Georg Friedrich Bernhard Riemann

Total Page：16

File Type：pdf, Size：1020Kb

Georg Friedrich Bernhard Riemann By: Supervised: Sandra Hanbo Dr. Vágó Zsuzsanna Biography • Born in 17 September 1826 in Breselenz, Kingdom of Hannover (Germany now) • Father : Friedrich Bernhard Riemann. Poor Lutheran pastor • Mother: Charlotte Ebell • Wife: Elise Koch and they had a daughter Elda • Universities : University of Göttingen (1846) studying math under Gauss, University of Berlin (1847) continuing study by : Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein • Doctoral supervisor : Carl Friedrich Gauss • Honors awarded to Bernhard Riemann 1. Fellow of the Royal Society: 1866 2. Lunar features: Crater Riemann 3. Popular biographies list: Number 18 • Dead : 20 July 1866 in Selasca (Italy) Some of his Articles • Basics for a general theory of functions of a changeable complex Size.- 1851 • On the Number of Primes Less Than a Given –1859 • On the laws of the distribution of voltage electricity in ponderable bodies, if these are not regarded as perfect conductors or non-conductors, but as reluctant to contain voltage electricity with finite force- - 1854 • On the theory of Nobili's color rings -1855 • Contributions to the theory of the functions that can be represented by the Gaussian series F (α, β, γ,( .- 1857 • Theory of Abel's functions -1857 • About the disappearance of the theta functions - 1866 Some topics named after Riemann • Cauchy–Riemann equations • Riemann form • Riemann Geometry • Riemann mapping theorem • Riemann problem • Riemann surface • Riemann solver • Riemann's differential equation • Riemann's explicit formula Riemann surface for f(z) = z1/2. Image by Leonid 2. • Riemann's minimal surface Famous Scientists – Bernard Riemann And others … References: • Articles: • "Bernhard Riemann." Famous Scientists. famousscientists.org. 28 May. 2017. Web. 4/7/2020 - <Link>. • Chai, Ching-Li , Riemann Forms, 12 March 2014- <Link> • E. PORTNOY ,Riemann’s contribution to differential geometry, Doi: 10.1016/0315-0860(82)90134-3 • Feldmann, Joel - The Cauchy–Riemann Equations - 19 January 2012- <Link> • Garrett, Paul- Riemann’s Explicit/Exact formula - 02 October 2010 - <Link> • H. FREUDENTHAL, Biography in Dictionary of Scientific Biography (New York 1970-1990). • Kim, Kang-Tae, The Riemann mapping theorem from Riemann's viewpoint- April 2016, Doi: 10.1186/s40627- 016-0009-7 • Meeks III, William H. , Pérez, Joaquín, The Riemann minimal examples,12 March 2018 ,<Link> • Books: • Laugwitz, Detlef, Bernard Riemann Book , ISBN-13: 978-0-8176-4776-6 • Toro, E.F. , Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, ISBN-13: 9783540498346 • Riemannian Geometry , <Link> • Websites: • Britannica Encyclopedia • http://totallyhistory.com/bernhard-riemann/ • Wikipedia • https://www.claymath.org/sites/default/files/zeta.pdf • https://dlmf.nist.gov/15.11 • https://www.emis.de/classics/Riemann/ • http://mathshistory.st-andrews.ac.uk/Biographies/Riemann.html Thank you for attention Sandra Hanbo.

Copy Link

Recommended publications

The Cambridge Mathematical Journal and Its Descendants: the Linchpin of a Research Community in the Early and Mid-Victorian Age ✩
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematica 31 (2004) 455–497 www.elsevier.com/locate/hm The Cambridge Mathematical Journal and its descendants: the linchpin of a research community in the early and mid-Victorian Age ✩ Tony Crilly ∗ Middlesex University Business School, Hendon, London NW4 4BT, UK Received 29 October 2002; revised 12 November 2003; accepted 8 March 2004 Abstract The Cambridge Mathematical Journal and its successors, the Cambridge and Dublin Mathematical Journal,and the Quarterly Journal of Pure and Applied Mathematics, were a vital link in the establishment of a research ethos in British mathematics in the period 1837–1870. From the beginning, the tension between academic objectives and economic viability shaped the often precarious existence of this line of communication between practitioners. Utilizing archival material, this paper presents episodes in the setting up and maintenance of these journals during their formative years. 2004 Elsevier Inc. All rights reserved. Résumé Dans la période 1837–1870, le Cambridge Mathematical Journal et les revues qui lui ont succédé, le Cambridge and Dublin Mathematical Journal et le Quarterly Journal of Pure and Applied Mathematics, ont joué un rôle essentiel pour promouvoir une culture de recherche dans les mathématiques britanniques. Dès le début, la tension entre les objectifs intellectuels et la rentabilité économique marqua l’existence, souvent précaire, de ce moyen de communication entre professionnels. Sur la base de documents d’archives, cet article présente les épisodes importants dans la création et l’existence de ces revues. 2004 Elsevier Inc.
[Show full text]
Elliptic Systems on Riemann Surfaces
Lecture Notes of TICMI, Vol.13, 2012 Giorgi Akhalaia, Grigory Giorgadze, Valerian Jikia, Neron Kaldani, Giorgi Makatsaria, Nino Manjavidze ELLIPTIC SYSTEMS ON RIEMANN SURFACES © Tbilisi University Press Tbilisi Abstract. Systematic investigation of Elliptic systems on Riemann surface and new results from the theory of generalized analytic functions and vectors are pre- sented. Complete analysis of the boundary value problem of linear conjugation and Riemann-Hilbert monodromy problem for the Carleman-Bers-Vekua irregular systems are given. 2000 Mathematics Subject Classi¯cation: 57R45 Key words and phrases: Generalized analytic function, Generalized analytic vector, irregular Carleman-Bers-Vekua equation, linear conjugation problem, holo- morphic structure, Beltrami equation Giorgi Akhalaia, Grigory Giorgadze*, Valerian Jikia, Neron Kaldani I.Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University 2 University st., Tbilisi 0186, Georgia Giorgi Makatsaria St. Andrews Georgian University 53a Chavchavadze Ave., Tbilisi 0162, Georgia Nino Manjavidze Georgian Technical University 77 M. Kostava st., Tbilisi 0175, Georgia * Corresponding Author. email: [email protected] Contents Preface 5 1 Introduction 7 2 Functional spaces induced from regular Carleman-Bers-Vekua equations 15 2.1 Some functional spaces . 15 2.2 The Vekua-Pompej type integral operators . 16 2.3 The Carleman-Bers-Vekua equation . 18 2.4 The generalized polynomials of the class Ap;2(A; B; C), p > 2 . 20 2.5 Some properties of the generalized power functions . 22 2.6 The problem of linear conjugation for generalized analytic functions . 26 3 Beltrami equation 28 4 The pseudoanalytic functions 36 4.1 Relation between Beltrami and holomorphic disc equations . 38 4.2 The periodicity of the space of generalized analytic functions .
[Show full text]
Early History of the Riemann Hypothesis in Positive Characteristic
The Legacy of Bernhard Riemann c Higher Education Press After One Hundred and Fifty Years and International Press ALM 35, pp. 595–631 Beijing–Boston Early History of the Riemann Hypothesis in Positive Characteristic Frans Oort∗ , Norbert Schappacher† Abstract The classical Riemann Hypothesis RH is among the most prominent unsolved prob- lems in modern mathematics. The development of Number Theory in the 19th century spawned an arithmetic theory of polynomials over finite fields in which an analogue of the Riemann Hypothesis suggested itself. We describe the history of this topic essentially between 1920 and 1940. This includes the proof of the ana- logue of the Riemann Hyothesis for elliptic curves over a finite field, and various ideas about how to generalize this to curves of higher genus. The 1930ies were also a period of conflicting views about the right method to approach this problem. The later history, from the proof by Weil of the Riemann Hypothesis in charac- teristic p for all algebraic curves over a finite field, to the Weil conjectures, proofs by Grothendieck, Deligne and many others, as well as developments up to now are described in the second part of this diptych: [44]. 2000 Mathematics Subject Classification: 14G15, 11M99, 14H52. Keywords and Phrases: Riemann Hypothesis, rational points over a finite field. Contents 1 From Richard Dedekind to Emil Artin 597 2 Some formulas for zeta functions. The Riemann Hypothesis in characteristic p 600 3 F.K. Schmidt 603 ∗Department of Mathematics, Utrecht University, Princetonplein 5, 3584 CC
[Show full text]
Bernhard Riemann 1826-1866
Modern Birkh~user Classics Many of the original research and survey monographs in pure and applied mathematics published by Birkh~iuser in recent decades have been groundbreaking and have come to be regarded as foun- dational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain ac- cessible to new generations of students, scholars, and researchers. BERNHARD RIEMANN (1826-1866) Bernhard R~emanno 1826 1866 Turning Points in the Conception of Mathematics Detlef Laugwitz Translated by Abe Shenitzer With the Editorial Assistance of the Author, Hardy Grant, and Sarah Shenitzer Reprint of the 1999 Edition Birkh~iuser Boston 9Basel 9Berlin Abe Shendtzer (translator) Detlef Laugwitz (Deceased) Department of Mathematics Department of Mathematics and Statistics Technische Hochschule York University Darmstadt D-64289 Toronto, Ontario M3J 1P3 Gernmany Canada Originally published as a monograph ISBN-13:978-0-8176-4776-6 e-ISBN-13:978-0-8176-4777-3 DOI: 10.1007/978-0-8176-4777-3 Library of Congress Control Number: 2007940671 Mathematics Subject Classification (2000): 01Axx, 00A30, 03A05, 51-03, 14C40 9 Birkh~iuser Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkh~user Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter de- veloped is forbidden.
[Show full text]
1 Complex Theory of Abelian Varieties
1 Complex Theory of Abelian Varieties Definition 1.1. Let k be a field. A k-variety is a geometrically integral k-scheme of finite type. An abelian variety X is a proper k-variety endowed with a structure of a k-group scheme. For any point x X(k) we can defined a translation map Tx : X X by 2 ! Tx(y) = x + y. Likewise, if K=k is an extension of fields and x X(K), then 2 we have a translation map Tx : XK XK defined over K. ! Fact 1.2. Any k-group variety G is smooth. Proof. We may assume that k = k. There must be a smooth point g0 G(k), and so there must be a smooth non-empty open subset U G. Since2 G is covered by the G(k)-translations of U, G is smooth. ⊆ We now turn our attention to the analytic theory of abelian varieties by taking k = C. We can view X(C) as a compact, connected Lie group. Therefore, we start off with some basic properties of complex Lie groups. Let X be a Lie group over C, i.e., a complex manifold with a group structure whose multiplication and inversion maps are holomorphic. Let T0(X) denote the tangent space of X at the identity. By adapting the arguments from the C1-case, or combining the C1-case with the Cauchy-Riemann equations we get: Fact 1.3. For each v T0(X), there exists a unique holomorphic map of Lie 2 δ groups φv : C X such that (dφv)0 : C T0(X) satisfies (dφv)0( 0) = v.
[Show full text]
Riemann Forms
Riemann Forms Ching-Li Chai∗ The notion of Riemann forms began as a coordinate-free reformulation of the Riemann period relations for abelian functions discovered by Riemann. This concept has evolved with the progress in mathematics in the 150 years after Riemann. Nowadays it is often viewed from the perspective of abstract Hodge theory: a Riemann form gives an alternating pairing on a (pure) Hodge Z-structure of Hodge type f(0;−1);(−1;0)g and weight −1, with values in the “Tate twisted” version Z(1) of Z. Along more algebraic lines, the concept of Riemann forms was extended to abelian varieties over arbitrary base fields, after Weil [8] developed the theory of abelian varieties over an arbitrary base fields. For any (co)homology theory, for instance de Rham, Hodge, etale´ or crystalline cohomology theory, a Riemann form for an abelian variety is an alternating pairing induced by a polarization of A, on the first homology group H1(A) of an abelian variety A, with values in H1(Gm), the first Tate twist of the homology of a point. A synopsis for the case of etale´ cohomology is given in x2. x1. From Riemann matrices to Riemann forms A version of theorem 1.1 appeared in [3, pp. 148–150], preceded by the high praise “il est extremement` remarquable est c’est a` M. le Dr. Riemann, de Gottingue,¨ qu’on doit cette decouverte´ analytique . ” from Hermite. This theorem is stated in terms of period matrices for abelian functions in the article on Riemann bilinear relations; see [7, Ch.
[Show full text]
Of Triangles, Gas, Price, and Men
OF TRIANGLES, GAS, PRICE, AND MEN Cédric Villani Univ. de Lyon & Institut Henri Poincaré « Mathematics in a complex world » Milano, March 1, 2013 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemannian (= non-Euclidean) geometry At each location, the units of length and angles may change Shortest path (= geodesics) are curved!! Geodesics can tend to get closer (positive curvature, fat triangles) or to get further apart (negative curvature, skinny triangles) Hyperbolic surfaces Bernhard Riemann 1826-1866 List of topics named after Bernhard Riemann From Wikipedia, the free encyclopedia Riemann singularity theorem Cauchy–Riemann equations Riemann solver Compact Riemann surface Riemann sphere Free Riemann gas Riemann–Stieltjes integral Generalized Riemann hypothesis Riemann sum Generalized Riemann integral Riemann surface Grand Riemann hypothesis Riemann theta function Riemann bilinear relations Riemann–von Mangoldt formula Riemann–Cartan geometry Riemann Xi function Riemann conditions Riemann zeta function Riemann curvature tensor Zariski–Riemann space Riemann form Riemannian bundle metric Riemann function Riemannian circle Riemann–Hilbert correspondence Riemannian cobordism Riemann–Hilbert problem Riemannian connection Riemann–Hurwitz formula Riemannian cubic polynomials Riemann hypothesis Riemannian foliation Riemann hypothesis for finite fields Riemannian geometry Riemann integral Riemannian graph Bernhard
[Show full text]
Chapter 6 Riemann Solvers I
Chapter 6 Riemann solvers I The numerical hydrodynamics algorithms we have devised in Chapter 5 were based on the idea of operator splitting between the advection and pressure force terms. The advection was done, for all conserved quantities, using the gas velocity, while the pressure force and work terms were treated as source terms. From Chapter 2 we know, however, thatthecharacteristicsoftheEuler equations are not necessarily all equal to the gas velocity. We have seen that there exist an eigen- vector which indeed has the gas velocity as eigenvectors, λ0 = u,buttherearetwoeigenvectors which have eigenvalues λ = u C which belong to the forward and backward sound propa- ± ± s gation. Mathematically speaking one should do the advectioninthesethreeeigenvectors,using their eigenvalues as advection velocity. The methods in Chapter 5 do not do this. By extracting the pressure terms out of the advection part and adding them asasourceterm,theadvectionpart has been reduced essentially to Burger’s equation,andthepropagationofsoundwavesisentirely driven by the addition of the source terms. Such methods therefore do not formally propagate the sound waves using advection, even though mathematically they should be. All the effort we have done in Chapters 3 and 4 to create the best advection schemes possible will therefore have no effect on the propagation of sound waves. Once could say that for two out of three characteristics our ingeneous advection scheme is useless. Riemann solvers on the other hand keep the pressure terms within the to-be-advected sys- tem. There is no pressure source term in these equations. The mathematical character of the equations remains intact. Such solvers therefore propagateallthecharacteristicsonequalfoot- ing.
[Show full text]
Quasiconformal Mappings, from Ptolemy's Geography to the Work Of
Quasiconformal mappings, from Ptolemy’s geography to the work of Teichmüller Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Quasiconformal mappings, from Ptolemy’s geography to the work of Teich- müller. 2016. hal-01465998 HAL Id: hal-01465998 https://hal.archives-ouvertes.fr/hal-01465998 Preprint submitted on 13 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. QUASICONFORMAL MAPPINGS, FROM PTOLEMY'S GEOGRAPHY TO THE WORK OF TEICHMULLER¨ ATHANASE PAPADOPOULOS Les hommes passent, mais les œuvres restent. (Augustin Cauchy, [204] p. 274) Abstract. The origin of quasiconformal mappings, like that of confor- mal mappings, can be traced back to old cartography where the basic problem was the search for mappings from the sphere onto the plane with minimal deviation from conformality, subject to certain conditions which were made precise. In this paper, we survey the development of cartography, highlighting the main ideas that are related to quasicon- formality. Some of these ideas were completely ignored in the previous historical surveys on quasiconformal mappings. We then survey early quasiconformal theory in the works of Grötzsch, Lavrentieff, Ahlfors and Teichmüller,which are the 20th-century founders of the theory.
[Show full text]
LONG-TERM HISTORY and EPHEMERAL CONFIGURATIONS Catherine Goldstein
LONG-TERM HISTORY AND EPHEMERAL CONFIGURATIONS Catherine Goldstein To cite this version: Catherine Goldstein. LONG-TERM HISTORY AND EPHEMERAL CONFIGURATIONS. Interna- tional Congress of Mathematicians, Aug 2018, Rio de Janeiro, Brazil. pp.487-522. hal-02334505 HAL Id: hal-02334505 https://hal.archives-ouvertes.fr/hal-02334505 Submitted on 29 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LONG-TERM HISTORY AND EPHEMERAL CONFIGURATIONS CATHERINE GOLDSTEIN Abstract. Mathematical concepts and results have often been given a long history, stretching far back in time. Yet recent work in the history of mathe- matics has tended to focus on local topics, over a short term-scale, and on the study of ephemeral configurations of mathematicians, theorems or practices. The first part of the paper explains why this change has taken place: a renewed interest in the connections between mathematics and society, an increased at- tention to the variety of components and aspects of mathematical work, and a critical outlook on historiography itself. The problems of a long-term history are illustrated and tested using a number of episodes in the nineteenth-century history of Hermitian forms, and finally, some open questions are proposed.
[Show full text]
Gotthold Eisenstein and Philosopher John
Gotthold Eisenstein and Philosopher John Franz Lemmermeyer Abstract Before the recent publication of the correspondence between Gauss and Encke, nothing was known about the role that John Taylor, a cotton merchant from Liverpool, had played in the life of Gotthold Eisenstein. In this article, we will bring together what we have discovered about John Taylor’s life. Eisenstein’s Journey to England Gotthold Eisenstein belonged, together with Dirichlet, Jacobi and Kummer, to the generation after Gauss that shaped the theory of numbers in the mid- 19th century, and like Galois, Abel, Riemann, Roch and Clebsch, Eisenstein died young. Today, Eisenstein’s name can be found in the Eisenstein series, Eisenstein sums, the Eisenstein ideal, Eisenstein’s reciprocity law and in his irreducibility criterion, and he is perhaps best known for his ingenious proofs of the quadratic, cubic and biquadratic reciprocity laws. Eisenstein’s father Jo- hann Konstantin Eisenstein emigrated to England in 1840; Eisenstein and his mother followed in June 1842, although Eisenstein’s few remarks on this episode in his autobiography [3] belie the dramatic events that he experienced in Eng- land. On their journey to England, the Eisensteins passed through Hamburg; during the Great Fire in May 1842 about a third of the houses in the Altstadt had burned down. What we learn from Eisenstein’s account is that he was impressed by the sight of railroad lines running right under the foundations of houses (in London?) and by the Menai suspension bridge in Wales: Eisenstein mentions that he undertook six sea voyages, and that on one of them they sailed arXiv:2101.03344v1 [math.HO] 9 Jan 2021 under the tremendous suspension bridge in Anglesey, which was so high that the Berlin Palace would easily have ﬁtted under its main arch.
[Show full text]
Riemann, Schottky, and Schwarz on Conformal Representation
Appendix 1 Riemann, Schottky, and Schwarz on Conformal Representation In his Thesis [1851], Riemann sought to prove that if a function u satisfies cer- tain conditions on the boundary of a surface T, then it is the real part of a unique complex function which can be defined on the whole of T. He was then able to show that any two simply connected surfaces (other than the complex plane itself Riemann was considering bounded surfaces) can be mapped conformally onto one another, and that the map is unique once the images of one boundary point and one interior point are specified; he claimed analogous results for any two surfaces of the same connectivity [w To prove that any two simply connected surfaces are conformally equivalent, he observed that it is enough to take for one surface the unit disc K = {z : Izl _< l} and he gave [w an account of how this result could be proved by means of what he later [1857c, 103] called Dirichlet's principle. He considered [w (i) the class of functions, ~., defined on a surface T and vanishing on the bound- ary of T, which are continuous, except at some isolated points of T, for which the integral t= rx + ry dt is finite; and (ii) functions ct + ~. = o9, say, satisfying dt=f2 < oo Ox Oy + for fixed but arbitrary continuous functions a and ft. 224 Appendix 1. Riemann, Schottky, and Schwarz on Conformal Representation He claimed that f2 and L vary continuously with varying ,~. but cannot be zero, and so f2 takes a minimum value for some o9.
[Show full text]