# Max Noether

### Quick Info

Born
24 September 1844
Died
13 December 1921
Erlangen, Germany

Summary
Max Noether was a German mathematician who was one of the leaders of nineteenth century algebraic geometry. Although himself a very distinguished mathematician, his daughter Emmy Noether was to bring greater innovation to mathematics than did her father.

### Biography

Max Noether's mother was Amalia Würzburger and his father was Hermann Noether. The family were Jewish so a little explanation is required as to why they had German names. In fact Max's paternal grandfather was Elias Samuel, the founder of a business in Bruchsal. Elias had nine children, one being a son Hertz Samuel. In 1809 the State of Baden made the Tolerance Edict which required Jews to adopt Germanic names. Elias Samuel chose the surname Nöther, becoming Elias Nöther, but also changed the given names of his children, giving Hertz the name Hermann. When he was eighteen years old, Hermann Nöther left his home town of Bruchsal and studied theology at the University of Mannheim. Then in 1837, together with his brother Joseph, he set up a wholesale business in iron hardware. Hermann Nöther and his wife Amalia had five children, the third of which was Max. The two children older than Max were Sarah (born 6 November 1839) and Emil. It is worth noting at this point that the Nöther iron-wholesaling business remained a family firm for exactly one hundred years, until the Nazis removed Jewish families from their own businesses in 1937. One other comment is necessary at this point. Although the family name was chosen to be Nöther by Max's grandfather, Max and his family always used the form Noether (except on Max's wedding certificate where the form Nöther appears).

Max attended school in Mannheim but his studies at the gymnasium were interrupted in 1858. He suffered an attack of polio when he was 14 years old and it left him with a handicap for the rest of his life. For two years he was unable to walk and was unable to attend the Gymnasium. However, his parents arranged for him to receive lessons at home and so he was able to complete the Gymnasium curriculum without returning to school. At this stage Noether was interested in astronomy, so before beginning his university studies he spent a short period at Mannheim Observatory.

He entered the University of Heidelberg in 1865 and spent three semesters there before obtaining a doctorate on 5 March 1868. At Heidelberg he was taught by Jacob Lüroth who was awarded a doctorate in 1865, but mainly he was influenced by Gustav Kirchhoff who was the professor of physics. His doctorate from Heidelberg was on astronomy, and Noether was not required to write a dissertation. He was given an oral examination in the Dean's room, with the only requirement being that the candidate had to supply the wine for the examiners. After the award of his doctorate Noether spent time in Giessen working with Alfred Clebsch. Like Noether, Clebsch had begun his career working on physics, but had collaborated with Paul Gordan on Theorie der Abelschen Funktionen in 1866. While at Giessen, Noether met Alexander von Brill who was a privatdozent there. The two became friends and collaborators, writing important papers together. Clebsch left Giessen to take up an appointment in Göttingen in 1969 and Noether went with him. In 1870 he submitted his habilitation thesis Über Flachen welche Schaaren rationaler Curven besitzen to Heidelberg and was appointed there as a privatdozent.

The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. Given Noether's friendship with Clebsch it is not surprising that he would make his journal the main outlet for his publication. He puublished Zur Theorie des eindeutigen Entsprechens algebraischer Gebilde von beliebig vielen Dimensionen in the second volume of the journal published in 1870, and in the third volume, also published in 1870, he published his habilitation thesis Über Flächen, welche Schaaren rationaler Curven besitzen , and another paper Über die eindeutigen Raumtransformationen, insbesondere in ihrer Anwendung auf die Abbildung algebraischer Flächen . Noether published many papers in Mathematische Annalen with a major contribution appearing almost every year between 1870 and 1921. He joined the editorial staff of the journal in 1893.

Appointed a privatdozent in Heidelberg in 1870, Noether was promoted to extraordinary professor there on 25 September 1874. In the following year he moved to Erlangen where he was appointed as an extraordinary professor, being named as an ordinary professor there on 16 April 1888. He married Ida Amalia Kaufmann (born 1852, died 1915), the daughter of a wealthy Jewish merchant family from Cologne, on 28 August 1880. Ida had a brother who was a professor at the University of Berlin. Max and Ida had four children, one of whom was the famous mathematician Emmy Noether; we give further details below of their children.

Max Noether was one of the leaders of nineteenth century algebraic geometry. He was influenced by Abel, Riemann, Cayley and Cremona. Following Cremona, Max Noether studied the invariant properties of an algebraic variety under the action of birational transformations. Macaulay writes [3]:-
He contributed greatly to the advancement of mathematical science in three distinct ways: by the new and fruitful ideas contained in his original researches, by the patient investigation and encouragement he gave to other writers, and by his acutely critical and detailed historical work.
In 1873 Noether proved an important result on the intersection of two algebraic curves in the paper Über einen Satz aus der Theorie der algebraischen Functionen published in volume 6 of Mathematische Annalen. This result showed that given two algebraic curves $f (x, y) = 0, g(x, y) = 0$ which intersect in a finite number of isolated points, then the equation of an algebraic curve which passes through all those points of intersection can be expressed in the form $af + bg = 0$, where $a$ and $b$ are polynomials in $x$ and $y$, is and only if certain conditions are satisfied. These conditions are now known as "Notherian conditions". This theorem gives [1]:-
... necessary and sufficient conditions for the case where the curves have common multiple points with contact of any degree of complexity.
Volume 7 of Mathematische Annalen contains the important paper Über die algebraischen Functionen und ihre Anwendung in der Geometrie which was written jointly by Noether and Brill.

Let us also note that Noether wrote many obituaries. For example, he wrote obituaries of Otto Hesse (1875), Arthur Cayley (1895), James Joseph Sylvester (1898), Francesco Brioschi (1898), Sophus Lie (1900), Charles Hermite (1901), Luigi Cremona (1904), George Salmon (1905), Jacob Lüroth (1911), Paul Gordan (1914), and Hieronymus Georg Zeuthen (9121). He was also an editor of Ludwig Otto Hesse's Complete Works, published in 1897. Many of the articles in this archive use material from Noether's obituaries.

In 1882 his daughter Emmy Noether was born. Emmy became interested in many similar topics to her father and generalised some of his theorems. In 1883 Max and Ida Noether had a son named Alfred, who later studied chemistry. He died before his father in 1918. Fritz, who was born in 1884, went on to become a mathematician. Forced to leave Germany under the Nazi anti-Semitic policies, he went to the Soviet Union and was appointed as a professor of mathematics at the University of Tomsk. He was arrested under Stalin's 1937-38 Great Purge for being a German spy, sentenced to 25 years in prison, then accused of anti-Soviet propaganda while in prison and shot in 1941. Max and Ida Noether's fourth child was Gustav Robert, born in 1889. He suffered ill health for most of his life and died in 1928.

Macaulay gives an interesting insight into Max Noether's thinking in [3]:-
Noether's mind was naturally intuitive, but he mistrusted intuition, and was apt to let anything suggested by it pass out of his thoughts. It seemed as if this cramped his powers to some extent in his later years. He found perhaps in his subject that intuition was often liable to mislead. He was of course never content without algebraic or arithmetic proof but had sometimes to be satisfied with an incomplete proof. Although naturally impatient he would take infinite pains to understand the thoughts of others, and to give them abundant help out of his own ample resources. There are many, including the writer of this note, who are grateful to him for his help. He had searching and peculiar methods of his own for testing the truth of things.

### References (show)

1. E A Kramer, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. G Castelnuovo, F Enriques and F Severi, Max Noether (Italian), Math. Ann. 93 (1) (1925), 161-181.
3. F S Macaulay, Max Noether, Proc. London Math. Soc. 21 (1920-23), 37-42.
4. M Menghini, Notes on the correspondence between Luigi Cremona and Max Noether, Historia Mathematica 13 (4) (1986), 341-351.
5. A W von Brill, Max Noether, Jahresberichte der Deutschen Mathematiker-Vereinigung 32 (1923), 211-233.