Centre for Advanced Study Newsletter no. 1 G May 2002 G 10th year The Myths about the Constitution I It is a myth that Norwegian democracy rests safely on the Constitution from 1814. In reali- ty the Constitution does not protect central political rights such as the freedom of associ- ation, the freedom of assembly, the freedom to demonstrate and the right to strike. I "In practical political life Nor- way protects human rights to a far greater degree than follows from the Constitution. Let’s hope that things continue that way," say the experts on the Constitution, Eivind Smith and Bjørn Erik Rasch. Pages 4-5 The lions in front of the Storting are guarding Norwegian democracy and the proud Norwegian Constitution – which looks more like a little pussycat Mathematics is older than the Greeks "Pythagoras and the Gre- eks have been given undeservedly much of the credit for having invented classical mathematics. Many of their ideas can be traced back to anonymous Mesopotamian mathema- The Genius and ticians," says Professor Jöran Friberg (picture). the Grammar The proof is to be found in such places as on the School Teacher small tablets of clay in the The Norwegian mathematician Niels Henrik Schøyen Collection. Abel (right) became world-famous in 1824. Ludwig Sylow (left) is one among Norwegian Pages 4-5 mathematicians who assumed the mantle after Abel. Pages 2-3 Mathematical growth in Abel’s footsteps Niels Henrik Abel was one of the greatest mathematical geniuses the world has ever fostered, and this year’s 200th jubilee is being used to mark his achievement. It is not quite so well known that along the path Abel left behind him there was flourishing growth of considerable importance. "You don’t have to open many textbooks in pletely but not managed to complete the writing. called Lie groups. They are to this day a cen- advanced mathematics before you come Sylow worked a great deal on interpretati- tral object of mathematical research and a across Axel Thue’s theorem, Sophus Lie’s ons and improvements of Abel’s works, and central aid in theoretical physics. groups or Ludvig Sylow’s theories. As a nati- was concerned with elliptical functions and In the 1890s he was ill a great deal. The on Norway was over-represented with out- theories of equations. But it was first and writer Bjørnstjerne Bjørnson took the initiati- standing mathematicians at the end of the foremost a publication from 1872, with des- ve to have a professorship established for him 19th century," says Geir Ellingsrud, who is a criptions of the three Sylow theorems within in Oslo, and in 1898 Lie moved home. But he professor of Mathematics at the University of group theory, which made him immortal. was at that time seriously ill with pernicious Oslo and a member of this year’s research Sylow was never given an appointment at anaemia and he died in the early part of 1899. group in mathematics at the CAS. the University of Kristiania, but remained a "Also Thoralf Skolem, Viggo Brun, Ernst secondary school teacher at Fredrikshald Axel Thue Selmer, Vilhelm Ljunggren and not least Sel- (Halden) for 40 years from 1858. He was (1863 – 1922) berg made important contributions to the Gol- granted leave to undertake a study tour to Axel Thue wor- den Age of number theory," says Ellingsrud. Paris and Berlin in 1861, to substitute for ked on number Broch in the period 1862-1863, and to edit theory, logic, A systematic development Abel’s works in co-operation with Sophus Lie geometry and Professor Jens Erik Fenstad in the Depart- from 1873 to 1877. In 1894 he was made a mechanics. He is ment of Mathematics at the University of doctor honoris causa of the University of most famous for Oslo agrees that Norway has fostered unusu- Copenhagen, and in 1898 the Storting nomi- his works on ally many outstanding mathematicians. "On nated him professor extraordinary with an arithmetical pro- both sides of the year 1900 we had many mat- emolument of 3000 kroner per year instead of perties of alge- hematicians who were right at the top of the his secondary school teacher’s pension. braic numbers, international research league, but on the other and theorems of hand we had no profession around them. The Marius the (un)solvability of Diophantine equations, great change came after the Second World Sophus Lie i.e. equations where the solution is a whole War, when a more systematic development (1842-1899) number. He is also famous for his pioneer was started. Today we don’t perhaps have Sophus Lie deve- work on what he called "Zeichenreihen" or many researchers in the forefront internatio- loped original "word problems". nally, but what we have got is a broad profes- and innovative Thue was known to go his own ways, and sion with points of impact in many areas of theories for he preferred developing his own ideas to Norwegian society, says Professor Fenstad. transformations making a study of other people’s works. He of geometrical became a teacher at the Institute of Technolo- Peter Ludvig objects (lines, gy in Trondheim in 1894 and in 1903 he Meidell spherical surfa- became Professor of Applied Mathematics in Sylow (1832- ces etc) and for Oslo. 1918) the integration of Thue thoroughly reformed the lectures on Ludvig Sylow ordinary and partial differential equations. He mechanics. It is said that he dictated his lectu- qualified as a was appointed as an extraordinary professor res, stopped at the nearest comma immediate- teacher of scien- at the University of Oslo in 1872, and in 1886 ly the time was up, and carried on from there ce and mathema- he became a professor in Leipzig as the on the next occasion. tics in 1856 and famous mathematician Felix Klein’s succes- became a pupil sor. The point of departure for Lie’s works Thoralf Albert Skolem of Ole Jacob was his own and Klein’s idea that geometry (1887 – 1963) Broch, who star- and analysis ought to be built up around the Thoralf Skolem published as many as 177 ted him off on concept of group, as Galois had built up his papers in the course of his long career. The Abel’s works. Sylow became extremely con- theory of algebraic equations. Lie made a stu- most important of his works were done within cerned with an unfinished Abel manuscript on dy of differential equations from this point of logic and Diophantine equations. the theory of equations, and he gradually docu- view and built up a general theory of "trans- Skolem obtained his doctoral degree in mented that Abel had solved the problem com- formation groups" or what have since been 1926 for a work on integral solutions of cer- 2 Newsletter no. 1 G May 2002 G 10th year tain algebraic equations and inequalities. Then he was a resear- cher at the Chris- tian Michelsen Institute in Bergen from 1930 to 1938, after which he became a professor in Oslo. When he reached the age of retirement in 1957 he was for a couple of years Visiting Professor at Notre Dame Uni- versity in the United States. His works in logic broke new ground (inter alia the "Skolem-Löwenheim Theorem"), and his results with respect to Diophantine equati- ons and the "Skolem-Nöther Theorem" in algebra are outstanding. His commitment to his work led among other things to his being commissioned to write about Diophantine equations for the German Springer-Verlag’s series Ergebnisse der Mathematik, and to- gether with Viggo Brun he edited the second impression of Eugen Netto’s Lehrbuch der Kombinatorik. Viggo Brun (1885 – 1978) Viggo Brun is best known for his work on prime number theory, but he also made a great contribution within continued fractions, generali- sations and combi- natorics. Among other things he developed a famous sieve met- hod, which he later used to develop two hypotheses in number theory that had previ- ously been considered impossible to prove. One of these hypotheses had been formulated by Goldbach and stated that every even num- An outstanding number theorist ber can be written as a sum of two odd prime numbers. Atle Selberg (born 1917) is considered appointed a professor in 1951. The sieve method has been taken further by to be one of the world’s most outstanding Selberg is also famous for his elementary among others Gelfond in Moscow and Atle number theorists of all times. His most proof of the prime number theorem, with its Selberg at Princeton and it has shown itself to famous work is his elaboration of what is generalisation to prime numbers in a random be very effective. Brun was also interested in known as Selberg’s trace formula. Selberg’s arithmetical series. When Selberg’s collected the history of mathematics, and in 1952 he doctorate from 1943 with amplifications on papers were published in 1989 and 1991, the found the lost manuscript of Abel’s Paris dis- what is called the Riemann zeta function critics were in agreement that the author is a sertation in a library in Florence. remained for at least 30 years as the most out- living classic who has exerted considerable in- He was made a professor at the Norwegian standing work within its field. fluence on his subject for more than 50 years. College of Advanced Technology in Trond- Selberg took his doctoral degree at the Uni- heim in 1924, and in 1945 he moved to Oslo versity of Oslo and became a research fellow where he worked until he reached retirement in 1942.