Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

, known for his work in

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

Life

Schwarz was born in Hermsdorf,

Silesia Silesia (see names #Etymology, below) is a historical region of Central Europe that lies mostly within Poland, with small parts in the Czech Silesia, Czech Republic and Germany. Its area is approximately , and the population is estimated at 8, ...

(now Sobieszów,

Poland Poland, officially the Republic of Poland, is a country in Central Europe. It extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukrai ...

). In 1868 he married Marie Kummer, who was the daughter to the mathematician Ernst Eduard Kummer and Ottilie née Mendelssohn (a daughter of Nathan Mendelssohn's and granddaughter of

Moses Mendelssohn Moses Mendelssohn (6 September 1729 – 4 January 1786) was a German-Jewish philosopher and theologian. His writings and ideas on Jews and the Jewish religion and identity were a central element in the development of the ''Haskalah'', or 'J ...

). Schwarz and Kummer had six children, including his daughter Emily Schwarz. Schwarz originally studied

chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...

in Berlin but Ernst Eduard Kummer and Karl Theodor Wilhelm Weierstrass persuaded him to change to

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

. He received his Ph.D. from the Universität Berlin in 1864 and was advised by Kummer and Weierstrass. Between 1867 and 1869 he worked at the

University of Halle Martin Luther University Halle-Wittenberg (), also referred to as MLU, is a public research university in the cities of Halle and Wittenberg. It is the largest and oldest university in the German state of Saxony-Anhalt. MLU offers German and i ...

, then at the Swiss Federal Polytechnic. From 1875 he worked at

Göttingen University Göttingen (, ; ; ) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. According to the 2022 German census, t ...

, dealing with the subjects of

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

and the

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

. He died in

Berlin Berlin ( ; ) is the Capital of Germany, capital and largest city of Germany, by both area and List of cities in Germany by population, population. With 3.7 million inhabitants, it has the List of cities in the European Union by population withi ...

Work

Schwarz's works include ''Bestimmung einer speziellen Minimalfläche'', which was crowned by the Berlin Academy in 1867 and printed in 1871, and ''Gesammelte mathematische Abhandlungen'' (1890). Among other things, Schwarz improved the proof of the

Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane \mathbb which is not all of \mathbb, then there exists a biholomorphic mapping f (i.e. a bijective hol ...

, developed a special case of the

Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...

, and gave a proof that the ball has less surface area than any other body of equal volume. His work on the latter allowed

Émile Picard Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924. Life He was born in Paris on 24 July 1856 and educated there at th ...

to show solutions of differential equations exist (the

Picard–Lindelöf theorem In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy– ...

). In 1892 he became a member of the

Academy of Science and a professor at the

University of Berlin The Humboldt University of Berlin (, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany. The university was established by Frederick William III on the initiative of Wilhelm von Humbol ...

, where his students included Lipót Fejér, Paul Koebe and

Ernst Zermelo Ernst Friedrich Ferdinand Zermelo (; ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel set theory, Z ...

. In total, he advised at least 22 Ph. D students. In 1914 Schwarz's friends and former students published a volume with 34 articles in celebration of the 50th anniversary of his doctoral dissertation. His name is attached to many ideas in mathematics, including the following: * Abstract additive Schwarz method * Additive Schwarz method * Schwarz alternating method * * Schwarzian derivative * Schwarz function * Schwarz lantern * Schwarz lemma *

Schwarz's list In the mathematical theory of special functions, Schwarz's list or the Schwarz table is the list of 15 cases found by when hypergeometric functions can be expressed algebraically. More precisely, it is a listing of parameters determining the case ...

* Schwarz minimal surface * Schwarz theorem (also known as Clairaut's theorem) * Schwarz integral formula * Schwarz–Christoffel mapping * Schwarz–Ahlfors–Pick theorem * Schwarz reflection principle * Schwarz triangle * Schwarz triangle function *

* Theorem of Pohlke and Schwarz

Publications

* *

Notes

External links

* * {{DEFAULTSORT:Schwarz, Hermann 1843 births 1921 deaths 20th-century German mathematicians 19th-century German mathematicians Academic staff of ETH Zurich Academic staff of the University of Göttingen Academic staff of the Martin Luther University of Halle-Wittenberg Academic staff of the Humboldt University of Berlin Corresponding members of the Saint Petersburg Academy of Sciences People from the Province of Silesia People from Głogów County Members of the Royal Society of Sciences in Uppsala