Person: Schwarz, Hermann Amandus
Hermann Schwarz worked on the conformal mapping of polyhedral surfaces onto the spherical surface and on a problem of the calculus of variation, namely surfaces of least area.
Mathematical Profile (Excerpt):
- Hermann studied at the Gymnasium in Dortmund where his favourite subject was chemistry.
- Schwarz began his study of chemistry at Berlin but it was not long before Kummer and Weierstrass had influenced him to change to mathematics.
- Through him Schwarz became interested in geometry.
- Schwarz attended Weierstrass's lectures on The integral calculus in 1861 and the notes that Schwarz took at these lectures still exist.
- While in Berlin, Schwarz worked on minimal surfaces (surfaces of least area), a characteristic problem of the calculus of variations.
- Schwarz had made an important contribution in 1865 when he discovered what is now known as the Schwarz minimal surface.
- Schwarz continued studying in Berlin for his teacher's training qualification which he completed by 1867.
- Perhaps surprisingly after Schwarz succeeded Weierstrass accepting a professorship in Berlin in 1892, the balance in favour of the most eminent university in Germany for mathematics, which had undoubtedly been Berlin, began to shift towards Göttingen.
- Firstly Schwarz failed to keep up his output of mathematical research after his move.
- That this was the case should not have come as a complete surprise to those making the appointment for Schwarz had published his Complete Works in 1890, two years earlier.
- We should not give the impression that the only reason for Berlin moving down from being the leading German university for mathematics to become its second university was due to Schwarz.
- The other effect was Klein whose dynamic leadership in Göttingen made it prosper at the expense of Berlin where Frobenius and Schwarz could not provide the same inspired approach.
- Perhaps the final sign that Göttingen had overtaken Berlin came in 1902 when Frobenius and Schwarz chose Hilbert to succeed to the Berlin chair which had become vacant on the death of Fuchs.
- The Berlin chair was then filled by Schottky but, like Schwarz before him, he had moved to Berlin after his best days for mathematical research were behind him.
- Schwarz continued teaching at Berlin until 1918.
- One important area which Schwarz worked on was that of conformal mappings.
- Schwarz's gave a method to conformally map polygonal regions to the circle.
- Schwarz also gave the alternating method for solving the Dirichlet problem which soon became a standard technique.
- Schwarz answered the question of whether a given minimal surface really yields a minimal area.
- The fact that Schwarz should have come up with a special case of the general result now known as the Cauchy-Schwarz inequality (or the Cauchy-Bunyakovsky-Schwarz inequality) is not surprising for much of his work is characterised by looking at rather specific and narrow problems but solving them using methods of great generality which have since found widespread applications.
- For example the Cauchy-Schwarz inequality appears in arithmetic, geometric and function-theoretic formulations in works of mathematicians such as Bunyakovsky, Cauchy, Grassmann, von Neumann and Weyl.
- In answering the problem of when Gauss's hypergeometric series was an algebraic function Schwarz, as he had done so many times, developed a method which would lead to much more general results.
- It was in this work that he defined a conformal mapping of a triangle with arcs of circles as sides onto the unit disc which is now known as the 'Schwarz function'.
- This function is an early example of an automorphic function and in this work Schwarz was looking at ideas which led Klein and Poincaré to develop the theory of automorphic functions.
- From their correspondence one finds that Schwarz addressed his teacher often with an accuracy going down to the last detail, sometimes almost timidly.
- Schwarz's demeanour has been described as naive, dramatic, coarse.
Born 25 January 1843, Hermsdorf am Kynast, Silesia (now Sobieszów, Jelenia Góra, Poland). Died 30 November 1921, Berlin, Germany.
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Tags relevant for this person:
Origin Poland, Set Theory
Thank you to the contributors under CC BY-SA 4.0!
Adapted from other CC BY-SA 4.0 Sources:
- O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive