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John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
John Charles Fields: A Sketch of His Life and Mathematical Work
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John Charles Fields: A Sketch of His Life and Mathematical Work

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The slides from Marcus Emmanuel Barnes' M.Sc. thesis defense presentation. …

The slides from Marcus Emmanuel Barnes' M.Sc. thesis defense presentation.

Abstract:

Every four years at the International Congress of Mathematicians the prestigious Fields medals, the mathematical equivalent of a Nobel prize, are awarded. The following question is often asked: who was Fields and what did he do mathematically? This question will be addressed by sketching the life and mathematical work of John Charles Fields (1863 – 1932), the Canadian mathematician who helped establish the awards and after whom the medals are named.

You can download a PDF version of the thesis here:

http://www.marcusebarnes.com/168/john-charles-fields-a-sketch-of-his-life-and-mathematical-work/

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  • 1. John Charles Fields: A Sketch of His Life and Mathematical Work Marcus Emmanuel Barnes Simon Fraser University Thursday, November 29, 2007 Marcus Emmanuel Barnes Simon Fraser University
  • 2. John Charles Fields (1863-1932) Figure: John Charles Fields (1863-1932) Marcus Emmanuel Barnes Simon Fraser University
  • 3. Early Years Born May 14th, 1863, in Hamilton, Ontario (then Canada West), the son of the leather shop operator J. C. Fields and his wife Harriet Bowes. Marcus Emmanuel Barnes Simon Fraser University
  • 4. Early Years Born May 14th, 1863, in Hamilton, Ontario (then Canada West), the son of the leather shop operator J. C. Fields and his wife Harriet Bowes. Fields had at least one brother Marcus Emmanuel Barnes Simon Fraser University
  • 5. Early Years Born May 14th, 1863, in Hamilton, Ontario (then Canada West), the son of the leather shop operator J. C. Fields and his wife Harriet Bowes. Fields had at least one brother Attended Hamilton Collegiate where he showed early talent in mathematics. Marcus Emmanuel Barnes Simon Fraser University
  • 6. Undergraduate Years Continued his studies at the University of Toronto in 1880, studying mathematics. Marcus Emmanuel Barnes Simon Fraser University
  • 7. Undergraduate Years Continued his studies at the University of Toronto in 1880, studying mathematics. Typical mathematical requirements for a BA degree in Canada in the second half of the nineteenth century: Marcus Emmanuel Barnes Simon Fraser University
  • 8. Undergraduate Years Continued his studies at the University of Toronto in 1880, studying mathematics. Typical mathematical requirements for a BA degree in Canada in the second half of the nineteenth century: Euclid’s Elements books 1-4, 6 Marcus Emmanuel Barnes Simon Fraser University
  • 9. Undergraduate Years Continued his studies at the University of Toronto in 1880, studying mathematics. Typical mathematical requirements for a BA degree in Canada in the second half of the nineteenth century: Euclid’s Elements books 1-4, 6 algebra to the binomial theorem Marcus Emmanuel Barnes Simon Fraser University
  • 10. Undergraduate Years Continued his studies at the University of Toronto in 1880, studying mathematics. Typical mathematical requirements for a BA degree in Canada in the second half of the nineteenth century: Euclid’s Elements books 1-4, 6 algebra to the binomial theorem trigonometry Marcus Emmanuel Barnes Simon Fraser University
  • 11. Undergraduate Years Continued his studies at the University of Toronto in 1880, studying mathematics. Typical mathematical requirements for a BA degree in Canada in the second half of the nineteenth century: Euclid’s Elements books 1-4, 6 algebra to the binomial theorem trigonometry basic mechanics and hydrostatics Marcus Emmanuel Barnes Simon Fraser University
  • 12. Undergraduate Years (Continued) Honours students could go beyond the basic material to study: conic sections Marcus Emmanuel Barnes Simon Fraser University
  • 13. Undergraduate Years (Continued) Honours students could go beyond the basic material to study: conic sections differential and integral calculus Marcus Emmanuel Barnes Simon Fraser University
  • 14. Undergraduate Years (Continued) Honours students could go beyond the basic material to study: conic sections differential and integral calculus differential equations Marcus Emmanuel Barnes Simon Fraser University
  • 15. Undergraduate Years (Continued) Honours students could go beyond the basic material to study: conic sections differential and integral calculus differential equations various topics in applied mathematics Marcus Emmanuel Barnes Simon Fraser University
  • 16. Undergraduate Years (Continued 2) Fields had a distinguished undergraduate career, winning a gold medal upon graduating in 1884 with a BA. Marcus Emmanuel Barnes Simon Fraser University
  • 17. Undergraduate Years (Continued 2) Fields had a distinguished undergraduate career, winning a gold medal upon graduating in 1884 with a BA. What next? Marcus Emmanuel Barnes Simon Fraser University
  • 18. The State of Mathematical Education in N.A. in the 1880s One option for further study was the PhD, but it was not possible to obtain a PhD in mathematics in Canada in the 1880s Marcus Emmanuel Barnes Simon Fraser University
  • 19. The State of Mathematical Education in N.A. in the 1880s One option for further study was the PhD, but it was not possible to obtain a PhD in mathematics in Canada in the 1880s Possibilities: Marcus Emmanuel Barnes Simon Fraser University
  • 20. The State of Mathematical Education in N.A. in the 1880s One option for further study was the PhD, but it was not possible to obtain a PhD in mathematics in Canada in the 1880s Possibilities: Marcus Emmanuel Barnes Simon Fraser University
  • 21. The State of Mathematical Education in N.A. in the 1880s One option for further study was the PhD, but it was not possible to obtain a PhD in mathematics in Canada in the 1880s Possibilities: a handful of places in the United States or go to Europe. Marcus Emmanuel Barnes Simon Fraser University
  • 22. The State of Mathematical Education in N.A. in the 1880s One option for further study was the PhD, but it was not possible to obtain a PhD in mathematics in Canada in the 1880s Possibilities: a handful of places in the United States or go to Europe. Fields chose the newly established Johns Hopkins University Marcus Emmanuel Barnes Simon Fraser University
  • 23. Johns Hopkins University established in 1876 Marcus Emmanuel Barnes Simon Fraser University
  • 24. Johns Hopkins University established in 1876 the mathematics departments was put together by J. J. Sylvester (1814 - 1897) Marcus Emmanuel Barnes Simon Fraser University
  • 25. Johns Hopkins University established in 1876 the mathematics departments was put together by J. J. Sylvester (1814 - 1897) From the start, the research productivity of the faculty was of high importance (something rather unique in North America at the time). Marcus Emmanuel Barnes Simon Fraser University
  • 26. Johns Hopkins University established in 1876 the mathematics departments was put together by J. J. Sylvester (1814 - 1897) From the start, the research productivity of the faculty was of high importance (something rather unique in North America at the time). Even though Sylvester left before Fields attended Johns Hopkins, he would influence Fields’ early mathematical work, specifically with regards to the “symbolical method in analysis”. We see this influence in his early papers on differential equations and differential coefficients. Marcus Emmanuel Barnes Simon Fraser University
  • 27. Fields at Johns Hopkins At Hopkins Fields took courses on the theory of functions, linear differential equations, elliptic and Abelian functions, among other topics. As well, he participated in several topic seminars. Most of the courses Fields took were either taught by W. Story or T. Craig. Marcus Emmanuel Barnes Simon Fraser University
  • 28. Fields at Johns Hopkins At Hopkins Fields took courses on the theory of functions, linear differential equations, elliptic and Abelian functions, among other topics. As well, he participated in several topic seminars. Most of the courses Fields took were either taught by W. Story or T. Craig. Fields received his PhD in 1887 with a thesis entitled Symbolic Finite Solutions and Solutions by Definite Integrals of the Equation d n y /dx n = x m y . Marcus Emmanuel Barnes Simon Fraser University
  • 29. Fields’ PhD thesis I n The equation d y = x m y , which is the focus of Fields’ thesis, is dx n similar to certain Riccati equations. Jacopo Riccati (1676-1754), an Italian nobleman and mathematician, studied certain second order differential equations. The differential equation dy = Ay 2 + Bx n , dx A and B constant, became known as Riccati’s equation. In generalized form it is usually written as dy = a0 (x) + a1 (x)y + a2 (x)y 2 . dx Marcus Emmanuel Barnes Simon Fraser University
  • 30. Fields’ PhD thesis II Using “symbolic methods” Fields’ was able to give the following n general solutions to d y = x m y : dx n −9i Case 1: m = 3i+1 „ «i „ «2i 3 d 3i 3 d 2 y =x x 1− 3i+1 x 1+ 3i+1 x 1− 3i+1 x − 3i+1 dx dx 1 1 1 „ « 3i+1 3i+1 3i+1 × C1 e −(3i+1)λ1 x + C2 e −(3i+1)λ2 x + C3 e −(3i+1)λ3 x ; −3(3i+1) Case 2: m = 3i+2 „ «i „ «2i+1 3 d 3i 3 d 1 y =x x 1− 3i+2 x 1− 3i+2 x 1− 3i+2 x − 3i+2 dx dx 1 1 1 „ « −(3i+2)λ1 x 3i+2 −(3i+2)λ2 x 3i+2 −(3i+2)λ3 x 3i+2 × C1 e + C2 e + C3 e . Marcus Emmanuel Barnes Simon Fraser University
  • 31. Fields’ PhD thesis III In the second portion of his thesis, Fields gives a generalization of results obtained by E. Kummer (1810 - 1893) and Spitzer for the n solution of d y = x m y by definite integrals. Furthermore, by using dx n some of the methods from the first part of his thesis, Fields is able to give particular solutions in definite integrals for given values of n m. For example, Fields shows that the equation d y = x −m y has a dx n solution given by Z ∞ Z ∞ m−n m−n 1 +···+un−1 +(xu1 ···un−1 )n−m ) y = x n−1 ··· u2 u3 · · · un−1 e n−m (u1 2 n−2 du1 · · · dun−1 0 0 where m is any positive quantity greater than n. Marcus Emmanuel Barnes Simon Fraser University
  • 32. Immediately after Fields’ PhD Fields became a fellow at Hopkins (which entailed a certain amount of undergraduate teaching) until 1889. Marcus Emmanuel Barnes Simon Fraser University
  • 33. Immediately after Fields’ PhD Fields became a fellow at Hopkins (which entailed a certain amount of undergraduate teaching) until 1889. In 1889 Fields became professor of mathematics at Allegheny College in Meadville, Pennsylvania. Marcus Emmanuel Barnes Simon Fraser University
  • 34. Immediately after Fields’ PhD Fields became a fellow at Hopkins (which entailed a certain amount of undergraduate teaching) until 1889. In 1889 Fields became professor of mathematics at Allegheny College in Meadville, Pennsylvania. During this time, Fields published several papers, including proofs of the fundamental theorem of algebra, the elliptic function addition theorem, as well as some papers on number theory and some further work on differential coefficients (using the symbolic method). Many of these papers appeared in the young American Journal of Mathematics. Marcus Emmanuel Barnes Simon Fraser University
  • 35. Heading to Europe Fields’ resigned from Allegheny in 1892 as a result of coming into his modest inheritance from his father and mother who had passed away when Fields was a teenager. He decided to use the money to continue his mathematical studies in Europe. Marcus Emmanuel Barnes Simon Fraser University
  • 36. Post-Doctoral Studies in Europe The standard obituary by J. L. Synge (1897-1995) states that Fields spent 5 years in Paris and 5 years in Berlin. Marcus Emmanuel Barnes Simon Fraser University
  • 37. Post-Doctoral Studies in Europe The standard obituary by J. L. Synge (1897-1995) states that Fields spent 5 years in Paris and 5 years in Berlin. There is only documentary evidence regarding Fields’ stay in Germany. Marcus Emmanuel Barnes Simon Fraser University
  • 38. Post-Doctoral Studies in Europe The standard obituary by J. L. Synge (1897-1995) states that Fields spent 5 years in Paris and 5 years in Berlin. There is only documentary evidence regarding Fields’ stay in Germany. Fields enrolled in G¨ttingen in November of 1894 where he o had the opportunity to attend lectures by Felix Klein (1849-1925) on number theory, as well as an introductory course on the theory of functions of a complex variable offered by the Privatdozent Ritter. Marcus Emmanuel Barnes Simon Fraser University
  • 39. Post-doctoral studies in Berlin Fields stayed at G¨ttingen until May of 1895, at which point o he travelled to Berlin. Marcus Emmanuel Barnes Simon Fraser University
  • 40. Post-doctoral studies in Berlin Fields stayed at G¨ttingen until May of 1895, at which point o he travelled to Berlin. Berlin was a natural choice for Fields given his early interest in linear differential equaitons and the fact that L. Fuchs (1833-1902) and G. Frobenius (1849-1917) were there. Marcus Emmanuel Barnes Simon Fraser University
  • 41. Post-doctoral studies in Berlin II Fields’ notebooks contain notes from lecture courses by: G. Frobenius: 2 courses on number theory; 1 on analytic geometry; 2 on algebraic equations. Marcus Emmanuel Barnes Simon Fraser University
  • 42. Post-doctoral studies in Berlin II Fields’ notebooks contain notes from lecture courses by: G. Frobenius: 2 courses on number theory; 1 on analytic geometry; 2 on algebraic equations. L. Fuchs: 9 courses including the theory of hyperelliptic and Abelian functions, and topics in differential equations. Marcus Emmanuel Barnes Simon Fraser University
  • 43. Post-doctoral studies in Berlin II Fields’ notebooks contain notes from lecture courses by: G. Frobenius: 2 courses on number theory; 1 on analytic geometry; 2 on algebraic equations. L. Fuchs: 9 courses including the theory of hyperelliptic and Abelian functions, and topics in differential equations. K. Hensel (1861-1941): 6 courses, including algebraic functions of one and two variables, a course on Abelian integrals, and a course on number theory Marcus Emmanuel Barnes Simon Fraser University
  • 44. Post-doctoral studies in Berlin III H. A. Schwarz (1843-1921): 15 courses. Topics included elliptic functions, variational calculus, theory of functions of one complex variable, synthetic projective geometry, number theory, and integral calculus. Marcus Emmanuel Barnes Simon Fraser University
  • 45. Post-doctoral studies in Berlin III H. A. Schwarz (1843-1921): 15 courses. Topics included elliptic functions, variational calculus, theory of functions of one complex variable, synthetic projective geometry, number theory, and integral calculus. Hetner: a course on definite integrals and a course on Fourier series; Knoblauch: 2 courses on curves and surfaces; Steinitz: a course on Cantor’s set theory. Marcus Emmanuel Barnes Simon Fraser University
  • 46. Post-doctoral studies in Berlin III H. A. Schwarz (1843-1921): 15 courses. Topics included elliptic functions, variational calculus, theory of functions of one complex variable, synthetic projective geometry, number theory, and integral calculus. Hetner: a course on definite integrals and a course on Fourier series; Knoblauch: 2 courses on curves and surfaces; Steinitz: a course on Cantor’s set theory. In addition, there is almost the entire series of lectures by M. Planck which would evolve into his famous course on theoretical physics, two courses on inorganic chemistry, and one on the history of philosophy. Marcus Emmanuel Barnes Simon Fraser University
  • 47. Some Images from Fields’ Berlin notebooks Marcus Emmanuel Barnes Simon Fraser University
  • 48. Some Images from Fields’ Berlin notebooks II Marcus Emmanuel Barnes Simon Fraser University
  • 49. Fields leaving certificate Marcus Emmanuel Barnes Simon Fraser University
  • 50. Research 1894-1900 Fields did not publish any papers during the years 1894 to 1900, though he seems to have continued to do research, presenting a talk at the meeting of the American Mathematical Society held in Toronto in 1897 on the reduction of the general Abelian integral. He would publish a paper based on this talk in 1901. It is not really surprising that Fields’ failed to publish during this time given the number of courses he apparently attended, as can be ascertained from large number of notebooks full of lecture notes he accumulated. Marcus Emmanuel Barnes Simon Fraser University
  • 51. Professor Fields I Fields took up a position as special lecturer at the University of Toronto in 1902. At that time the mathematics department had roughly five members including Fields. Marcus Emmanuel Barnes Simon Fraser University
  • 52. Professor Fields I Fields took up a position as special lecturer at the University of Toronto in 1902. At that time the mathematics department had roughly five members including Fields. By 1905 he had gained a regular position as Associate Professor and would later become Professor in 1914 and Research Professor in 1923. Marcus Emmanuel Barnes Simon Fraser University
  • 53. Professor Fields I Fields took up a position as special lecturer at the University of Toronto in 1902. At that time the mathematics department had roughly five members including Fields. By 1905 he had gained a regular position as Associate Professor and would later become Professor in 1914 and Research Professor in 1923. Among the honours that Fields received was being elected to the Royal Society of Canada in 1909 and to the Royal Society of London in 1913. Marcus Emmanuel Barnes Simon Fraser University
  • 54. Professor Fields II On a local level, Fields was active in the life of the university, often giving talks to the mathematics and physics student club. He also successfully lobbied the Ontario legislature for monetary support for scientific research being carried out at the University. Marcus Emmanuel Barnes Simon Fraser University
  • 55. Professor Fields II On a local level, Fields was active in the life of the university, often giving talks to the mathematics and physics student club. He also successfully lobbied the Ontario legislature for monetary support for scientific research being carried out at the University. Fields was involved with scientific organization on the national level. For example, he was President of the Royal Canadian Institute from 1919 to 1925. Marcus Emmanuel Barnes Simon Fraser University
  • 56. Professor Fields II On a local level, Fields was active in the life of the university, often giving talks to the mathematics and physics student club. He also successfully lobbied the Ontario legislature for monetary support for scientific research being carried out at the University. Fields was involved with scientific organization on the national level. For example, he was President of the Royal Canadian Institute from 1919 to 1925. On the international level, Fields was Vice-President of both the British Association for the Advancement of Science in 1924 and the American Association for the Advancement of Science, Section A in 1924. Marcus Emmanuel Barnes Simon Fraser University
  • 57. Professor Fields II On a local level, Fields was active in the life of the university, often giving talks to the mathematics and physics student club. He also successfully lobbied the Ontario legislature for monetary support for scientific research being carried out at the University. Fields was involved with scientific organization on the national level. For example, he was President of the Royal Canadian Institute from 1919 to 1925. On the international level, Fields was Vice-President of both the British Association for the Advancement of Science in 1924 and the American Association for the Advancement of Science, Section A in 1924. Fields is also well known for organizing the 1924 International Congress of Mathematicians held in Toronto. It was during a cross country train trip with the conference delegates that Fields’ health began to deteriorate. Marcus Emmanuel Barnes Simon Fraser University
  • 58. Fields’ theory of algebraic functions At the turn of the twentieth century, there were several approaches to the theory, often categorized as either transcendental, geometric, or arithmetic. Fields’ approach seems to be an outgrowth of Hensel’s push for a purely “algebraic” approach (i.e., arithmetic) to the theory of algebraic functions. However, Fields’ approach, contrary to that presented by Hensel and Landsberg in their 1904 memoir seems to have retained some of the Weierstrassian function theoretic methods, in that it avoids the use of Riemann surfaces and the theory of divisors entirely. Marcus Emmanuel Barnes Simon Fraser University
  • 59. Fields’ theory of algebraic functions II One of Fields’ primary goals was to study rational functions subject to the condition defined by an algebraic function F (x, y ) = 0. We would now describe these as rational functions on a variety. Marcus Emmanuel Barnes Simon Fraser University
  • 60. Fields’ theory of algebraic functions III Fields’ theory is built up from the following concept: Definition (Order of Coincidence) The order of coincidence (at a point) of a rational function H(x, y ) with respect to a branch y − P = 0 is the smallest exponent of the series expansion for H(x, P). Marcus Emmanuel Barnes Simon Fraser University
  • 61. Fields’ theory of algebraic functions IIII Example Consider the rational function H(x, y ) = y 2 + x subject to F (x, y ) = y 3 + x 3 y + x = 0. Then the order of coincidence can by found by substituting y = Pi into H, resulting in 1 ω 8 2 ω 4 16 H(x, Pi ) = (ωx 3 + x 3 + · · · )2 + x = ω 2 x 3 + x + x 3 + ··· . 3 9 Thus the order of coincidence of H(x, y ) with respect to the branch y − Pi , i = 1, 2, 3, is 2 , the least exponent in the series 3 expansion of H(x, Pi ). Marcus Emmanuel Barnes Simon Fraser University
  • 62. Fields’ theory of algebraic functions IV Using the basic machinery provided by the concept of order of coincidence, Fields is able to build up a theory that recovers the core results of algebraic function theory, like the Riemann-Roch theorem. Marcus Emmanuel Barnes Simon Fraser University
  • 63. Fields’ theory of algebraic functions V In order to give state one version of the Riemann-Roch theorem that Fields gives, we need to clearify some terminology. Definition (Adjoint Curve) A curve C is said to be adjoint to a curve C when the multiple points of C are ordinary or cusps and if C has a point of multiplicity of order k − 1 at every multiple point of C of order k. Definition (Strength of a Set of Multiple Points) Given a curve F (x, y ) = 0 of order n, the strength of a set of Q (multiple) points used in determining an adjoint curve of degree n − 3 is defined to be the number of q conditions to which the coefficients of the general adjoint curve of degree n − 3 must be subjected in order that it may pass through these Q points. Marcus Emmanuel Barnes Simon Fraser University
  • 64. Fields’ theory of algebraic functions VI An algebraic equation F (x, y ) = 0 can be factored into a product of ρ irreducible factors. In stating the Riemann-Roch theorem, Fields uses the following notations. He indicates the poles ci of the first order by ci−1 and uses the term “coincidences” to indicate singularities such as the ci s. Marcus Emmanuel Barnes Simon Fraser University
  • 65. Fields’ theory of algebraic functions VII The following is one version of the theorem which Fields gives in his 1906 memoir: Theorem (Riemann-Roch) The most general rational function of (x, y ) whose infinities are −1 −1 included under a certain set of Q infinities c1 , . . . , cQ , depends upon Q − q + ρ arbitrary constants where q is the strength of the set of Q coincidences c1 , ..., cQ . Marcus Emmanuel Barnes Simon Fraser University
  • 66. Reception of Fields’ work on algebraic functions I G. Landsberg, who had also undertaken work to find new algebraic proofs of the Riemann-Roch theorem, in reviews in the Jahrbuch uber die Fortschritte der Mathematik, a reviewing and abstracting ¨ journal based in Germany, stated that he had reservations about Fields’ earlier works on algebraic function theory, particularly with regards to simplifying assumptions that were made based on geometric arguments — that is, Fields’ approach was not algebraic enough for Landsberg Marcus Emmanuel Barnes Simon Fraser University
  • 67. Reception of Fields’ work on algebraic functions II Given that much of Fields’ later writings were reworkings of various parts of his 1906 monograph, we can surmise that the reception of his monograph was lukewarm. Marcus Emmanuel Barnes Simon Fraser University
  • 68. Reception of Fields’ work on algebraic functions III Consider Fields’ paper of 1910 entitled “The Complementary Theorem” which appeared in the pages of the American Journal of Mathematics. G. Faber of the University of K¨nigsberg, in his o reviewing the paper in the Jahrbuch wrote that “the paper purports to give a proof of the so-called ‘Weierstrass Preparation Theorem’ that the author gave in the 11th chapter of his Theory of Algebraic Functions, by a shorter and simple one,” however “the proof still seems to me long and hard to understand.” Marcus Emmanuel Barnes Simon Fraser University
  • 69. Reception of Fields’ work on algebraic functions IV In another review, on Fields’ paper entitled “Direct derivation of the complementary theorem from elementary properties of the rational functions,” which was published in the proceedings of the fifth International Congress of Mathematicians in 1913, Prof. Lampe of Berlin writes, after quoting Fields’ own introduction to a paper in the Philosophical Transactions of the Royal Society where Fields’ claims to have achieved simplification, that “perhaps he [Fields] could try for even more simplification.” Marcus Emmanuel Barnes Simon Fraser University
  • 70. Decline in research productivity Fields’ research productivity started to subside as he spent more and more time as a scientific organizer in the 1920s and as his health began to deteriorate. Marcus Emmanuel Barnes Simon Fraser University
  • 71. John Charles Fields, 1863-1932. Fields’ life ended on August 9th, 1932, apparently from stroke. He is buried in Hamilton Cemetery which overlooks the western end of Lake Ontario (at “Cootes Paradise” where McMaster University now sits). Figure: Fields’ gravestone:“John Charles Fields, Born May 14, 1863, Died August 9, 1932.” Marcus Emmanuel Barnes Simon Fraser University
  • 72. Fields’ estate He left an estate of $45071, a large part of which was used toward what would become the international medals in mathematics he was in the process of organizing. He left his brother with a small annuity, and his maid Julia Agnes Sinclair, widow, a small pension as long as she remained unmarried. Marcus Emmanuel Barnes Simon Fraser University
  • 73. The impact of Fields’ mathematical work Fields’ papers are badly written in the sense that they are hard to follow. Marcus Emmanuel Barnes Simon Fraser University
  • 74. The impact of Fields’ mathematical work Fields’ papers are badly written in the sense that they are hard to follow. So who exactly read his work? Marcus Emmanuel Barnes Simon Fraser University
  • 75. The impact of Fields’ mathematical work Fields’ papers are badly written in the sense that they are hard to follow. So who exactly read his work? Clearly, his student S. Beatty (1881-1970) read his work; those who reviewed Fields’ work must have read portions of it; the American mathematician Bliss mentions Fields’ book in the preface of his book on algebraic functions, but does not use Fields’ approach in his book. Marcus Emmanuel Barnes Simon Fraser University
  • 76. The impact of Fields’ mathematical work II His work was well regarded, as can be seen by his election to the Royal Society in 1913 and to other societies and academies. Marcus Emmanuel Barnes Simon Fraser University
  • 77. The impact of Fields’ mathematical work II His work was well regarded, as can be seen by his election to the Royal Society in 1913 and to other societies and academies. Fields’ work was “old fashioned”, did not afford easy generalization, and would be subsumed by approaches utilizing the machinery afforded by modern abstract algebra, so the ultimate influence of his work is small, maybe mostly expressed through the work of Fields’ student S. Beatty. Marcus Emmanuel Barnes Simon Fraser University
  • 78. The impact of Fields’ mathematical work II His work was well regarded, as can be seen by his election to the Royal Society in 1913 and to other societies and academies. Fields’ work was “old fashioned”, did not afford easy generalization, and would be subsumed by approaches utilizing the machinery afforded by modern abstract algebra, so the ultimate influence of his work is small, maybe mostly expressed through the work of Fields’ student S. Beatty. However, consider this... Marcus Emmanuel Barnes Simon Fraser University
  • 79. The impact of Fields’ mathematical work III The research legitimized Fields as a scientific authority. Marcus Emmanuel Barnes Simon Fraser University
  • 80. The impact of Fields’ mathematical work III The research legitimized Fields as a scientific authority. Fields’ authority surely must have played a key role in the his push, along with others, to get governmental support for scientific research, a goal that would eventually come to fruition, as can be seen in the funding structures that exist today. Marcus Emmanuel Barnes Simon Fraser University
  • 81. The impact of Fields’ mathematical work III The research legitimized Fields as a scientific authority. Fields’ authority surely must have played a key role in the his push, along with others, to get governmental support for scientific research, a goal that would eventually come to fruition, as can be seen in the funding structures that exist today. According to S. Beatty, writing around 1930, Fields “by his insistence on the value of research as well as by the importance of his published papers, has, perhaps, done most of all Canadians to advance the cause of mathematics in Canada.” Marcus Emmanuel Barnes Simon Fraser University
  • 82. Thank you for your attention Thank you for your attention! Marcus Emmanuel Barnes Simon Fraser University
  • 83. Fields’ love affair [The Star, August 10, 1932: “Death Claims Noted Savant at University”] “Dr. Fields was a bachelor, and an amusing story is told of his ‘love affairs’ abroad. On one occasion, while abroad, he arranged to meet Professor Love at a certain hotel. When Dr. Fields arrived there, the girl clerk presented him with a telegram reading: (Sorry, I cannot meet you, Love). The doctor treasured this evidence of his ‘love affairs.’” Marcus Emmanuel Barnes Simon Fraser University

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