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Sophus M. Lie LA GACETA DE LA RSME, Vol. 5.1 (2002), Pags.´ 121{130 121 HISTORIA Secci´on a cargo de Antonio J. Dur´an1 Sophus M. Lie por Juan Nu´nez~ Vald´es y Angel´ F. Tenorio Villal´on Los dos matem´aticos m´as insignes con los que el pueblo noruego ha con- tribuido a la ciencia son, sin ninguna duda, Niels Henrik Abel (Isla de Finnoy, 1802 - Arendal, 1829) y Sophus Marius Lie. Sophus Marius Lie (en adelante, S. Lie) naci´o el 17 de diciembre de 1842 en la aldea de Eid, situada en el fondo del Eidfjord, una de las ramas del Nordfjord, que penetra profundamente en tierra firme a doscientos kil´ometros al norte de Bergen y a unos ciento veinte kil´ometros al sudoeste de Andalsnes. Su padre, Johann Herman Lie, era un pastor lu- terano de la aldea de Eid, que fue llamado en 1851 a ejercer su ministerio en la pequena~ ciudad de Moss, situada a orillas del fiordo de Oslo, no lejos de la ca- pital, llamada por entonces Christiania. Johann Lie ten´ıa seis hijos, siendo Sophus el menor de todos. El pequeno~ Sophus asisti´o a la escuela comunal Sophus Lie de la ciudad, en la que curs´o Primaria y Secundaria. Cuando cumpli´o los 15 anos,~ en 1857, ingres´o en un colegio privado de Chris- tiania llamado Nissen's Private Latin School, en el que decidi´o seguir la carrera militar, si bien por problemas de visi´on tuvo que dejar esa idea. De todas for- mas, en 1859 entr´o en la universidad de Christiania para estudiar Matem´aticas y Ciencias. Aunque era un buen alumno en general, no lleg´o a mostrar aptitu- 1 Los interesados en colaborar con esta secci´on pueden dirigir sus contribuciones a la siguiente direcci´on: Antonio J. Dur´an; Secci´on Historia Gaceta RSME; Departamento de An´alisis Matem´atico; Facultad de Matem´aticas; Universidad de Sevilla; Aptdo. 1160; 41080{ Sevilla; [email protected] 122 HISTORIA des especiales en ninguna disciplina en concreto. Se podr´ıa decir que careci´o de la temprana capacidad matem´atica que acompan´~o desde su juventud a ma- tem´aticos como Pascal, Clairaut o el mismo Abel. En esta etapa universitaria, Lie asisti´o a las clases de Ludwig Sylow (1832-1918) sobre los trabajos de Abel y Galois en ecuaciones algebraicas y a las de Carl Anton Bjerknes (1825-1903) sobre matem´aticas. Tras obtener su diploma de licenciado en ciencias en 1865, sin recibir distinci´on alguna, S. Lie, que deseaba continuar una carrera acad´emica, ejerci´o de tutor de otros estudiantes d´andoles clases particulares. Al tiempo, persigui´o objetivos propios de astronom´ıa y mec´anica, hasta 1868. En ese ano~ y tras leer los trabajos de los ge´ometras Poncelet (1788-1867) y Pluc¨ ker (1801-1868), S. Lie se sinti´o muy atra´ıdo por ambos. De hecho, su admiraci´on por estos dos grandes maestros, a los cuales nunca lleg´o a conocer, se mantuvo toda su vida. En concreto, Pluc¨ ker fue uno de los que m´as le motiv´o debido a que indica- ba nuevas geometr´ıas que usaban l´ıneas y curvas en lugar de puntos, como los elementos de un espacio dado. En 1869, S. Lie public´o su primer trabajo, que tuvo que pagar de su bolsillo, basado en esa idea de Pluc¨ ker que tanto le llam´o la atenci´on. Posteriormente escribi´o una exposici´on m´as detallada, con ideas revolucionarias para la ´epoca, que no fue aceptada por algunos matem´aticos de su tiempo. Incluso la Academia de Ciencias de Christiania tuvo sus dudas a la hora de publicar su trabajo. Para intentar arreglar esta situaci´on, S. Lie lleg´o a escribir cartas a dos matem´aticos prusianos, Carl Theodor Reye (1838-1919) y Alfred Clebsch (1833-1872), pidi´endoles que se le reconocieran sus ideas. No obstante, fue su amigo Motzfeldt el que le anim´o a publicar sus ideas matem´aticas y el resultado apareci´o a fines de 1869, al ser publicado su art´ıculo en el Journal de Crelle. Ese art´ıculo le vali´o para obtener una beca del Estado que le permiti´o viajar y conocer a matem´aticos importantes. De este modo, S. Lie pudo permanecer en el extranjero una larga temporada y aprender de los mejores matem´aticos del momento. As´ı, en el invierno de 1869 fue a Prusia, visitando G¨ottingen y luego Berl´ın. En Berl´ın, conoci´o a Leopold Kronecker (1823-1891), Erns Edward Kummer (1810-1893) y Karl Weierstrass (1815- 1897). A S. Lie no le atra´ıa mucho el estilo de Weierstrass, que era el que dominaba en Berl´ın. Sus intereses se acercaban m´as a Kummer, leyendo en el seminario que ´este dirig´ıa sus propios resultados y llegando incluso a corregir algunos errores que el propio Kummer cometi´o en un trabajo sobre l´ıneas de congruencias de grado 3. En esa ´epoca, fue importante para S. Lie conocer a un alumno de Pluc¨ ker llamado F´elix Klein (1849-1925). Entre ambos surgi´o una amistad (que durar´ıa hasta la muerte de S. Lie), que les llev´o a trabajar juntos y a intercambiar ideas de una forma igual de fruct´ıfera para los dos. Fue en Berl´ın donde S. Lie desarroll´o una nueva autoconfianza en sus ha- bilidades matem´aticas. Recibi´o alabanzas tanto de Kummer como de otros matem´aticos que antes le hab´ıan ignorado, entre ellos los anteriormente ci- tados Reye y Clebsch, que le animaban a seguir. Al respecto, S. Lie escribi´o LA GACETA 123 a su amigo Motzfeldt en Christiania, dici´endole: \ : : : en los anos~ 1864-68, realmente subestim´e mi propio poder mental". En la primavera de 1870, S. Lie y Klein decidieron ir a Par´ıs y despu´es a Inglaterra, por lo que Klein pidi´o cartas de recomendaci´on para ´el y para S. Lie al Ministerio de Educaci´on de Berl´ın, pero la respuesta fue negativa. De todos modos se fueron juntos a Par´ıs, donde conocieron a Camille Jordan (1838-1922), a Gast´on Darboux (1842-1917) y a Michel Chasles (1793-1880), entre otros. Fue en Par´ıs donde S. Lie descubri´o la c´elebre transformaci´on que lleva su nombre y que establece una relaci´on entre las rectas y las esferas del espacio, por una parte, y entre las l´ıneas asint´oticas y las l´ıneas de curvatura de la superficie, por otra. En esta estancia en Par´ıs, S. Lie y Klein (que tuvieron habitaciones ado- sadas) empezaron a interesarse en los trabajos de Evariste Galois (1811-1832), gracias al \Trait´e des sustitutions et des ´equations alg´ebriques" publicado en 1868 por Jordan. Ambos amigos conocieron el aprendizaje de las matem´aticas que defend´ıa Charles Hermite (1822-1901) y la metodolog´ıa que ´este practica- ba con sus estudiantes. Tambi´en aprendieron la virtuosidad de la escuela de geometr´ıa de Michel Chasles. Es en esta ´epoca en la que Klein y S. Lie publi- caron sus tres volumenes´ Gesemmelte mathematische Abhandlungen. Aunque en ese momento ambos decidieron viajar a Inglaterra, tuvieron que abando- nar esa idea al declararse la guerra franco-alemana en 1870. Klein, por ser de procedencia alemana, tuvo que volver de inmediato a Berl´ın, mientras que S. Lie prefiri´o quedarse en Francia (pudo hacerlo por ser noruego) ya que las discusiones matem´aticas que all´ı encontraba le resultaban estimulantes. Sin embargo, cuando en agosto de ese ano~ el ej´ercito alem´an atac´o a parte del ej´ercito franc´es en Metz, S. Lie decidi´o que era tiempo de partir y salir hacia Italia, viajando a pie atravesando Francia. Desgraciadamente, los franceses le detuvieron al atravesar el bosque de Fontainebleau y le hicieron pasar cuatro semanas en la prisi´on de dicha localidad, de la que sali´o gracias a la media- ci´on apurada de Darboux, que consigui´o convencer a las autoridades francesas de que las cartas en alem´an que ten´ıa S. Lie en su poder eran inofensivas y trataban s´olo de matem´aticas. Despu´es, desde Italia, a donde lleg´o sin proble- mas, S. Lie volvi´o a Christiania pasando por Suiza y Alemania, con el fin de encontrarse de nuevo con Klein. En 1871, la Universidad de Christiania le otorg´o una beca de investigaci´on para la obtenci´on del doctorado en ciencias. Durante ese tiempo fue profesor asistente de esta misma Universidad y tambi´en fue profesor del Nissen's Private Latin School, por lo que dio clases en dos de los centros en los que antes hab´ıa estado como alumno. En la universidad, S. Lie present´o un trabajo sobre una clase de transformaciones geom´etricas (en noruego) para su doctorado, que consigui´o en julio de 1872. Esta disertaci´on consist´ıa en las ideas de sus primeros resultados publicados en el Journal de Crelle y adem´as en el trabajo en transformaciones de contacto, un caso especial de transformaciones que descubri´o en Par´ıs. Tras obtener su tesis, S. Lie ensen´~o durante un ano~ en la Universidad de Lund, en Suecia (por entonces Noruega estaba bajo soberan´ıa 124 HISTORIA sueca), obteniendo posteriormente una c´atedra que el Parlamento noruego hab´ıa creado en la Universidad de Christiania. Es en ese momento, cuando puede entregarse de lleno al desarrollo de sus ideas.
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