The ICM Through History
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Algebraic Generality Vs Arithmetic Generality in the Controversy Between C
Algebraic generality vs arithmetic generality in the controversy between C. Jordan and L. Kronecker (1874). Frédéric Brechenmacher (1). Introduction. Throughout the whole year of 1874, Camille Jordan and Leopold Kronecker were quarrelling over two theorems. On the one hand, Jordan had stated in his 1870 Traité des substitutions et des équations algébriques a canonical form theorem for substitutions of linear groups; on the other hand, Karl Weierstrass had introduced in 1868 the elementary divisors of non singular pairs of bilinear forms (P,Q) in stating a complete set of polynomial invariants computed from the determinant |P+sQ| as a key theorem of the theory of bilinear and quadratic forms. Although they would be considered equivalent as regard to modern mathematics (2), not only had these two theorems been stated independently and for different purposes, they had also been lying within the distinct frameworks of two theories until some connections came to light in 1872-1873, breeding the 1874 quarrel and hence revealing an opposition over two practices relating to distinctive cultural features. As we will be focusing in this paper on how the 1874 controversy about Jordan’s algebraic practice of canonical reduction and Kronecker’s arithmetic practice of invariant computation sheds some light on two conflicting perspectives on polynomial generality we shall appeal to former publications which have already been dealing with some of the cultural issues highlighted by this controversy such as tacit knowledge, local ways of thinking, internal philosophies and disciplinary ideals peculiar to individuals or communities [Brechenmacher 200?a] as well as the different perceptions expressed by the two opponents of a long term history involving authors such as Joseph-Louis Lagrange, Augustin Cauchy and Charles Hermite [Brechenmacher 200?b]. -
The History of the Abel Prize and the Honorary Abel Prize the History of the Abel Prize
The History of the Abel Prize and the Honorary Abel Prize The History of the Abel Prize Arild Stubhaug On the bicentennial of Niels Henrik Abel’s birth in 2002, the Norwegian Govern- ment decided to establish a memorial fund of NOK 200 million. The chief purpose of the fund was to lay the financial groundwork for an annual international prize of NOK 6 million to one or more mathematicians for outstanding scientific work. The prize was awarded for the first time in 2003. That is the history in brief of the Abel Prize as we know it today. Behind this government decision to commemorate and honor the country’s great mathematician, however, lies a more than hundred year old wish and a short and intense period of activity. Volumes of Abel’s collected works were published in 1839 and 1881. The first was edited by Bernt Michael Holmboe (Abel’s teacher), the second by Sophus Lie and Ludvig Sylow. Both editions were paid for with public funds and published to honor the famous scientist. The first time that there was a discussion in a broader context about honoring Niels Henrik Abel’s memory, was at the meeting of Scan- dinavian natural scientists in Norway’s capital in 1886. These meetings of natural scientists, which were held alternately in each of the Scandinavian capitals (with the exception of the very first meeting in 1839, which took place in Gothenburg, Swe- den), were the most important fora for Scandinavian natural scientists. The meeting in 1886 in Oslo (called Christiania at the time) was the 13th in the series. -
On the Origin and Early History of Functional Analysis
U.U.D.M. Project Report 2008:1 On the origin and early history of functional analysis Jens Lindström Examensarbete i matematik, 30 hp Handledare och examinator: Sten Kaijser Januari 2008 Department of Mathematics Uppsala University Abstract In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18th and 19th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of infinite systems of equations and determinants by Helge von Koch were used in Ivar Fredholm’s 1900 paper on the integral equation b Z ϕ(s) = f(s) + λ K(s, t)f(t)dt (1) a which resulted in a vast study of integral equations. One of the most enthusiastic followers of Fredholm and integral equation theory was David Hilbert, and we will see how he further developed the theory of integral equations and spectral theory. The concept introduced by Fredholm to study sets of transformations, or operators, made Maurice Fr´echet realize that the focus should be shifted from particular objects to sets of objects and the algebraic properties of these sets. This led him to introduce abstract spaces and we will see how he introduced the axioms that defines them. Finally, we will investigate how the Lebesgue theory of integration were used by Frigyes Riesz who was able to connect all theory of Fredholm, Fr´echet and Lebesgue to form a general theory, and a new discipline of mathematics, now known as functional analysis. -
Transcendental Numbers
INTRODUCTION TO TRANSCENDENTAL NUMBERS VO THANH HUAN Abstract. The study of transcendental numbers has developed into an enriching theory and constitutes an important part of mathematics. This report aims to give a quick overview about the theory of transcen- dental numbers and some of its recent developments. The main focus is on the proof that e is transcendental. The Hilbert's seventh problem will also be introduced. 1. Introduction Transcendental number theory is a branch of number theory that concerns about the transcendence and algebraicity of numbers. Dated back to the time of Euler or even earlier, it has developed into an enriching theory with many applications in mathematics, especially in the area of Diophantine equations. Whether there is any transcendental number is not an easy question to answer. The discovery of the first transcendental number by Liouville in 1851 sparked up an interest in the field and began a new era in the theory of transcendental number. In 1873, Charles Hermite succeeded in proving that e is transcendental. And within a decade, Lindemann established the tran- scendence of π in 1882, which led to the impossibility of the ancient Greek problem of squaring the circle. The theory has progressed significantly in recent years, with answer to the Hilbert's seventh problem and the discov- ery of a nontrivial lower bound for linear forms of logarithms of algebraic numbers. Although in 1874, the work of Georg Cantor demonstrated the ubiquity of transcendental numbers (which is quite surprising), finding one or proving existing numbers are transcendental may be extremely hard. In this report, we will focus on the proof that e is transcendental. -
A Survey of the Development of Geometry up to 1870
A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics. -
On “Discovering and Proving That Π Is Irrational”
On \Discovering and Proving that π Is Irrational" Li Zhou 1 A needle and a haystack. Once upon a time, there was a village with a huge haystack. Some villagers found the challenge of retrieving needles from the haystack rewarding, but also frustrating at times. Instead of looking for needles directly, one talented villager had the great idea and gift of looking for threads, and found some sharp and useful needles, by their threads, from the haystack. Decades passed. A partic- ularly beautiful golden needle he found was polished, with its thread removed, and displayed in the village temple. Many more decades passed. The golden needle has been admired in the temple, but mentioned rarely together with its gifted founder and the haystack. Some villagers of a new generation start to claim and believe that the needle could be found simply by its golden color and elongated shape. 2 The golden needle. Let me first show you the golden needle, with a different polish from what you may be used to see. You are no doubt aware of the name(s) of its polisher(s). But you will soon learn the name of its original founder. Theorem 1. π2 is irrational. n n R π Proof. Let fn(x) = x (π − x) =n! and In = 0 fn(x) sin x dx for n ≥ 0. Then I0 = 2 and I1 = 4. For n ≥ 2, it is easy to verify that 00 2 fn (x) = −(4n − 2)fn−1(x) + π fn−2(x): (1) 2 Using (1) and integration by parts, we get In = (4n−2)In−1 −π In−2. -
CHARLES HERMITE's STROLL THROUGH the GALOIS FIELDS Catherine Goldstein
Revue d’histoire des mathématiques 17 (2011), p. 211–270 CHARLES HERMITE'S STROLL THROUGH THE GALOIS FIELDS Catherine Goldstein Abstract. — Although everything seems to oppose the two mathematicians, Charles Hermite’s role was crucial in the study and diffusion of Évariste Galois’s results in France during the second half of the nineteenth century. The present article examines that part of Hermite’s work explicitly linked to Galois, the re- duction of modular equations in particular. It shows how Hermite’s mathemat- ical convictions—concerning effectiveness or the unity of algebra, analysis and arithmetic—shaped his interpretation of Galois and of the paths of develop- ment Galois opened. Reciprocally, Hermite inserted Galois’s results in a vast synthesis based on invariant theory and elliptic functions, the memory of which is in great part missing in current Galois theory. At the end of the article, we discuss some methodological issues this raises in the interpretation of Galois’s works and their posterity. Texte reçu le 14 juin 2011, accepté le 29 juin 2011. C. Goldstein, Histoire des sciences mathématiques, Institut de mathématiques de Jussieu, Case 247, UPMC-4, place Jussieu, F-75252 Paris Cedex (France). Courrier électronique : [email protected] Url : http://people.math.jussieu.fr/~cgolds/ 2000 Mathematics Subject Classification : 01A55, 01A85; 11-03, 11A55, 11F03, 12-03, 13-03, 20-03. Key words and phrases : Charles Hermite, Évariste Galois, continued fractions, quin- tic, modular equation, history of the theory of equations, arithmetic algebraic analy- sis, monodromy group, effectivity. Mots clefs. — Charles Hermite, Évariste Galois, fractions continues, quintique, équa- tion modulaire, histoire de la théorie des équations, analyse algébrique arithmétique, groupe de monodromie, effectivité. -
An Overview of Definitions and Their Relationships
INTERNATIONAL JOURNAL OF GEOMETRY Vol. 10 (2021), No. 2, 50 - 66 A SURVEY ON CONICS IN HISTORICAL CONTEXT: an overview of definitions and their relationships. DIRK KEPPENS and NELE KEPPENS1 Abstract. Conics are undoubtedly one of the most studied objects in ge- ometry. Throughout history different definitions have been given, depending on the context in which a conic is seen. Indeed, conics can be defined in many ways: as conic sections in three-dimensional space, as loci of points in the euclidean plane, as algebraic curves of the second degree, as geometrical configurations in (desarguesian or non-desarguesian) projective planes, ::: In this paper we give an overview of all these definitions and their interre- lationships (without proofs), starting with conic sections in ancient Greece and ending with ovals in modern times. 1. Introduction There is a vast amount of literature on conic sections, the majority of which is referring to the groundbreaking work of Apollonius. Some influen- tial books were e.g. [49], [13], [51], [6], [55], [59] and [26]. This paper is an attempt to bring together the variety of existing definitions for the notion of a conic and to give a comparative overview of all known results concerning their connections, scattered in the literature. The proofs of the relationships can be found in the references and are not repeated in this article. In the first section we consider the oldest definition for a conic, due to Apollonius, as the section of a plane with a cone. Thereafter we look at conics as loci of points in the euclidean plane and as algebraic curves of the second degree going back to Descartes and some of his contemporaries. -
Steiner's Theorems on the Complete Quadrilateral
Forum Geometricorum b Volume 4 (2004) 35–52. bbb FORUM GEOM ISSN 1534-1178 Steiner’s Theorems on the Complete Quadrilateral Jean-Pierre Ehrmann Abstract. We give a translation of Jacob Steiner’s 1828 note on the complete quadrilateral, with complete proofs and annotations in barycentric coordinates. 1. Steiner’s note on the complete quadrilateral In 1828, Jakob Steiner published in Gergonne’s Annales a very short note [9] listing ten interesting and important theorems on the complete quadrilateral. The purpose of this paper is to provide a translation of the note, to prove these theorems, along with annotations in barycentric coordinates. We begin with a translation of Steiner’s note. Suppose four lines intersect two by two at six points. (1) These four lines, taken three by three, form four triangles whose circum- circles pass through the same point F . (2) The centers of the four circles (and the point F ) lie on the same circle. (3) The perpendicular feet from F to the four lines lie on the same line R, and F is the only point with this property. (4) The orthocenters of the four triangles lie on the same line R. (5) The lines R and R are parallel, and the line R passes through the midpoint of the segment joining F to its perpendicular foot on R. (6) The midpoints of the diagonals of the complete quadrilateral formed by the four given lines lie on the same line R (Newton). (7) The line R is a common perpendicular to the lines R and R. (8) Each of the four triangles in (1) has an incircle and three excircles. -
The Rise of Projective Geometry II
The Rise of Projective Geometry II The Renaissance Artists Although isolated results from earlier periods are now considered as belonging to the subject of projective geometry, the fundamental ideas that form the core of this area stem from the work of artists during the Renaissance. Earlier art appears to us as being very stylized and flat. The Renaissance Artists Towards the end of the 13th century, early Renaissance artists began to attempt to portray situations in a more realistic way. One early technique is known as terraced perspective, where people in a group scene that are further in the back are drawn higher up than those in the front. Simone Martini: Majesty The Renaissance Artists As artists attempted to find better techniques to improve the realism of their work, the idea of vertical perspective was developed by the Italian school of artists (for example Duccio (1255-1318) and Giotto (1266-1337)). To create the sense of depth, parallel lines in the scene are represented by lines that meet in the centerline of the picture. Duccio's Last Supper The Renaissance Artists The modern system of focused perspective was discovered around 1425 by the sculptor and architect Brunelleschi (1377-1446), and formulated in a treatise a few years later by the painter and architect Leone Battista Alberti (1404-1472). The method was perfected by Leonardo da Vinci (1452 – 1519). The German artist Albrecht Dürer (1471 – 1528) introduced the term perspective (from the Latin verb meaning “to see through”) to describe this technique and illustrated it by a series of well- known woodcuts in his book Underweysung der Messung mit dem Zyrkel und Rychtsscheyed [Instruction on measuring with compass and straight edge] in 1525. -
Jemma Lorenat Pitzer College [email protected]
Jemma Lorenat Pitzer College [email protected] www.jemmalorenat.com EDUCATION PhD, Mathematics (2010 – 2015) Simon Fraser University (Canada) and Université Pierre et Marie Curie (France) Doctoral programs in mathematics at the Department of Mathematics at SFU and at the Institut de mathématiques de Jussieu, Paris Rive Gauche at UPMC Thesis: “Die Freude an der Gestalt: Methods, Figures, and Practices in Early Nineteenth Century Geometry.” Advisors: Prof. Thomas Archibald and Prof. Catherine Goldstein MA, Liberal Studies (2008 – May 2010) City University of New York Graduate Center Thesis: “The development and reception of Leopold Kronecker’s philosophy of mathematics” BA (Summa Cum Laude), Mathematics (2005 – May 2007) San Francisco State University Undergraduate Studies (2004 – 2005) University of St Andrews, St Andrews, Scotland EMPLOYMENT 2015 – Present, Pitzer College (Claremont, CA), Assistant Professor of Mathematics 2013 – 2015, Pratt Institute (Brooklyn), Visiting Instructor 2013 – 2015, St Joseph’s College (Brooklyn), Visiting Instructor 2010 – 2012, Simon Fraser University, Teaching Assistant 2009 – 2010, College Now, Hunter College (New York), Instructor 2007 – 2010, Middle Grades Initiative, City University of New York PUBLISHED RESEARCH “Certain modern ideas and methods: “geometric reality” in the mathematics of Charlotte Angas Scott” Review of Symbolic Logic (forthcoming). “Actual accomplishments in this world: the other students of Charlotte Angas Scott” Mathematical Intelligencer (forthcoming). “Je ne point ambitionée d'être neuf: modern geometry in early nineteenth-century French textbooks” Interfaces between mathematical practices and mathematical eduation ed. Gert Schubring. Springer, 2019, pp. 69–122. “Radical, ideal and equal powers: naming objects in nineteenth century geometry” Revue d’histoire des mathématiques 23 (1), 2017, pp. -
Saikat Mazumdar Curriculum Vitae Department of Mathematics Office: 115-C Indian Institute of Technology Bombay Phone: +91 22 2576 9475 Mumbai, Maharashtra 400076, India
Saikat Mazumdar Curriculum Vitae Department of Mathematics Office: 115-C Indian Institute of Technology Bombay Phone: +91 22 2576 9475 Mumbai, Maharashtra 400076, India. Email: [email protected], [email protected] Positions • May 2019 − Present: Assistant Professor, Indian Institute of Technology Bombay, Mumbai, India. • September 2018 −April 2019: Postdoctoral Fellow, McGill University, Montr´eal,Canada. Supervisors: Pengfei Guan, Niky Kamran and J´er^omeV´etois. • September 2016 −August 2018: Postdoctoral Fellow, University of British Columbia, Vancouver, Canada. Supervisor: Nassif Ghoussoub. • October 2015 −August 2016 : ATER-Doctorat, Universit´ede Lorraine, Nancy, France. • November 2016 −September 2015: Doctorant, Institut Elie´ Cartan de Lorraine, Universit´ede Lorraine, Nancy, France. Funding: F´ed´erationCharles Hermite and R´egionLorraine. Education • 2016: Ph.D in Mathematics, Universit´ede Lorraine, Institut Elie´ Cartan de Lorraine, Nancy, France. Advisors: Fr´ed´ericRobert and Dong Ye. Thesis Title: Polyharmonic Equations on Manifolds and Asymptotic Analysis of Hardy-Sobolev Equations with Vanishing Singularity. Defended: June 2016. Ph.D. committee: Emmanuel Hebey (Universit´ede Cergy-Pontoise), Patrizia Pucci (Universit`adegli Studi di Perugia), Tobias Weth (Goethe-Universit¨atFrankfurt), Yuxin Ge (Universit´ePaul Sabatier, Toulouse), David Dos Santos Ferreira (Universit´ede Lorraine), Fr´ed´ericRobert (Universit´ede Lor- raine) and Dong Ye (Universit´ede Lorraine). • 2012: M.Sc and M.Phil in Mathematics, Tata Institute Of Fundamental Research-CAM, Bangalore, India. Advisor: K Sandeep. Thesis Title: On A Variational Problem with Lack of Compactness: The Effect of the Topology of the Domain . Defended: September, 2012. • 2009: B.Sc (Hons) in Mathematics, University of Calcutta, Kolkata, India. Research Interests Geometric analysis and Nonlinear partial differential equations : Blow-up analysis and Concentration phenomenon in Elliptic PDEs, Prescribing curvature problems, Higher-order conformally invariant PDEs.