**Hurwitz** studied the genus of the Riemann surface and worked on how class number relations could be derived from modular equations.

- Hurwitz entered the Realgymnasium Andreanum in Hildesheim in 1868.
- We note that this first paper by Hurwitz, written jointly with Schubert, was on Chasles's theorem.
- While at Berlin Hurwitz continued to keep in contact with Klein and assisted him with a paper on elliptic modular functions which he was writing.
- After three semesters at the University of Berlin, Hurwitz returned to the University of Munich in 1879 to continue working with Klein, so when Klein moved to the University of Leipzig in October 1880, Hurwitz went with him.
- It would have been natural for Hurwitz to become a Privatdozent at the University of Leipzig since he was a student of Klein, the professor of mathematics there.
- However there was a difficulty -- Hurwitz did not have sufficient knowledge of Greek to satisfy the Faculty requirements! Luckily Göttingen had no such requirement and Hurwitz became a Privatdozent at the University of Göttingen after submitting his habilitation thesis there in 1882.
- Hurwitz had not been at Munich during 1881-82, rather he had returned to Berlin where he attended further courses of lectures by Weierstrass and Kronecker.
- In 1884 Hurwitz accepted an invitation from Lindemann to become an extraordinary Professor at Königsberg and he was to remain there for eight years.
- In 1892 Frobenius left his chair at Eidgenössische Polytechnikum Zürich to return to Berlin and Hurwitz was appointed to the vacant chair at Zürich.
- Hurwitz remained at Zürich for the rest of his life, unfortunately continually suffering from ill health.
- Although, as we have pointed out, Hurwitz remained at Zürich for the rest of his life, that was not because he had not been offered in chair in Germany.
- Göttingen approached Hurwitz and offered him the vacant chair only weeks after he had accepted the Zürich chair, but he turned down the offer.
- This must have been a remarkably hard decision for Hurwitz since at that time a chair at a leading German university such as Göttingen would have been much more prestigious to any German than a chair in Switzerland.
- However Hurwitz was an extremely loyal person, and having given his word that he would accept the Zürich position he would not renege on his promise.
- Much of Hurwitz's mathematics can be seen as being strongly influenced by Klein (and also by Riemann whose ideas where transmitted to Hurwitz via Klein).
- Hurwitz studied the genus of the Riemann surface.
- Further topics studied by Hurwitz include complex function theory, the roots of Bessel functions, and difference equations.
- Hurwitz solved this problem completely showing that the condition held if and only if a certain sequence of determinants are all positive.
- This remarkably influential paper was reprinted 100 years later in the proceedings of the Hurwitz Symposium on Stability theory in Ascona in 1995.
- Hurwitz did excellent work in algebraic number theory.
- A full proof of Hurwitz's ideas appears in a booklet published in the year of his death.
- But Kakeya's theorem contained a mistake, which was corrected by A Hurwitz in 1913.
- Hurwitz informed E Landau about Kakeya's result (corrected); Landau needed the result in a proof of a theorem on infinite power series.
- We mention a generalization of Eneström's theorem and give an application to a similar result by Hurwitz.
- The reception committee seems to have had the highest workload out of the four sub-committees, and Hurwitz suggested that a publications committee be set up in order to relieve his committee of some of its duties.
- Together with Geiser and Minkowski, Hurwitz was responsible for choosing the plenary speakers (on Rudio's suggestion).
- Furthermore, Hurwitz wrote up the attendance list, which was given to every congress participant.
- On 8th August, Hurwitz spent the entire day 'at the train station, welcoming the arriving mathematicians, handing out the congress cards and organising accommodation for the arrivals if desired'.
- Migraine was not the extent of Hurwitz's health problems which became increasingly severe.

Born 26 March 1859, Hildesheim, Lower Saxony, Germany. Died 18 November 1919, Zürich, Switzerland.

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Origin Germany, Set Theory

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive