Sophus Lie: Unfortunate genius

Like the elder Niel Abel, Sophus Lie (1842-1899) emerged as a phenomenon in Nordic countries in the context that the region's science was almost unknown in the international arena. Throughout his life, he constantly brought out extremely bold ideas.

Sophus Lie is one of the big names in mathematical history, one of the visionaries. He also created a new subject: the Lie math group, math now developed and rooted in most math and physics-math disciplines.

Jean Dieudonné, one of the founders of the Bourbaki school, must not have been able to produce any valuable work without the Lie group mathematics. Later mathematical and physical works, especially the works of Élie Cartan and Hermann Weyl, were also thanks to Lie's mathematical theory.

During the 30-year period of research, Sophus Lie published works about 8,000 pages in length.

The start is not perfect

Sophus Lie was born on December 17, 1842 in Nordfjordeid, a small city in the west of Norway. When Sophus was 9, his family moved to Moss, which is located in the southeastern part of Oslo.

One year after arriving in Moss, Sophus's mother died. This misfortune made Sophus 10 years old almost lose his childhood and become a serious child. From the beginning of the first school year, Sophus has enjoyed a lot of subjects. Since the school in Moss did not grant a final diploma, when he was 15, Lie went to Hartvig Nissen school in the Norwegian capital. This famous school, founded by Nissen and Ole Jacob Broch, has a clear goal: to use modern pedagogical and pedagogical methods and to teach important sciences.

Sophus is also interested in Norway's educational reform process and is always involved in extensive social debates. He wrote many articles stating the need to renew the education system in Norway and he often compared books with other education such as France and Germany. He also emphasized the special importance of higher education. Although education made him interested, he eventually chose science as his career path. It should be remembered that at that time, among the 560 University of Oslo students, only about a dozen people followed the path of science.

Sophus Lie (1842-1899) - (Photo: cache.eb.com)

Studies conducted by Sophus Lie are in 3 areas. The most arduous is the field of mathematics including geometry, mechanics, technical design . In this area, Sophus Lie's professors are Broch and Carl Anton Bjerknes, mathematician Lugwig Sylow.

In Sylow's class from 1862 to 1863, there were three highly regarded people including Sophus and this class involved a new discipline: the theory of algebraic equations of Abel and Galois. This subject attracted so much Sophus that he neglected other subjects such as physics-chemistry and natural history. Therefore, his diploma was not as excellent as expected. In his final years, his personality became melancholy, odd and lacking in his own "intellectual ability" . The rigor in Catholic education does not allow people to be too weak. It was this rigidity that caused Sophus to fall into depression.

Luckily, he had a lot of friends and they shared his feelings with him during weekend trips in the bay and forests in Oslo. He also practiced sports, riding horses. In winter, he plays skiing and sleds. Lie is a very loyal friend. Ernst Motzfeldt, his classmate at Nissen School became a lawyer, and senior civil servants were always his friends and protected him.

Lie enjoys taking part in the Association réaliste discussions organized by science students. At that time, in every science in the university, there was a student association related to their field of study to offer intensive discussions to complement the courses. Lie is one of the most active members of the Society and has carried out a series of geometry talks here.

After receiving a diploma, Lie paid much attention to astronomy, he began to teach and spread this science. He participated in teaching at an observatory but did not earn an astronomical leg as desired. In order to earn a living, he took up the position of teaching for professors in mathematics, physics and astronomy sessions. He is a highly regarded professor.

Jump to math

Sports and sports movement was very developed at the time and Sophus Lie was a well-known mountain climber. He is also a good walker: on a normal day, he walks from 30 to 40 kilometers, the day of exercise, he goes for 70-80 kilometers. It has been said that one day he forgot a book in Moss and so he walked a round trip from the capital to Moss, which is about 100 kilometers a day. He walked so fast that many people thought he was a . thief.

However, Sophus's social integration ability is poor. It was this that caused him to be tormented and have a soul that was less peaceful. In March 1968, he wrote a letter to his close friend Motzfeldt: "When I said goodbye to you the day before Christmas, I thought it was far away from you forever. I always wanted to kill myself but I didn't There is enough courage, so I find ways to try to live. "

Not knowing what you want in life is a big defect. It was not until 1868 that Sophus Lie found his job when he was 26, after meeting the Danish mathematician Hyeronymus Zeuthen. At that time Zeuthen was studying in Paris under the guidance of geometric figure Michel Chasles. Thanks to Zeuthen, Lie sought and reveled in the book on the optical properties of Jean-Victor Poncelet's figures. One of Poncelet's inventions at the time was to introduce and use complex numbers in projective geometry. Besides this intriguing theme, Sophus is as passionate about the works of Allemand Julius Plucker.

In the early 1869, Lie completed his first book on realistic description of imaginary numbers. This 8-page article was immediately welcomed by two famous Danish math teachers, Broch and Bjerknes. In the same year, the article was re-published in German in the popular newspaper Journal de Crelle. Sophus Lie is a lot of attention and it is easy to get scholarships to study in Berlin, Göttingen and Paris. Upon returning home, he was immediately appointed a member of the university and the following year became a professor of the university.

The first trip abroad was very important for Lie. He was met by friends and colleagues who greatly appreciated his research. Especially in Berlin, he met Felix Klein, Alfred Clebsch in Göttingen and Gaston Darboux and Camille Jordan in Paris. Lie quickly became a close friend of Klein and together they announced three studies on geometry. Klein also visited Lie often and often visited Darboux and Jordan.

Unexpected hero

When the Franco-Prussian war broke out in July 1870, Klein left Paris for an appeal against France. He crossed the border but fell ill suddenly and could not participate in combat. At that time, Lie decided to walk from Paris to Milan to meet the mathematician Luigi Cremona. But just to Fointainebleau, he was imprisoned for being suspected of being a German spy. Immediately, on the front page a Norwegian daily was headlined: Norwegian scientist was imprisoned for suspected German spy.

Later Lie recounted: The guards then assumed that the mathematical symbols in Lie's notebook were spy codes. Fortunately, Darboux had a relationship with senior military figures and he sought a way to free Lie. Years later, Lie still considered that the time of captivity in Fontainebleau was the most calm and major period during this time, he wrote the main part of his doctoral thesis. This thesis is named 'A way of ranking geometric transformations' which he defended the following year in Oslo. According to E. Holst, Lie's first friend and biographer, the Norwegian jury was really incompetent at the same time as the French guards.

However, in Europe, this thesis is much more appreciated. Darboux commented that this is one of the most interesting discoveries of modern geometry. Nor did the Norwegian government create the best conditions for Lie to work. In 1872, Senate senators asked to give the title of special professor to Sophus Lie. They did not want to repeat the mistake they had made with Abel. Thus, the new 30-year-old Lie has become a true professor.

It was a happy time for Lie, he met and immediately fell in love with Anna Birch, then 18 and a cousin of Motzeldt. He sent a letter of confession to Anna. This letter made many people surprised because few people thought he could be "brave" to marry. Anna's grandfather was the uncle Abel and Sophus hoping it was a good thing. Although Anna was initially hesitant because of 30's age at the time, it was "too old" for her, but the couple's wedding continued 20 months later. After the wedding, Lie's life was summed up in three stages: the years in Oslo, then the next 12 years in Leipzig and a short time returning to Norway before his death.

Years in Oslo, he worked hard but almost no scientific research was remarkable. In return, his family life is very happy: Sophus and Anna are very compatible and they have 3 children.

In 1882, Lie worked with Klein and Adolph Mayer in Leipzig and traveled to Paris for two months in a row to talk to Darboux, Jordan, Hermite, Poincaré, Picard, Halphén and Lévy. These people fully understand the importance of Lie's studies. In 1884, Klein and Mayer wanted to help Lie improve his life and they sent student Friedrich Engel to guide and help him perfect his ideas. The joint study of Engel and Lie was grouped into a series of three books called Theorie der Transformationsgroppen (Theory of transforming groups), which was born in 1888, 1890 and 1893 respectively, with a total length 2,000 pages of paper.

In 1886, Lie was appointed professor at Leipzig and replaced Klein, a very important scientific position there. He became a bright face in the European math community and many domestic and foreign students came to be his students. However, the job of teaching and guiding students has appropriated all his time. He also felt depressed when he always had to work with many mediocre students. Leipzig does not seem to be heaven as he once wanted.

Lie's failure to fully master German and political theater and personal conflicts increasingly tired him. He also remembered his friends and home country of Norway. He wanted to work somewhere peaceful, but the relationship was always pushing him. Gradually, he felt that he had misunderstood and was abused too much. These conflicts caused him to lose sleep and fell into depression again. At this time, he was terribly disappointed.

In December 1889, he was transferred to a mental hospital near Hanovre. He stayed there for 7 months and was treated with opium and sleeping pills. But it seems that the habit of walking a lot has made him recover. Even so, he was completely unable to return completely healthy. He came back difficult to contact and doubted others. It is these that undermine his long-standing friendship with the soulful. He suffered from haunting and always accused friends who had stolen their ideas. His relationship with Klein was completely broken.

Unfortunate genius

Lie continued his creative research. What he announced continued to be spread and evaluated increasingly. In 1892, he joined the French Academy of Sciences. In 1893 Darboux and Tannery invited him to Paris and he was delighted to meet Élie Cartan. In April 1893, Lie and Cartan were seen sitting in the Café de la Source. Lie argues that before the transformation group theory can be used to solve differential equations, it is necessary to arrange variable group structures with a limited dimension. A year later, in 1894, Élie Cartan published the article 'On the structure of limited and non-stop groups'. Their studies seem to have quite similar results.

Lie was also highly valued in Norway and in 1905, the Senate of this country changed his title to Professor of transformative group theory. He also enjoyed a salary almost twice that of a regular professor. In 1897, Lie received the prestigious Lobatchevski award for his outstanding work on geometry, especially the geometry outside of the Euclid.

Unfortunately, when he returned to Norway in the summer of 1898, everyone felt his illness was severe. He had malignant anemia, an incurable disease at the time. In the autumn of that year, he was very difficult to go to class, even he sometimes had to lecture from the bed. On February 18, 1899, he died.

Many people believe that death made him forced to retire from many unfinished studies. Today, his most recognized work is the study of differential equations and this work later became an important branch of mathematical research: Lie group theory and Lie algebras.

* Arild Stubhaug is a historian of science and works at the mathematics department at the University of Oslo

Arild Stubhaug *

Source: Tia Sang Magazine - Translation from Pour la Science, October 2005