We continue here our attempt of a systematic historical account of Group Theory inspected by means of the lives and the works of its main actors. The aim is to bring the interested reader through orig- inal correspondences, published and unpublished works, historical perspectives, diatribes and friendships.

This issue contains the translation of a memory of Sophus Lie writ- ten by Ludwig Sylow. It was published in the 1899 issue of Archiv for Mathematik of Naturvidenskab soon after Lie’s death.

We are grateful to Gunnar Traustason for his translation from Nor- wegian

Francesco de Giovanni

Marco Trombetti

Mattia Brescia

Directory of Open Access Journals

Advances in Group Theory and Applicationsc© 2018 AGTA - www.advgrouptheory.com/journal6 (2018), pp. 129–134 ISSN: 2499-1287DOI: 10.32037/agta-2018-009A History of Group Theorythrough the Lives of Group TheoristsSophus Lie - Part 1We continue here our attempt of a systematic historical account ofGroup Theory inspected by means of the lives and the works of itsmain actors. The aim is to bring the interested reader through orig-inal correspondences, published and unpublished works, historicalperspectives, diatribes and friendships.This issue contains the translation of a memory of Sophus Lie writ-ten by Ludwig Sylow. It was published in the 1899 issue of Archiv forMathematik of Naturvidenskab soon after Lie’s death.We are grateful to Gunnar Traustason for his translation from Nor-wegian.Mattia BresciaFrancesco de GiovanniMarco Trombetti130 A history of group theory: Sophus Lie - Part 1Sophus LieMarius Sophus Lie was born on December 17th, 1842 in the vil-lage Nordfjordeid. His father was a priest that moved later to thetown Moss, whence the young Sophus was sent to Christiana tostudy at the private school Nissens Latin- og Realsskole. As a young ladhe was evidently, without comparison, the best mathematics studentin his class. However, while Abel neglected his other school subjectsin order to focus on higher mathematics, and at once started to workon a problem which later would be solved by him, Lie was good inall subjects and had no special calling for any particular discipline.He graduated with distinction in 1859 with outstanding marks inall subjects apart from Norwegian writing. He started with uncer-tainty as to what to study afterwards. He was not aware of his ex-ceptional mathematical gifts and quite on the contrary he later saidthat the road to Mathematics was long and difficult for him. Aftersome wavering he opted to study to obtain teaching qualificationsin mathematics and natural sciences, which he passed with excellentmarks in 1865. In 1862 he had followed a short course on the theoryof substitutions through which he became familiar with the conceptof a Galois group that had influence on his later development. Butwhat gave him the early stimulation and made him realise that therelurked a mathematician in him, to quote his own words, was howeverthe study of modern geometry, a subject which, before Lie, was givenlittle attention at our university, but that had profound impact on Lie.Lie was in fact a natural geometer; he worked essentially with geo-metrical settings and geometrical images, even when he was broadlyworking within analysis. His first work is in 1869 and involves arepresentation of planar geometrical objects, whereby theorems inplanar geometry are transformed into stereometric theorems, whereone replaces a point in the real or complex plane by a real straightline in the space. As his exceptional gifts were becoming clear to him-self and others, he applied for and obtained a stipend to go abroad,and spent the autumn 1860 in Germany, followed by a period inParis. This trip was important for the path that Lie would take later.Certainly he brought with him from home his core ideas, whereinhe had already started his research on transformation groups anddifferential equations, but the contact with many outstanding math-ematicians influenced to a large degree how these core ideas devel-oped. He spent a short time under the supervision of Prof. A. Cleb-Adv. Group Theory Appl. 6 (2018) 131sch in Götttingen, and made acquaintance with the brilliant youngmathematicians F. Klein, C. Jordan and G. Darboux and probablymany others through this group. In particular, he formed a friend-ship with Klein, who was Plücker’s student and publisher of hislatest papers. In collaboration with Klein he studied curves and sur-faces that are invariant under a group of linear transformations andpresented some of his results to the Paris academy. The outbreak ofthe Franco-German war at this moment put an end to their stay to-gether in Paris, but for many years there was a lively exchange ofideas between them. Lie, a strong walker, wanted to travel throughFrance on foot but didn’t get further than Fontainebleau, where hiswander was treated as suspicious and he was arrested and his papersonly made the situation worse. In vain he appealed to French math-ematicians; the blond giant looking too much like a German. At lastDarboux was found, and could easily convince the authorities thatthe suspicious notes were mathematical formulas and figures, andthat the German name belonged to a known mathematician. He wasthus finally released and continued his trip through Switzerland tothe north of Italy and returned home at the end of the year.Soon after his return Lie received one of the university researchscholarships. He continued his geometric studies, that had now de-veloped more and more into the study of geometric transformations.He had come up with a peculiar transformation that transformsstraight lines of the space into spherical surfaces, the foundation forspherical geometry, and this had led to a remarkable result, wherehe could determine the asymptotic curves of the fourth degree Kum-mer surface with 16 knot points, and show that they are algebraic;this was presented to the Berlin Academy in 1870. This transforma-tion now became a basis for a thesis that he submitted and defendedto obtain a PhD in 1871. In this thesis, and in particular in the subse-quent work, Lie used also his geometric transformations to study dif-ferential equations, that now became his central theme. Also in 1871he published jointly with Klein a work in Mathematische Annalenwhere transformations are again used to study differential equationsand the notion of a group as well as a number of group theoreticalterms are used for the transformations; now these were also doneinfinitesimally. Lie was now also working within the field of contin-uous transformations.During the winter 1871-1872, Lie applied for a vacant professor-ship in Lund. But the references about his scientific work that came132 A history of group theory: Sophus Lie - Part 1from many leading mathematicians of the time led to a special pro-fessorship being created for him at our own university. The appoint-ment was in July 1872. He now worked as a professor in Christianiauntil the year 1886; from now on he could devote his time on hisresearch and the result was an exceedingly rich product of originalwork.The problem of integrating partial differential equations of the firstorder became the subject of a series of papers, and he succeededin simplifying the older methods, due to mathematicians such asCauchy and Jacobi. In this context, he met his friend and later col-league Prof. A. Mayer in Leipzig, who at the same time and indepen-dently from Lie, studied the same questions; the two researchers hadcome to exactly the same results, and it led to an unusual collabora-tion between them, as one took up and worked out the ideas of theother.Soon after, especially since the year 1876, when he, together withProfessors Worm-Muller and G.O. Sars founded this journal, his pub-lications focused around his own studies, namely the theory of con-tinuous transformation groups. He expanded the original concept toinclude contact transformations, thereby creating a new and wideranging theory of importance for many, almost all the areas of math-ematics. Others contributed to the theory by using it to integratecertain partial differential equations of the second order, and evenclasses of these, later also general differential equations that are pre-served under a known transformation group, something that hadalready been touched on in one of his earlier joint work with Klein.Nor did he forget the actual geometry; he did extensive work onminimal surfaces, surfaces with constant curvature, translation sur-faces, and in each of these areas, he provided a plethora of new re-sults.In 1886, Lie was offered a professorship in Leipzig. As his profes-sorship in Christiania was personal, he was allowed to take a sabbat-ical while taking up the new position and was free to return backto his old post, something that probably had been his plan from thebeginning. The arrival in Leipzig marked the beginning of the lastepisode of Lie’s life that differed much from what his life had beenlike previously. In Germany, the conditions for a leading scientist arequite different than here; not only is the literature and other helpresources richer, the interests of the general public larger, but onecan count on being always surrounded by a circle of students, thatAdv. Group Theory Appl. 6 (2018) 133study science as something of value in itself. This did not happenimmediately to Lie, because in Germany the students tend to fol-low their lecturer as he moves from one university to another, andLie’s superior in the post was just one of Germany’s greatest math-ematicians, namely his old friend Klein. But soon his growing repu-tation brought a crowd of students around him; and these came notonly from Germany, as these were joined by students from Franceand America, and then of course young men with excellent gifts.Lie continued publishing high quality papers, that now appearedusually in the Saxon scientific Berichte; but in addition he gatheredhis work on transformation theory as a collection in three volumesTheorie der Transformationsgruppen, that was published by TeubnersForlag in Leipzig in the period 1888–1893. Here he was assisted bya younger German scientist, Prof. F. Engel, primarily for languageand style; but of course his assistance was not restricted to this. Prof.Engel had already met Lie in 1884 when, upon suggestion of theLeipzig mathematicians Mayer and Klein, he travelled to Christianiato spend some time there to absorb Lie’s theories and to assist him inthe preparation of a work which Lie had already decided to publish.Additionally, Lie’s lectures on differential equations and his lectureson continuous groups, both produced by his student Dr. G. Scheffers,the first closely monitored by Lie and both with his strong guidance,were published in 1891 and 1893, respectively. But Lie had more am-bitious plans. He wanted to publish a work, that would be a selfcontained account of the theory of differential invariants that wouldthen lead to the theory of infinite continuous groups, and in additionthere would be an application of these theories for the integrationof differential equations. Another work should include an elaboratepresentation of his studies in geometry. This has also resulted in afirst volume under the title Geometrie der Berührungstransformationendargestellt von Sophus Lie und Georg Scheffers (Leipzig, 1896). The intro-duction says that it contains a representation of Lie’s geometric workin the years 1869-1872. So the work is still far from being complete.Unfortunately, he would not live to see these plans go through.Despite strong constitution, the intense work had taken its toll. Oncehe was suffering so badly from lack of sleep that he needed to take abreak to seek medical treatment. He did regain some energy but stilldid not feel that well; the climate of Leipzig was not compatible withhis physical condition. The desire to return home became strongerand when this became known, the parliament generously augmented134 A history of group theory: Sophus Lie - Part 1his salary to 10.000 kronas, so that the return would not restrain himfinancially. He got released from his post in Leipzig and took up hisposition at the University of Christiania from the second semesterof 1898. But his health faltered upon arrival. It had been his intentionto give two series of lectures on differential equations, one for moreadvanced students and one for beginners. For the former there werethree American students, that had their lessons at his home, as soonhe was not able to leave home. The second series never got started.The condition deteriorated slowly; he was only short time bedriddenand passed away without pain on February 18th, 1899. The cause ofdeath was anaemia. He was buried at the expense of the universitywith a gracious ceremony; the king and the parliament sent theirrepresentatives and the general attendance was extraordinary. Hiswife Anna Sophie, born Birch and related to Nils Henrik Abel on themother side, survives him with two daughters and one son.Lie’s greatest legacy is for having created the theory of continuoustransformation groups. He has himself been involved with its appli-cation in Geometry, in the integration of differential equations, in thehalf philosophical chapter on the elementary foundation of geometry,and in the theory of complex numbers. Presumably, the future willshow that this is not all. He was widely regarded as one of the verygreatest mathematicians of the current era, and there has been nolack of academic honours. I only mention that he was a member orcorrespondent of the Royal Society of London, Académie des Sciences inParis, Accademia dei Lincei in Rome and the academy in Petersburg.But his greatest honour lies in the general influence that his theorieshave had, as a number of mathematicians, including some famousmen, have gone further along Lie’s new paths, and also some of hisresults are found in textbooks and used in higher studies.Lie’s death was a major loss for science, if one only looks at theplanned projects that were not completed; but he left us relativelyyoung, giving hope for new discoveries. For our science and uni-versity the loss is irreparable, so much more for the men, who areactually called upon to continue his research into the longer future,his own students, almost exclusively foreigners.L. Sylow

A History of Group Theory through the Lives of Group Theorists: Sophus Lie - Part 1

http://www.advgrouptheory.com/journal/Volumes/6/Sophus%20Lie.pdf

A History of Group Theory through the Lives of Group Theorists: Sophus Lie - Part 1

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals