And the Riemann Hypothesis
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FIELDS MEDAL for Mathematical Efforts R
Recognizing the Real and the Potential: FIELDS MEDAL for Mathematical Efforts R Fields Medal recipients since inception Year Winners 1936 Lars Valerian Ahlfors (Harvard University) (April 18, 1907 – October 11, 1996) Jesse Douglas (Massachusetts Institute of Technology) (July 3, 1897 – September 7, 1965) 1950 Atle Selberg (Institute for Advanced Study, Princeton) (June 14, 1917 – August 6, 2007) 1954 Kunihiko Kodaira (Princeton University) (March 16, 1915 – July 26, 1997) 1962 John Willard Milnor (Princeton University) (born February 20, 1931) The Fields Medal 1966 Paul Joseph Cohen (Stanford University) (April 2, 1934 – March 23, 2007) Stephen Smale (University of California, Berkeley) (born July 15, 1930) is awarded 1970 Heisuke Hironaka (Harvard University) (born April 9, 1931) every four years 1974 David Bryant Mumford (Harvard University) (born June 11, 1937) 1978 Charles Louis Fefferman (Princeton University) (born April 18, 1949) on the occasion of the Daniel G. Quillen (Massachusetts Institute of Technology) (June 22, 1940 – April 30, 2011) International Congress 1982 William P. Thurston (Princeton University) (October 30, 1946 – August 21, 2012) Shing-Tung Yau (Institute for Advanced Study, Princeton) (born April 4, 1949) of Mathematicians 1986 Gerd Faltings (Princeton University) (born July 28, 1954) to recognize Michael Freedman (University of California, San Diego) (born April 21, 1951) 1990 Vaughan Jones (University of California, Berkeley) (born December 31, 1952) outstanding Edward Witten (Institute for Advanced Study, -
The Riemann Mapping Theorem Christopher J. Bishop
The Riemann Mapping Theorem Christopher J. Bishop C.J. Bishop, Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651 E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary: 30C35, Secondary: 30C85, 30C62 Key words and phrases. numerical conformal mappings, Schwarz-Christoffel formula, hyperbolic 3-manifolds, Sullivan’s theorem, convex hulls, quasiconformal mappings, quasisymmetric mappings, medial axis, CRDT algorithm The author is partially supported by NSF Grant DMS 04-05578. Abstract. These are informal notes based on lectures I am giving in MAT 626 (Topics in Complex Analysis: the Riemann mapping theorem) during Fall 2008 at Stony Brook. We will start with brief introduction to conformal mapping focusing on the Schwarz-Christoffel formula and how to compute the unknown parameters. In later chapters we will fill in some of the details of results and proofs in geometric function theory and survey various numerical methods for computing conformal maps, including a method of my own using ideas from hyperbolic and computational geometry. Contents Chapter 1. Introduction to conformal mapping 1 1. Conformal and holomorphic maps 1 2. M¨obius transformations 16 3. The Schwarz-Christoffel Formula 20 4. Crowding 27 5. Power series of Schwarz-Christoffel maps 29 6. Harmonic measure and Brownian motion 39 7. The quasiconformal distance between polygons 48 8. Schwarz-Christoffel iterations and Davis’s method 56 Chapter 2. The Riemann mapping theorem 67 1. The hyperbolic metric 67 2. Schwarz’s lemma 69 3. The Poisson integral formula 71 4. A proof of Riemann’s theorem 73 5. Koebe’s method 74 6. -
Transnational Mathematics and Movements: Shiing- Shen Chern, Hua Luogeng, and the Princeton Institute for Advanced Study from World War II to the Cold War1
Chinese Annals of History of Science and Technology 3 (2), 118–165 (2019) doi: 10.3724/SP.J.1461.2019.02118 Transnational Mathematics and Movements: Shiing- shen Chern, Hua Luogeng, and the Princeton Institute for Advanced Study from World War II to the Cold War1 Zuoyue Wang 王作跃,2 Guo Jinhai 郭金海3 (California State Polytechnic University, Pomona 91768, US; Institute for the History of Natural Sciences, Chinese Academy of Sciences, Beijing 100190, China) Abstract: This paper reconstructs, based on American and Chinese primary sources, the visits of Chinese mathematicians Shiing-shen Chern 陈省身 (Chen Xingshen) and Hua Luogeng 华罗庚 (Loo-Keng Hua)4 to the Institute for Advanced Study in Princeton in the United States in the 1940s, especially their interactions with Oswald Veblen and Hermann Weyl, two leading mathematicians at the IAS. It argues that Chern’s and Hua’s motivations and choices in regard to their transnational movements between China and the US were more nuanced and multifaceted than what is presented in existing accounts, and that socio-political factors combined with professional-personal ones to shape their decisions. The paper further uses their experiences to demonstrate the importance of transnational scientific interactions for the development of science in China, the US, and elsewhere in the twentieth century. Keywords: Shiing-shen Chern, Chen Xingshen, Hua Luogeng, Loo-Keng Hua, Institute for 1 This article was copy-edited by Charlie Zaharoff. 2 Research interests: History of science and technology in the United States, China, and transnational contexts in the twentieth century. He is currently writing a book on the history of American-educated Chinese scientists and China-US scientific relations. -
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. -
The Intellectual Journey of Hua Loo-Keng from China to the Institute for Advanced Study: His Correspondence with Hermann Weyl
ISSN 1923-8444 [Print] Studies in Mathematical Sciences ISSN 1923-8452 [Online] Vol. 6, No. 2, 2013, pp. [71{82] www.cscanada.net DOI: 10.3968/j.sms.1923845220130602.907 www.cscanada.org The Intellectual Journey of Hua Loo-keng from China to the Institute for Advanced Study: His Correspondence with Hermann Weyl Jean W. Richard[a],* and Abdramane Serme[a] [a] Department of Mathematics, The City University of New York (CUNY/BMCC), USA. * Corresponding author. Address: Department of Mathematics, The City University of New York (CUN- Y/BMCC), 199 Chambers Street, N-599, New York, NY 10007-1097, USA; E- Mail: [email protected] Received: February 4, 2013/ Accepted: April 9, 2013/ Published: May 31, 2013 \The scientific knowledge grows by the thinking of the individual solitary scientist and the communication of ideas from man to man Hua has been of great value in this respect to our whole group, especially to the younger members, by the stimulus which he has provided." (Hermann Weyl in a letter about Hua Loo-keng, 1948).y Abstract: This paper explores the intellectual journey of the self-taught Chinese mathematician Hua Loo-keng (1910{1985) from Southwest Associ- ated University (Kunming, Yunnan, China)z to the Institute for Advanced Study at Princeton. The paper seeks to show how during the Sino C Japanese war, a genuine cross-continental mentorship grew between the prolific Ger- man mathematician Hermann Weyl (1885{1955) and the gifted mathemati- cian Hua Loo-keng. Their correspondence offers a profound understanding of a solidarity-building effort that can exist in the scientific community. -
THE IMPACT of RIEMANN's MAPPING THEOREM in the World
THE IMPACT OF RIEMANN'S MAPPING THEOREM GRANT OWEN In the world of mathematics, scholars and academics have long sought to understand the work of Bernhard Riemann. Born in a humble Ger- man home, Riemann became one of the great mathematical minds of the 19th century. Evidence of his genius is reflected in the greater mathematical community by their naming 72 different mathematical terms after him. His contributions range from mathematical topics such as trigonometric series, birational geometry of algebraic curves, and differential equations to fields in physics and philosophy [3]. One of his contributions to mathematics, the Riemann Mapping Theorem, is among his most famous and widely studied theorems. This theorem played a role in the advancement of several other topics, including Rie- mann surfaces, topology, and geometry. As a result of its widespread application, it is worth studying not only the theorem itself, but how Riemann derived it and its impact on the work of mathematicians since its publication in 1851 [3]. Before we begin to discover how he derived his famous mapping the- orem, it is important to understand how Riemann's upbringing and education prepared him to make such a contribution in the world of mathematics. Prior to enrolling in university, Riemann was educated at home by his father and a tutor before enrolling in high school. While in school, Riemann did well in all subjects, which strengthened his knowl- edge of philosophy later in life, but was exceptional in mathematics. He enrolled at the University of G¨ottingen,where he learned from some of the best mathematicians in the world at that time. -
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]. -
Speech by Honorary Degree Recipient
Speech by Honorary Degree Recipient Dear Colleagues and Friends, Ladies and Gentlemen: Today, I am so honored to present in this prestigious stage to receive the Honorary Doctorate of the Saint Petersburg State University. Since my childhood, I have known that Saint Petersburg University is a world-class university associated by many famous scientists, such as Ivan Pavlov, Dmitri Mendeleev, Mikhail Lomonosov, Lev Landau, Alexander Popov, to name just a few. In particular, many dedicated their glorious lives in the same field of scientific research and studies which I have been devoting to: Leonhard Euler, Andrey Markov, Pafnuty Chebyshev, Aleksandr Lyapunov, and recently Grigori Perelman, not to mention many others in different fields such as political sciences, literature, history, economics, arts, and so on. Being an Honorary Doctorate of the Saint Petersburg State University, I have become a member of the University, of which I am extremely proud. I have been to the beautiful and historical city of Saint Petersburg five times since 1997, to work with my respected Russian scientists and engineers in organizing international academic conferences and conducting joint scientific research. I sincerely appreciate the recognition of the Saint Petersburg State University for my scientific contributions and endeavors to developing scientific cooperations between Russia and the People’s Republic of China. I would like to take this opportunity to thank the University for the honor, and thank all professors, staff members and students for their support and encouragement. Being an Honorary Doctorate of the Saint Petersburg State University, I have become a member of the University, which made me anxious to contribute more to the University and to the already well-established relationship between Russia and China in the future. -
Richard Von Mises's Philosophy of Probability and Mathematics
“A terrible piece of bad metaphysics”? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic Lukas M. Verburgt If the true is what is grounded, then the ground is neither true nor false LUDWIG WITTGENSTEIN Whether all grow black, or all grow bright, or all remain grey, it is grey we need, to begin with, because of what it is, and of what it can do, made of bright and black, able to shed the former , or the latter, and be the latter or the former alone. But perhaps I am the prey, on the subject of grey, in the grey, to delusions SAMUEL BECKETT “A terrible piece of bad metaphysics”? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het College voor Promoties ingestelde commissie in het openbaar te verdedigen in de Agnietenkapel op donderdag 1 oktober 2015, te 10:00 uur door Lukas Mauve Verburgt geboren te Amersfoort Promotiecommissie Promotor: Prof. dr. ir. G.H. de Vries Universiteit van Amsterdam Overige leden: Prof. dr. M. Fisch Universitat Tel Aviv Dr. C.L. Kwa Universiteit van Amsterdam Dr. F. Russo Universiteit van Amsterdam Prof. dr. M.J.B. Stokhof Universiteit van Amsterdam Prof. dr. A. Vogt Humboldt-Universität zu Berlin Faculteit der Geesteswetenschappen © 2015 Lukas M. Verburgt Graphic design Aad van Dommelen (Witvorm) -
Life and Work of Egbert Brieskorn (1936-2013)
Life and work of Egbert Brieskorn (1936 – 2013)1 Gert-Martin Greuel, Walter Purkert Brieskorn 2007 Egbert Brieskorn died on July 11, 2013, a few days after his 77th birthday. He was an impressive personality who left a lasting impression on anyone who knew him, be it in or out of mathematics. Brieskorn was a great mathematician, but his interests, knowledge, and activities went far beyond mathematics. In the following article, which is strongly influenced by the authors’ many years of personal ties with Brieskorn, we try to give a deeper insight into the life and work of Brieskorn. In doing so, we highlight both his personal commitment to peace and the environment as well as his long–standing exploration of the life and work of Felix Hausdorff and the publication of Hausdorff ’s Collected Works. The focus of the article, however, is on the presentation of his remarkable and influential mathematical work. The first author (GMG) has spent significant parts of his scientific career as a arXiv:1711.09600v1 [math.AG] 27 Nov 2017 graduate and doctoral student with Brieskorn in Göttingen and later as his assistant in Bonn. He describes in the first two parts, partly from the memory of personal cooperation, aspects of Brieskorn’s life and of his political and social commitment. In addition, in the section on Brieskorn’s mathematical work, he explains in detail 1Translation of the German article ”Leben und Werk von Egbert Brieskorn (1936 – 2013)”, Jahresber. Dtsch. Math.–Ver. 118, No. 3, 143-178 (2016). 1 the main scientific results of his publications. -
The Measurable Riemann Mapping Theorem
CHAPTER 1 The Measurable Riemann Mapping Theorem 1.1. Conformal structures on Riemann surfaces Throughout, “smooth” will always mean C 1. All surfaces are assumed to be smooth, oriented and without boundary. All diffeomorphisms are assumed to be smooth and orientation-preserving. It will be convenient for our purposes to do local computations involving metrics in the complex-variable notation. Let X be a Riemann surface and z x iy be a holomorphic local coordinate on X. The pair .x; y/ can be thoughtD C of as local coordinates for the underlying smooth surface. In these coordinates, a smooth Riemannian metric has the local form E dx2 2F dx dy G dy2; C C where E; F; G are smooth functions of .x; y/ satisfying E > 0, G > 0 and EG F 2 > 0. The associated inner product on each tangent space is given by @ @ @ @ a b ; c d Eac F .ad bc/ Gbd @x C @y @x C @y D C C C Äc a b ; D d where the symmetric positive definite matrix ÄEF (1.1) D FG represents in the basis @ ; @ . In particular, the length of a tangent vector is f @x @y g given by @ @ p a b Ea2 2F ab Gb2: @x C @y D C C Define two local sections of the complexified cotangent bundle T X C by ˝ dz dx i dy WD C dz dx i dy: N WD 2 1 The Measurable Riemann Mapping Theorem These form a basis for each complexified cotangent space. The local sections @ 1 Â @ @ Ã i @z WD 2 @x @y @ 1 Â @ @ Ã i @z WD 2 @x C @y N of the complexified tangent bundle TX C will form the dual basis at each point. -
The Riemann Mapping Theorem
University of Toronto – MAT334H1-F – LEC0101 Complex Variables 18 – The Riemann mapping theorem Jean-Baptiste Campesato December 2nd, 2020 and December 4th, 2020 Theorem 1 (The Riemann mapping theorem). Let 푈 ⊊ ℂ be a simply connected open subset which is not ℂ. −1 Then there exists a biholomorphism 푓 ∶ 푈 → 퐷1(0) (i.e. 푓 is holomorphic, bijective and 푓 is holomorphic). We say that 푈 and 퐷1(0) are conformally equivalent. Remark 2. Note that if 푓 ∶ 푈 → 푉 is bijective and holomorphic then 푓 −1 is holomorphic too. Indeed, we proved that if 푓 is injective and holomorphic then 푓 ′ never vanishes (Nov 30). Then we can conclude using the inverse function theorem. Note that this remark is false for ℝ-differentiability: define 푓 ∶ ℝ → ℝ by 푓(푥) = 푥3 then 푓 ′(0) = 0 and 푓 −1(푥) = √3 푥 is not differentiable at 0. Remark 3. The theorem is false if 푈 = ℂ. Indeed, by Liouville’s theorem, if 푓 ∶ ℂ → 퐷1(0) is holomorphic then it is constant (as a bounded entire function), so it can’t be bijective. The Riemann mapping theorem states that up to biholomorphic transformations, the unit disk is a model for open simply connected sets which are not ℂ. Otherwise stated, up to a biholomorphic transformation, there are only two open simply connected sets: 퐷1(0) and ℂ. Formally: Corollary 4. Let 푈, 푉 ⊊ ℂ be two simply connected open subsets, none of which is ℂ. Then there exists a biholomorphism 푓 ∶ 푈 → 푉 (i.e. 푓 is holomorphic, bijective and 푓 −1 is holomorphic).