383). PN acknowledges support under the Aus- tralian Research Council's Discovery Projects funding scheme (project numbers DP170102028 and DP180103891). 1 2 MICHAEL K. MURRAY AND PAUL NORBURY 5.1. SO(3) action on JNR monopoles. 12 6. Projection to rational maps. 12 7. The energy density at infinity 14 References 16 1. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Moduli of Monopole Walls and Amoebas Arxiv:1202.1294V1 [Hep-Th]
Moduli of Monopole Walls and Amoebas Sergey A. Cherkis Richard S. Ward Department of Mathematics Department of Mathematical Sciences University of Arizona, University of Durham, Tucson AZ, 85721-0089, USA Durham DH1 3LE, UK [email protected] [email protected] Abstract We study doubly-periodic monopoles, also called monopole walls, determining their spectral data and computing the dimensions of their moduli spaces. Using spectral data we identify the moduli, and compare our results with a perturbative analysis. We also identify an SL(2; Z) action on monopole walls, in which the S transformation corresponds to the Nahm transform. arXiv:1202.1294v1 [hep-th] 6 Feb 2012 Contents 1 Introduction and Motivation 2 1.1 Monopole Wall . 2 1.1.1 Asymptotic Conditions . 3 1.1.2 Singularities . 4 1.2 Moduli and Parameters . 5 2 String Theory Dualities 7 3 Spectral Approach 11 3.1 Spectral data . 11 3.1.1 x-spectral data . 11 3.1.2 y-spectral data . 12 3.1.3 z-spectral data . 13 3.1.4 Examples . 13 3.2 Newton Polygon . 14 3.2.1 Amoebas . 16 3.2.2 Examples . 16 4 Moduli and Asymptotics 22 4.1 Newton Polygon from the Boundary Data . 22 4.2 Number of Moduli . 22 5 Nahm Transform and SL(2; Z) Action 26 5.1 The Nahm transform . 26 5.2 SL(2; Z) Action . 27 6 Perturbative Approach 29 6.1 The case (Q−;Q+) = (0; 1)........................... 29 6.2 The case Q± =1 ................................ 32 6.3 Kuranishi Complex . 36 7 Conclusions 37 A A four-dimensional detour 38 B Theta-function Relations 39 1 1 Introduction and Motivation The classical dynamics of monopoles was found to be intimately related to quantum gauge theories in three [1, 2, 3] and four [4] dimensions. -
An Interview with Martin Davis
Notices of the American Mathematical Society ISSN 0002-9920 ABCD springer.com New and Noteworthy from Springer Geometry Ramanujan‘s Lost Notebook An Introduction to Mathematical of the American Mathematical Society Selected Topics in Plane and Solid Part II Cryptography May 2008 Volume 55, Number 5 Geometry G. E. Andrews, Penn State University, University J. Hoffstein, J. Pipher, J. Silverman, Brown J. Aarts, Delft University of Technology, Park, PA, USA; B. C. Berndt, University of Illinois University, Providence, RI, USA Mediamatics, The Netherlands at Urbana, IL, USA This self-contained introduction to modern This is a book on Euclidean geometry that covers The “lost notebook” contains considerable cryptography emphasizes the mathematics the standard material in a completely new way, material on mock theta functions—undoubtedly behind the theory of public key cryptosystems while also introducing a number of new topics emanating from the last year of Ramanujan’s life. and digital signature schemes. The book focuses Interview with Martin Davis that would be suitable as a junior-senior level It should be emphasized that the material on on these key topics while developing the undergraduate textbook. The author does not mock theta functions is perhaps Ramanujan’s mathematical tools needed for the construction page 560 begin in the traditional manner with abstract deepest work more than half of the material in and security analysis of diverse cryptosystems. geometric axioms. Instead, he assumes the real the book is on q- series, including mock theta Only basic linear algebra is required of the numbers, and begins his treatment by functions; the remaining part deals with theta reader; techniques from algebra, number theory, introducing such modern concepts as a metric function identities, modular equations, and probability are introduced and developed as space, vector space notation, and groups, and incomplete elliptic integrals of the first kind and required. -
Mathematical Challenges of the 21St Century: a Panorama of Mathematics
comm-ucla.qxp 9/11/00 3:11 PM Page 1271 Mathematical Challenges of the 21st Century: A Panorama of Mathematics On August 7–12, 2000, the AMS held the meeting lenges was a diversity Mathematical Challenges of the 21st Century, the of views of mathemat- Society’s major event in celebration of World ics and of its connec- Mathematical Year 2000. The meeting took place tions with other areas. on the campus of the University of California, Los The Mathematical Angeles, and drew nearly 1,000 participants, who Association of America enjoyed the balmy coastal weather as well as held its annual summer the panorama of contemporary mathematics Mathfest on the UCLA Reception outside Royce Hall. provided in lectures by thirty internationally campus just prior to renowned mathematicians. the Mathematical Chal- This meeting was very different from other na- lenges meeting. On Sunday, August 6, a lecture by tional meetings organized by the AMS, with their master expositor Ronald L. Graham of the Uni- complicated schedules of lectures and sessions versity of California, San Diego, provided a bridge running in parallel. Mathematical Challenges was between the two meetings. Graham discussed a by comparison a streamlined affair, the main part number of unsolved problems that, like those of which comprised thirty plenary lectures, five per presented by Hilbert, have the intriguing combi- day over six days (there were also daily contributed nation of being simple to state while at the same paper sessions). Another difference was that the time being difficult to solve. Among the problems Mathematical Challenges speakers were encour- Graham talked about were Goldbach’s conjecture, aged to discuss the broad themes and major the “twin prime” conjecture, factoring of large outstanding problems in their areas rather than numbers, Hadwiger’s conjecture about chromatic their own research. -
EMS Newsletter No 38
CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS STEEN MARKVORSEN Department of Mathematics Technical University of Denmark Building 303 NEWSLETTER No. 38 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] December 2000 KRZYSZTOF CIESIELSKI Mathematics Institute Jagiellonian University EMS News: Reymonta 4 Agenda, Editorial, Edinburgh Summer School, London meeting .................. 3 30-059 Kraków, Poland e-mail: [email protected] KATHLEEN QUINN Joint AMS-Scandinavia Meeting ................................................................. 11 The Open University [address as above] e-mail: [email protected] The World Mathematical Year in Europe ................................................... 12 SPECIALIST EDITORS INTERVIEWS The Pre-history of the EMS ......................................................................... 14 Steen Markvorsen [address as above] SOCIETIES Krzysztof Ciesielski [address as above] Interview with Sir Roger Penrose ............................................................... 17 EDUCATION Vinicio Villani Interview with Vadim G. Vizing .................................................................. 22 Dipartimento di Matematica Via Bounarotti, 2 56127 Pisa, Italy 2000 Anniversaries: John Napier (1550-1617) ........................................... 24 e-mail: [email protected] MATHEMATICAL PROBLEMS Societies: L’Unione Matematica -
Taubes Receives NAS Award in Mathematics
Taubes Receives NAS Award in Mathematics Clifford H. Taubes has re- four-manifolds. The latter was, of course, crucial in ceived the 2008 NAS Award Donaldson’s celebrated application of gauge theory in Mathematics from the Na- to four-dimensional differential topology. Taubes tional Academy of Sciences. himself proved the existence of uncountably many He was honored “for ground- exotic differentiable structures on R4; he reinter- breaking work relating to preted Casson’s invariant in terms of gauge theory Seiberg-Witten and Gromov- and proved a homotopy approximation theorem Witten invariants of symplec- for Yang-Mills moduli spaces. Taubes also proved tic 4-manifolds, and his proof a powerful existence theorem for anti-self-dual of the Weinstein conjecture conformal structures on four-manifolds. for all contact 3-manifolds.” “When Witten proposed the study of the so- The NAS Award in Math- called Seiberg-Witten equations in 1994, Cliff ematics was established by Taubes was one of the handful of mathematicians the AMS in commemora- who quickly worked out the basics of the theory Clifford H. Taubes tion of its centennial, which and launched it on its meteoric path. From an was celebrated in 1988. The analyst’s point of view, the quasi-linear Seiberg- award is presented every four Witten equations may seem rather mundane, at years in recognition of excellence in research in least when compared to the challenges offered up the mathematical sciences published within the by the Yang-Mills equations. True to this spirit, past ten years. The award carries a cash prize of Taubes announced at the time that he would never US$5,000. -
AMS Officers and Committee Members
Of®cers and Committee Members Numbers to the left of headings are used as points of reference 1.1. Liaison Committee in an index to AMS committees which follows this listing. Primary All members of this committee serve ex of®cio. and secondary headings are: Chair Felix E. Browder 1. Of®cers Michael G. Crandall 1.1. Liaison Committee Robert J. Daverman 2. Council John M. Franks 2.1. Executive Committee of the Council 3. Board of Trustees 4. Committees 4.1. Committees of the Council 2. Council 4.2. Editorial Committees 4.3. Committees of the Board of Trustees 2.0.1. Of®cers of the AMS 4.4. Committees of the Executive Committee and Board of President Felix E. Browder 2000 Trustees Immediate Past President 4.5. Internal Organization of the AMS Arthur M. Jaffe 1999 4.6. Program and Meetings Vice Presidents James G. Arthur 2001 4.7. Status of the Profession Jennifer Tour Chayes 2000 4.8. Prizes and Awards H. Blaine Lawson, Jr. 1999 Secretary Robert J. Daverman 2000 4.9. Institutes and Symposia Former Secretary Robert M. Fossum 2000 4.10. Joint Committees Associate Secretaries* John L. Bryant 2000 5. Representatives Susan J. Friedlander 1999 6. Index Bernard Russo 2001 Terms of members expire on January 31 following the year given Lesley M. Sibner 2000 unless otherwise speci®ed. Treasurer John M. Franks 2000 Associate Treasurer B. A. Taylor 2000 2.0.2. Representatives of Committees Bulletin Donald G. Saari 2001 1. Of®cers Colloquium Susan J. Friedlander 2001 Executive Committee John B. Conway 2000 President Felix E. -
2018-06-108.Pdf
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Feature S E European Tensor Product and Semi-Stability M M Mathematical Interviews E S Society Peter Sarnak Gigliola Staffilani June 2018 Obituary Robert A. Minlos Issue 108 ISSN 1027-488X Prague, venue of the EMS Council Meeting, 23–24 June 2018 New books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House. Bogdan Nica (McGill University, Montreal, Canada) A Brief Introduction to Spectral Graph Theory (EMS Textbooks in Mathematics) ISBN 978-3-03719-188-0. 2018. 168 pages. Hardcover. 16.5 x 23.5 cm. 38.00 Euro Spectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions. -
Spectral Curves, and the Equations of Bogomolny and Nahm
Monopoles on R3, Spectral Curves, and the Equations of Bogomolny and Nahm Alexander B. Atanasov Abstract This paper introduces the Bogomolny equations, describing SU(n) monopoles of Yang- Mills-Higgs theory in three dimensional Euclidean space. After restricting to the case of SU(2), we describe the geometry of the monopole solution space via a scattering approach of Hitchin [1]. We demonstrate how monopole solutions give rise to a spectral curve of eigenval- 1 ues on T CP and how this may more clearly be understood in terms of a rational map idea of Donaldson [2]. We then introduce the Nahm equations of [3] as an application of the ADHM 3 idea to finding solutions to the self-duality conditions in the reduced case of R , and illustrate the equivalence of the Bogomolny and Nahm equations. Finally, we generalize these ideas by introducing the Nahm transform: a nonabelian generalization of the Fourier transform that relates the self-dual vector bundles on one space to vector bundles on another. Introduction The goal of this paper is to give the reader a gentle introduction to the notable discoveries in the study of monopoles in R3. In section 1, we give a review of the mathematics of gauge theory. We make use of of these techniques in section 2 to give two derivations of the Bogomolny equations. The first approach derives the equations directly from the anti-self-duality (ASD) conditions for instanton solutions in 4 R by treating the fourth component of the connection 1-form, A4, as a scalar field φ and ignoring translations @4 along the x4 direction. -
Dusa Mcduff, Grigory Mikhalkin, Leonid Polterovich, and Catharina Stroppel, Were Funded by MSRI‟S Eisenbud Endowment and by a Grant from the Simons Foundation
Final Report, 2005–2010 and Annual Progress Report, 2009–2010 on the Mathematical Sciences Research Institute activities supported by NSF DMS grant #0441170 October, 2011 Mathematical Sciences Research Institute Final Report, 2005–2010 and Annual Progress Report, 2009–2010 Preface Brief summary of 2005–2010 activities……………………………………….………………..i 1. Overview of Activities ............................................................................................................... 3 1.1 New Developments ............................................................................................................. 3 1.2 Summary of Demographic Data for 2009-10 Activities ..................................................... 7 1.3 Scientific Programs and their Associated Workshops ........................................................ 9 1.4 Postdoctoral Program supported by the NSF supplemental grant DMS-0936277 ........... 19 1.5 Scientific Activities Directed at Underrepresented Groups in Mathematics ..................... 21 1.6 Summer Graduate Schools (Summer 2009) ..................................................................... 22 1.7 Other Scientific Workshops .............................................................................................. 25 1.8 Educational & Outreach Activities ................................................................................... 26 a. Summer Inst. for the Pro. Dev. of Middle School Teachers on Pre-Algebra ....... 26 b. Bay Area Circle for Teachers .............................................................................. -
2008 Annual Report
Contents Clay Mathematics Institute 2008 Letter from the President James A. Carlson, President 2 Annual Meeting Clay Research Conference 3 Recognizing Achievement Clay Research Awards 6 Researchers, Workshops, Summary of 2008 Research Activities 8 & Conferences Profile Interview with Research Fellow Maryam Mirzakhani 11 Feature Articles A Tribute to Euler by William Dunham 14 The BBC Series The Story of Math by Marcus du Sautoy 18 Program Overview CMI Supported Conferences 20 CMI Workshops 23 Summer School Evolution Equations at the Swiss Federal Institute of Technology, Zürich 25 Publications Selected Articles by Research Fellows 29 Books & Videos 30 Activities 2009 Institute Calendar 32 2008 1 smooth variety. This is sufficient for many, but not all applications. For instance, it is still not known whether the dimension of the space of holomorphic q-forms is a birational invariant in characteristic p. In recent years there has been renewed progress on the problem by Hironaka, Villamayor and his collaborators, Wlodarczyck, Kawanoue-Matsuki, Teissier, and others. A workshop at the Clay Institute brought many of those involved together for four days in September to discuss recent developments. Participants were Dan Letter from the president Abramovich, Dale Cutkosky, Herwig Hauser, Heisuke James Carlson Hironaka, János Kollár, Tie Luo, James McKernan, Orlando Villamayor, and Jaroslaw Wlodarczyk. A superset of this group met later at RIMS in Kyoto at a Dear Friends of Mathematics, workshop organized by Shigefumi Mori. I would like to single out four activities of the Clay Mathematics Institute this past year that are of special Second was the CMI workshop organized by Rahul interest.