**Gustav Roch** was a German mathematician known for the *Riemann-Roch theorem* which relates the genus of a topological surface to algebraic properties of the surface.

- Gustav attended school in Dresden before moving to a school in Neustadt.
- A very important influence on Roch was Oscar Schlömilch who taught at the Polytechnic Institute.
- He quickly saw that Roch's talents lay in mathematics and its applications to physics, rather than in chemistry, and he soon persuaded Roch to seriously consider changing the main topic that he was studying.
- This was not a simple matter, for Roch did not have a particularly strong background in either mathematics or physics, so in order to prepare himself for advanced study he took courses at a private institute as well as at the Polytechnic Institute.
- In the spring of 1859, the same year as his first paper was published, Roch entered the University of Leipzig.
- Roch continued his research on the mathematical theory of electricity and magnetism, publishing four further articles Über magnetische Momente Ⓣ(On magnetic moments) (1859), Über Magnetismus Ⓣ(On magnetism) (1859), Bemerkung zur Theorie der electrischen Ströme Ⓣ(Remarks to the theory of the electrical currents) (1860), and Über Magnetismus Ⓣ(On magnetism) (1861), all of which appeared in Schlömilch's Zeitschrift.
- On 13 April 1861 Roch entered the University of Göttingen.
- Following his time in Göttingen, Roch went to Berlin where he made contact with L Kronecker, E E Kummer, K Weierstrass and K W Borchardt.
- Roch was awarded a Master's Degree by the University of Leipzig in 1862 and later that year, on 28 May, he was awarded his doctorate for his thesis Über die Darstellung von Functionen dreier Variablen durch Potentialausdrücke ...
- Roch submitted his habilitation thesis De theoremate quodam circa functiones Abelianas Ⓣ(The theory of Abelian functions) to the University of Halle on 13 October 1863.
- It appear in print in the following year and contains the theorem now known as the Riemann-Roch theorem.
- It was not the first of Roch's papers to appear in Crelle's Journal for he had published Über eine Transformation des Potentials Ⓣ(On a transformation of potentials) in it in 1864.
- Several other papers by Roch such as Über Functionen complexer Grössen Ⓣ(On the size of complex functions) (1863), Functionen complexer Grössen Ⓣ(The size of complex functions) (1865), and Über die Ausdrücke elliptischer Integrale zweiter und dritter Gattung durch Theta-Functionen Ⓣ(On the terms of elliptic integrals, second and third class by theta functions) (1865) had appeared in Schlömilch's Zeitschrift für Mathematik und Physik.
- As presented by Roch, the Riemann-Roch theorem related the topological genus of a Riemann surface to purely algebraic properties of the surface.
- The Riemann-Roch theorem was so named by Max Noether and Alexander von Brill in a paper they jointly wrote 1874 when they refined the information obtained from the theorem.
- It was extended to algebraic curves in 1929 and then in the 1950s an nnn-dimensional version, the Hirzebruch-Riemann-Roch theorem, was proved by Hirzebruch and a version for a morphism between two varieties, the Grothendieck-Riemann-Roch theorem, was proved by Grothendieck.
- Over the three academic years 1863-64, 1864-65 and 1865-66 Roch gave a number of courses at Halle.
- Up to this time Roch was still a privatdozent at Halle but in the spring of 1866 the University began to take up referees' reports with a view to appointing him as an extraordinary professor.
- Heine wrote a strong letter of support and Roch was appointed extraordinary professor at the University of Halle-Wittenberg on 21 August.
- However Roch's health was failing and on 13 October he was granted leave for the winter semester of 1866-67 to allow him to regain his health.
- Roch went to Venice where he hoped the warmer weather would aid his recovery.
- Roch's name will live on through the fundamental Riemann-Roch theorem, but it is a tragedy that the young man with so much mathematical promise died when he had only just commenced his career.

Born 9 December 1839, Dresden, Germany. Died 21 November 1866, Venice, Italy.

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Origin Germany

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