Background
Marius Sophus Lie was born on December 17, 1842, in Nordfjordeid, Norway, the sixth and youngest child of a Lutheran pastor, Johann Herman Lie.
Problemveien 7, 0315 Oslo, Norway
Lie first attended school in Moss (Kristianiafjord), then, from 1857 to 1859, Nissen’s Private Latin School in Christiania. He studied at Christiania University (now University of Oslo) from 1859 to 1865, mainly mathematics and sciences. Although mathematics was taught by such people as Bjerknes and Sylow, Lie was not much impressed. After his examination in 1865, he gave private lessons, became slightly interested in astronomy, and tried to learn mechanics; but he could not decide what to do. The situation changed when, in 1868, he hit upon Poncelet’s and Plücker’s writings.
2019
Nordfjordeid, Norway
Monument in Nordfjordeid
Problemveien 7, 0315 Oslo, Norway
Lie first attended school in Moss (Kristianiafjord), then, from 1857 to 1859, Nissen’s Private Latin School in Christiania. He studied at Christiania University (now University of Oslo) from 1859 to 1865, mainly mathematics and sciences. Although mathematics was taught by such people as Bjerknes and Sylow, Lie was not much impressed. After his examination in 1865, he gave private lessons, became slightly interested in astronomy, and tried to learn mechanics; but he could not decide what to do. The situation changed when, in 1868, he hit upon Poncelet’s and Plücker’s writings.
(Sophus Lie had a tremendous impact in several areas of ma...)
Sophus Lie had a tremendous impact in several areas of mathematics. His work centered on understanding continuous transformation groups and showing how these groups supply an organizing principle for different areas of mathematics. One of those areas is differential equations, and this book is his magnum opus on the subject. One of Lie's major interests was to develop a theory for differential equations in analogy to Galois theory for polynomial equations. He showed how one could naturally associate a continuous group to a differential equation, so that the solvability of the group (in the sense of algebra) is related to the solvability of the differential equation (in the sense of ''quadrature'', meaning integration and algebraic manipulations). The book also discusses Lie's remarkable classification of all three-dimensional groups and their possible actions on the plane. The exposition in the book is elementary and contains numerous examples. This is a textbook on the integration of ordinary and partial differential equations in which the Lie theory for solving such equations is expounded.
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Marius Sophus Lie was born on December 17, 1842, in Nordfjordeid, Norway, the sixth and youngest child of a Lutheran pastor, Johann Herman Lie.
Lie first attended school in Moss (Kristianiafjord), then, from 1857 to 1859, Nissen’s Private Latin School in Christiania. He studied at Christiania University (now University of Oslo) from 1859 to 1865, mainly mathematics and sciences. Although mathematics was taught by such people as Bjerknes and Sylow, Lie was not much impressed. After his examination in 1865, he gave private lessons, became slightly interested in astronomy, and tried to learn mechanics; but he could not decide what to do. The situation changed when, in 1868, he hit upon Poncelet’s and Plücker’s writings.
Later, he called himself a student of Plücker’s, although he had never met him. Plücker’s momentous idea to create new geometries by choosing figures other than points - in fact, straight lines - as elements of space pervaded all of Lie’s work.
Lie’s first published paper brought him a scholarship to study abroad. He spent the winter of 1869-1870 in Berlin, where he met Felix Klein, whose interest in geometry also had been influenced by Plücker’s work. This acquaintance developed into a friendship that, although seriously troubled in later years, proved crucial for the scientific progress of both men. Lie and Klein had quite different characters as humans and mathematicians: the algebraist Klein was fascinated by the peculiarities of charming problems; the analyst Lie, parting from special cases, sought to understand a problem in its appropriate generalization.
Lie and Klein spent the summer of 1870 in Paris, where they became acquainted with Darboux and Camille Jordan. Her Lie, influenced by the ideas of the French “anallagmatic” school, discovered his famous contact transformation, which maps straight lines into spheres and principal tangent curves into curvature lines. He also became familiar with Monge’s theory of differential equations. At the outbreak of the Franco-Prussian war in July, Klein left Paris; Lie, as a Norwegian, stayed. In August he decided to hike to Italy but was arrested near Fontainebleau as a spy. After a month in prison, he was freed through Darboux’s Intervention. Just before the Germans blockaded Paris, he escaped to Italy. From there he returned to Germany, where he again met Klein.
In 1871 Lie was awarded a scholarship to Kristiania University. He also taught at Nissen’s Private Latin School. In July 1872 he received his Ph.D. During this period he developed the integration theory of partial differential equations now found in many textbooks, although rarely under his name. Lie’s results were found at the same time by Adolph Mayer, with whom he conducted a lively correspondence. Lie’s letters are a valuable source of knowledge about his development.
In 1872 a chair in mathematics was created for him at Kristiania University. In 1873 Lie turned from the invariants of contact transformations to the principles of the theory of transformation groups. Together with Sylow he assumed the editorship of Niels Abel’s works.
His main interest turned to transformation groups, his most celebrated creation, although in 1876 he returned to differential geometry. In the same year, he joined G. O. Sars and Worm Müller in founding the Archir för mathenratik og naturvidenskab. In 1882 the work of Halphen and Laguerre on differential invariants led Lie to resume his investigations on transformation groups.
Lie was quite isolated in Kristiania. He had no students interested in his research. Abroad, except for Klein, Mayer, and somewhat later Picard, nobody paid attention to his work. In 1884 Klein and Mayer induced F. Engel, who had just received his Ph.D., to visit Lie in order to learn about transformation groups and to help him write a comprehensive book on the subject. Engel stayed nine months with Lie. Thanks to his activity the work was accomplished, its three parts being published between 1888 and 1893, whereas Lie’s other great projects were never completed. F. Hausdorff, whom Lie had chosen to assist him in preparing a work on contact transformations and partial differential equations, got interested in quite different subjects.
This happened after 1886 when Lie had succeeded Klein at Leipzig, where, indeed, he found students, among whom was G. Seheffers. With him, Lie published textbooks on transformation groups and on differential equations, and a fragmentary geometry of contact transformations. In the last years of his life, Lie turned to foundations of geometry, which at that time meant the Helmholtz space problem.
In 1889 Lie was struck by what was then called neurasthenia. Treatment in a mental hospital led to his recovery, and in 1890 lie could resume his work. His character, however, had changed greatly. He became increasingly sensitive, irascible, suspicious, and misanthropic, despite the many tokens of recognition that were heaped upon him.
Meanwhile, his Norwegian friends sought to lure him back to Norway. Another special chair in mathematics was created for him at Kristiania University, and in September 1898 he moved there. He died of pernicious anemia the following February. His papers have been edited, with excellent annotations, by F. Engel and P. Heegaard.
(Sophus Lie had a tremendous impact in several areas of ma...)
(German Edition)
1896Lie’s first papers dealt with very special subjects in geometry, more precisely, in differential geometry. In comparison with his later performances, they seem like classroom exercises; but they are actually the seeds from which his great theories grew. Change of the space element and related mappings, the lines of a complex considered as solutions of a differential equation, special contact transformations, and trajectories of special groups prepared his theory of partial differential equations, contact transformations, and transformation groups. He often returned to this less sophisticated differential geometry. His best-known discoveries of this kind during his later years concern minimal surfaces.
The crucial idea that emerged from his preliminary investigations was a new choice of space element, the contact element: an incidence pair of point and line or, in n dimensions, of point and hyperplane, The manifold of these elements was now studied, not algebraically, as Klein would have done - and actually did - but analytically or, rather, from the standpoint of differential geometry.
Lie’s integration theory was the result of marvelous geometric intuitions. The preceding short account is the most direct way to present it. The usual way is a rigmarole of formulas, even in the comparatively excellent book of Engel and Faber. Whereas transformation groups have become famous as Lie groups, his integration theory is not as well known as it deserves to be. To a certain extent, this is Lie’s own fault. The nineteenth-century mathematical public often could not understand lucid abstract ideas if they were not expressed in the analytic language of that time, even if this language would not help to make things clearer. So Lie, a poor analyst in comparison with his ablest contemporaries, had to adapt and express in a host of formulas, ideas which would have been said better without them. It was Lie’s misfortune that by yielding to this urge, he rendered his theories obscure to the geometricians and failed to convince the analysts.
Lie was described as an open-hearted man of gigantic stature and excellent physical health. After he was struck by what was then called neurasthenia, his character changed greatly. He became increasingly sensitive, irascible, suspicious, and misanthropic, despite the many tokens of recognition that were heaped upon him.
Lie married Anna Birch in 1874. The couple had three children: Marie, Dagny, and Herman.