THE INSTITUTE for ADVANCED STUDY Prinoeton, New Jersey
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European Mathematical Society
CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF MARTIN RAUSSEN Department of Mathematical Sciences, Aalborg University Fredrik Bajers Vej 7G DK-9220 Aalborg, Denmark e-mail: [email protected] ASSOCIATE EDITORS VASILE BERINDE Department of Mathematics, University of Baia Mare, Romania NEWSLETTER No. 52 e-mail: [email protected] KRZYSZTOF CIESIELSKI Mathematics Institute June 2004 Jagiellonian University Reymonta 4, 30-059 Kraków, Poland EMS Agenda ........................................................................................................... 2 e-mail: [email protected] STEEN MARKVORSEN Editorial by Ari Laptev ........................................................................................... 3 Department of Mathematics, Technical University of Denmark, Building 303 EMS Summer Schools.............................................................................................. 6 DK-2800 Kgs. Lyngby, Denmark EC Meeting in Helsinki ........................................................................................... 6 e-mail: [email protected] ROBIN WILSON On powers of 2 by Pawel Strzelecki ........................................................................ 7 Department of Pure Mathematics The Open University A forgotten mathematician by Robert Fokkink ..................................................... 9 Milton Keynes MK7 6AA, UK e-mail: [email protected] Quantum Cryptography by Nuno Crato ............................................................ 15 COPY EDITOR: KELLY -
THE RIEMANN HYPOTHESIS BERNARD RUSSO University Of
THE RIEMANN HYPOTHESIS BERNARD RUSSO University of California, Irvine PART I THE PRIME NUMBER THEOREM JULY 29, 2010 PACIFIC SUMMER UNSOLVED PROBLEMS SEMINAR PART II THE RIEMANN HYPOTHESIS SEPTEMBER 14, 2010 MATH COLLOQUIUM FULLERTON COLLEGE DEPARTMENT OF MATHEMATICS BEGINNING OF PART I “book report” “PRIME OBSESSION” JOHN DERBYSHIRE 2003 PRESENTATION OF CREDENTIALS Bernard Russo 1965 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 L. Fejer 1902 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 L. Fejer 1902 H. A. Schwarz 1864 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 L. Fejer 1902 H. A. Schwarz 1864 Karl Weierstrass 1854—E. E. Kummer 1831 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 L. Fejer 1902 H. A. Schwarz 1864 Karl Weierstrass 1854—E. E. Kummer 1831 C. Gudermann 1841—H. Scherk 1823 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 L. Fejer 1902 H. A. Schwarz 1864 Karl Weierstrass 1854—E. E. Kummer 1831 C. Gudermann 1841—H. -
Irving Kaplansky
Portraying and remembering Irving Kaplansky Hyman Bass University of Michigan Mathematical Sciences Research Institute • February 23, 2007 1 Irving (“Kap”) Kaplansky “infinitely algebraic” “I liked the algebraic way of looking at things. I’m additionally fascinated when the algebraic method is applied to infinite objects.” 1917 - 2006 A Gallery of Portraits 2 Family portrait: Kap as son • Born 22 March, 1917 in Toronto, (youngest of 4 children) shortly after his parents emigrated to Canada from Poland. • Father Samuel: Studied to be a rabbi in Poland; worked as a tailor in Toronto. • Mother Anna: Little schooling, but enterprising: “Health Bread Bakeries” supported (& employed) the whole family 3 Kap’s father’s grandfather Kap’s father’s parents Kap (age 4) with family 4 Family Portrait: Kap as father • 1951: Married Chellie Brenner, a grad student at Harvard Warm hearted, ebullient, outwardly emotional (unlike Kap) • Three children: Steven, Alex, Lucy "He taught me and my brothers a lot, (including) what is really the most important lesson: to do the thing you love and not worry about making money." • Died 25 June, 2006, at Steven’s home in Sherman Oaks, CA Eight months before his death he was still doing mathematics. Steven asked, -“What are you working on, Dad?” -“It would take too long to explain.” 5 Kap & Chellie marry 1951 Family portrait, 1972 Alex Steven Lucy Kap Chellie 6 Kap – The perfect accompanist “At age 4, I was taken to a Yiddish musical, Die Goldene Kala. It was a revelation to me that there could be this kind of entertainment with music. -
HPM Newsletter 56 July 2004
ICME-10 Satellite meeting of HPM, to be No. 56 July 2004 held in Uppsala (authors: Florence Fasanelli HPM Advisory Board: and John Fauvel, 2004). Fulvia Furinghetti, Chair Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy Peter Ransom, Editor ([email protected]), The Mountbatten School and Language College, The interest for the use of history in Romsey, SO51 5SY, UK mathematics education has remote roots in the Masami Isoda, Webmaster work of famous historians such as Florian ([email protected]), Institute of Cajori, David Eugene Smith, Gino Loria and Education, University of Tsukuba, 305-8572 JAPAN Hieronymus Georg Zeuthen. In recent times the ideas outlined in a theoretical way by Jan van Maanen, The Netherlands, (former chair); those important historians of the past had Florence Fasanelli, USA, (former chair); Ubiratan D’Ambrosio, Brazil, (former chair) interesting applications in the classroom. Evelyne Barbin, France Teaching experiments are discussed in Luis Radford, Canada specific studies and doctoral dissertations are Gert Schubring, Germany written all over the world. All that assists in making the links of history and mathematics Report by HPM: The International education more rigorous and fruitful. Study Group on the Relations between the History and At the end of my four years (2000-2004) as Pedagogy of Mathematics the chairperson of HPM, I browse through my memories and the HPM Newsletter issues to pick up information on the activities of HPM HPM Activities 2000-2004 and, more generally, on relevant events related to the links between history and Together with the PME group (International pedagogy in mathematics. -
Irving Ezra Segal (1918–1998)
mem-segal.qxp 5/12/99 12:57 PM Page 659 Irving Ezra Segal (1918–1998) John C. Baez, Edwin F. Beschler, Leonard Gross, Bertram Kostant, Edward Nelson, Michèle Vergne, and Arthur S. Wightman Irving Segal died suddenly on August 30, 1998, After the war while taking an evening walk. He was seventy-nine Segal spent two and was vigorously engaged in research. years at the Insti- Born on September 13, 1918, in the Bronx, he tute for Advanced grew up in Trenton and received his A.B. from Study, where he Princeton in 1937. What must it have been like to held the first of the be a member of the Jewish quota at Princeton in three Guggenheim the 1930s? He told me once that a fellow under- Fellowships that he graduate offered him money to take an exam in his was to win. Other stead and was surprised when Irving turned him honors included down. election to the Na- He received his Ph.D. from Yale in 1940. His the- tional Academy of sis was written under the nominal direction of Sciences in 1973 Einar Hille, who suggested that Segal continue his and the Humboldt Award in 1981. At and Tamarkin’s investigation of the ideal theory the University of of the algebra of Laplace-Stieltjes transforms ab- Chicago from 1948 solutely convergent in a fixed half-plane. But, Segal to 1960, he had fif- wrote, “For conceptual clarification and for other teen doctoral stu- reasons, an investigation of the group algebra of dents, and at MIT, a general [locally compact] abelian group was of where he was pro- Irving Segal interest.” And the thesis was not restricted to fessor from 1960 abelian groups. -
The Legacy of Norbert Wiener: a Centennial Symposium
http://dx.doi.org/10.1090/pspum/060 Selected Titles in This Series 60 David Jerison, I. M. Singer, and Daniel W. Stroock, Editors, The legacy of Norbert Wiener: A centennial symposium (Massachusetts Institute of Technology, Cambridge, October 1994) 59 William Arveson, Thomas Branson, and Irving Segal, Editors, Quantization, nonlinear partial differential equations, and operator algebra (Massachusetts Institute of Technology, Cambridge, June 1994) 58 Bill Jacob and Alex Rosenberg, Editors, K-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara, July 1992) 57 Michael C. Cranston and Mark A. Pinsky, Editors, Stochastic analysis (Cornell University, Ithaca, July 1993) 56 William J. Haboush and Brian J. Parshall, Editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, Editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, Editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, Editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, Editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, Editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, Editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, Editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. -
Quantization, Nonlinear Partial Differential Equations,And Operator
http://dx.doi.org/10.1090/pspum/059 Other Titles in This Series 59 William Arveson, Thomas Branson, and Irving Segal, editors, Quantization, nonlinear partial differential equations, and operator algebra (Massachusetts Institute of Technology, Cambridge, June 1994) 58 Bill Jacob and Alex Rosenberg, editors, K-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara, July 1992) 57 Michael C. Cranston and Mark A. Pinsky, editors, Stochastic analysis (Cornell University, Ithaca, July 1993) 56 William J. Haboush and Brian J. Parshall, editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. O. Wells, Jr., editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987) 47 Paul Fong, editor, The Arcata conference on representations of finite groups (Humboldt State University, Arcata, California, July 1986) 46 Spencer J. -
Publications of Irving Segal Papers
Publications of Irving Segal Papers [1] Fiducial distribution of several parameters with application to a normal system, Proc. Camb. Philos. Soc. , 34 (1938), 41–47. Zbl 018.15703 [2] The automorphisms of the symmetric group, Bull. Amer. Math. Soc. 46 (1940), 565. MR 2 #1c Zbl 061.03301 [3] The group ring of a locally compact group. I, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 348–352. MR 3 #36b Zbl 063.06858 [4] The span of the translations of a function in a Lebesgue space, Proc. Nat. Acad. Sci. U. S. A. 30 (1944), 165–169. MR 6 #49c Zbl 063.06859 [5] Topological groups in which multiplication of one side is differen- tiable, Bull. Amer. Math. Soc. 52 (1946), 481–487. MR 8 #132d Zbl 061.04411 [6] Semi-groups of operators and the Weierstrass theorem, Bull. Amer. Math. Soc. 52 (1946), 911–914. (with Nelson Dunford) MR 8 #386e Zbl 061.25307 [7] The group algebra of a locally compact group, Trans. Amer. Math. Soc. 61 (1947), 69–105. MR 8 #438c Zbl 032.02901 [8] Irreducible representations of operator algebras, Bull. Amer. Math. Soc. 53 (1947), 73–88. MR 8 #520b Zbl 031.36001 [9] The non-existence of a relation which is valid for all finite groups, Bol. Soc. Mat. S˜aoPaulo 2 (1947), no. 2, 3–5 (1949). MR 13 #316b [10] Postulates for general quantum mechanics, Ann. of Math. (2) 48 (1947), 930–948. MR 9 #241b Zbl 034.06602 [11] Invariant measures on locally compact spaces, J. Indian Math. Soc. -
The 3-Fold Way and Consciousness Studies K. Korotkov, A. Levichev
© K. Korotkov, A. Levichev The 3-fold Way and Consciousness Studies K. Korotkov, A. Levichev Contents Part I. Biological fields and Quantum Mechanical representations I.1. Conventional Quantum Mechanical representations. I.2. Chronometric development of QM and the DLF-perspective. I.3. Fields of biological subjects. Part II. Penrose-Hameroff approach to quantum mechanics as the foundation for a theory of consciousness II.1. Discussion of some quantum-mechanical topics involved II.2. Quantum coherence, quantum computation, and where to seek the physical basis of mind II.3. The Penrose-Hameroff Orchestrated Objective Reduction model II.4. DLF-approach implanted into Penrose-Hameroff model Part III. Segal’s Chronometry and its LF-development III.1. Segal’s Chronometric Theory: a brief overview. III.2. Space-times L, F are on equal footing with D; the list is now complete III.3. Can the New Science be based on the DLF-triad? Part IV. Emergence of New Science and the GDV Bioelectrography IV.1. The three worlds have been known to humanity since ancient times IV.2. Is Direct Vision an example of L-phenomenon? I.1. Conventional Quantum Mechanical representations. Accordingly to quantum mechanics, each object is described by its state, or wave function. We prefer to use “state”, since (initially, at least) it is neither numerical-, nor vector-valued. Rather, it is a section of an induced vector bundle over space- time. It can be converted into a function (with values in a prescribed “spin space”) but one needs to go through the “parallelization” procedure (see our III.2, III.3 for more details). -
Who Is Lennart Carleson?
The Abel Prize for 2006 has been awarded Lennart Carleson, The Royal Institute of Technology, Stockholm, Sweden Professor Lennart Carleson will accept the Abel Prize for 2006 from His Majesty King Harald in a ceremony at the University of Oslo Aula, at 2:00 p.m. on Tuesday, 23 May 2006. In this background paper, we shall give a description of Carleson and his work. This description includes precise discussions of his most important results and attempts at popular presentations of those same results. In addition, we present the committee's reasons for choosing the winner and Carleson's personal curriculum vitae. This paper has been written by the Abel Prize's mathematics spokesman, Arne B. Sletsjøe, and is based on the Abel Committee's deliberations and prior discussion, relevant technical literature and discussions with members of the Abel Committee. All of this material is available for use by the media, either directly or in an adapted version. Oslo, Norway, 23 March 2006 Contents Introduction .................................................................................................................... 2 Who is Lennart Carleson? ............................................................................................... 3 Why has he been awarded the Abel Prize for 2006? ........................................................ 4 Popular presentations of Carleson's results ...................................................................... 6 Convergence of Fourier series.................................................................................... -
The Measure Algebra of the Heisenberg Group* L
Journal of Functional Analysis 161, 509525 (1999) Article ID jfan.1998.3354, available online at http:ÂÂwww.idealibrary.com on The Measure Algebra of the Heisenberg Group* L. A. Coburn Department of Mathematics, State University of New York, Buffalo, New York 14214 Communicated by Irving Segal Received June 16, 1998; revised August 28, 1998; accepted August 28, 1998 Irreducible representations of the convolution algebra M(Hn) of bounded regular complex Borel measures on the Heisenberg group Hn are analyzed. For the SegalBargmann representation \, the C*-algebra generated by \[M(Hn)] is just the C*-algebra generated by BerezinToeplitz operators with positive-definite ``symbols.'' This algebra is a deformation of the sup norm algebra generated by n View metadata,positive-definite citation and functions similar on papers complex at core.ac.ukn-space C . 1999 Academic Press brought to you by CORE provided by Elsevier - Publisher Connector 1. INTRODUCTION The Heisenberg group Hn is a nilpotent Lie group which plays a critical role in quantum mechanics and a significant role in representation theory. The irreducible infinitedimensional unitary representations of Hn satisfy the ``canonical commutation relations.'' Recently, [BC1 ,BC2 ,BC3 ,] provided an analysis of the C*-algebra generated by these unitary operators using the complex-function theory tools available in the SegalBargmann repre- sentation space. This work suggests an interesting connection between the convolution algebra M(Hn) of bounded regular Borel measures on Hn and the BerezinToeplitz quantization. In particular, for the SegalBargmann representation \, the C*-algebra generated by \[M(Hn)] is just the C*- algebra generated by BerezinToeplitz operators with positive-definite ``symbols.'' This algebra is a deformation of the sup norm algebra generated by positive-definite functions on complex n-space Cn. -
Projective Loop Quantum Gravity II. Searching for Semi-Classical States
Projective Loop Quantum Gravity II. Searching for1,2 Semi-Classical States1 Suzanne Lanéry and homas hiemann 1 Institute for Quantum Gravity, Friedrich-Alexander University Erlangen-Nürnberg, Germany 2 Mathematics and #heoretical Physics Laboratory, François-Rabelais University of #ours, France October 7, 2015 Abstract In [13] an extension of the Ashtekar-Lewandowski (AL) state space [2] of %oop Quantum Gravity was set up with the help a projective formalism introduced by 5ijowski [12, 21, 15]. #he motivation for this work was to achieve a more balanced treatment of the position and momentum variables (aka. holonomies and fluxes). Indeed, states in the AL Hilbert spaces describe discrete quantum excitations on top of a vacuum which is an eigenstate of the flux variables (a ‘no-geometry’ state): in such states, most holonomies are totally spread, making it difficult to approximate a smooth, classical 4-geometry. However, going beyond the AL sector does not fully resolve this difficulty: one uncovers a deeper issue hindering the constructionSU of states semi-classical with respect to a full set of observables. In the present article, we analyze this issue in the case of real-valued holonomies 1we will briefly comment on the heuristic implications for other gauge groups, eg. (2) 26 Specifically, we sho 0 that, in this case, there does not exist any state on the holonomy-flux algebra in which the variances of the holonomies and fluxes observables would all be finite, let alone small6 It is important to note that this obstruction cannot be bypassed by further enlarging the quantum state space, for it arises from the structure of the algebra itself: as there are too many (uncountably many) non-vanishing commutators between the holonomy and flux operators, the corresponding Heisenberg inequalities force the quantum uncertainties to blow up uncontrollably.