Carl Ludwig Siegel (31 December 1896 – 4 April 1981) 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, structure, space, models, and change.
History
On ...
specialising in
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
. He is known for, amongst other things, his contributions to the
Thue–Siegel–Roth theorem
In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half a ...
in
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by r ...
, Siegel's method,
Siegel's lemma In mathematics, specifically in transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence of these polynomials ...
and the
Siegel mass formula for quadratic forms. He was named as one of the most important mathematicians of the 20th century.
[Pérez, R. A. (2011]
''A brief but historic article of Siegel''
NAMS 58(4), 558–566.
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
, without hesitation, named Siegel as the greatest mathematician of the first half of the 20th century.
Atle Selberg
Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded t ...
said of Siegel and his work:
Biography
Siegel was born in
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
, where he enrolled at the
Humboldt University
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative of ...
in Berlin in 1915 as a student in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. Amongst his teachers were
Max Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical p ...
and
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous ...
, whose influence made the young Siegel abandon astronomy and turn towards number theory instead. His best-known student was
Jürgen Moser
Jürgen Kurt Moser (July 4, 1928 – December 17, 1999) was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations.
Life
Moser's mother Ilse Strehl ...
, one of the founders of
KAM theory (
Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
–
Arnold–Moser), which lies at the foundations of
chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
. Another notable student was
Kurt Mahler
Kurt Mahler FRS (26 July 1903, Krefeld, Germany – 25 February 1988, Canberra, Australia) was a German mathematician who worked in the fields of transcendental number theory, diophantine approximation, ''p''-adic analysis, and the geometry of ...
, the number theorist.
Siegel was an
antimilitarist
Antimilitarism (also spelt anti-militarism) is a doctrine that opposes war, relying heavily on a critical theory of imperialism and was an explicit goal of the First and Second International. Whereas pacifism is the doctrine that disputes (especi ...
, and in 1917, during
World War I
World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fightin ...
he was committed to a psychiatric institute as a
conscientious objector
A conscientious objector (often shortened to conchie) is an "individual who has claimed the right to refuse to perform military service" on the grounds of freedom of thought, conscience, or religion. The term has also been extended to object ...
. According to his own words, he withstood the experience only because of his support from
Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
Biography
Edmund Landau was born to a Jewish family in Berlin. His father was Leopold ...
, whose father had a clinic in the neighborhood. After the end of
World War I
World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fightin ...
, he enrolled at the
University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
, studying under Landau, who was his doctoral thesis supervisor (
Ph.D.
A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...
in 1920). He stayed in Göttingen as a teaching and research assistant; many of his groundbreaking results were published during this period. In 1922, he was appointed professor at the
Johann Wolfgang Goethe-Universität
Goethe University (german: link=no, Johann Wolfgang Goethe-Universität Frankfurt am Main) is a university located in Frankfurt am Main, Germany. It was founded in 1914 as a citizens' university, which means it was founded and funded by the wealt ...
of
Frankfurt am Main
Frankfurt, officially Frankfurt am Main (; Hessian: , "Frank ford on the Main"), is the most populous city in the German state of Hesse. Its 791,000 inhabitants as of 2022 make it the fifth-most populous city in Germany. Located on its na ...
as the successor of
Arthur Moritz Schönflies
Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology.
Schoenflies ...
. Siegel, who was deeply opposed to Nazism, was a close friend of the
docent
The title of docent is conferred by some European universities to denote a specific academic appointment within a set structure of academic ranks at or below the full professor rank, similar to a British readership, a French " ''maître de conf ...
s
Ernst Hellinger
Ernst David Hellinger (September 30, 1883 – March 28, 1950) was a German mathematician.
Early years
Ernst Hellinger was born on September 30, 1883 in Striegau, Silesia, Germany (now Strzegom, Poland) to Emil and Julie Hellinger. He grew up in ...
and
Max Dehn
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...
and used his influence to help them. This attitude prevented Siegel's appointment as a successor to the chair of
Constantin Carathéodory
Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
in Munich. In Frankfurt he took part with Dehn, Hellinger,
Paul Epstein, and others in a seminar on the history of mathematics, which was conducted at the highest level. In the seminar they read only original sources. Siegel's reminiscences about the time before World War II are in an essay in his collected works.
In 1936 he was a Plenary Speaker at the
ICM in Oslo. In 1938, he returned to
Göttingen
Göttingen (, , ; nds, Chöttingen) 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. At the end of 2019, t ...
before emigrating in 1940 via
Norway
Norway, officially the Kingdom of Norway, is a Nordic country in Northern Europe, the mainland territory of which comprises the western and northernmost portion of the Scandinavian Peninsula. The remote Arctic island of Jan Mayen and t ...
to the
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
, where he joined the
Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
in
Princeton
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ni ...
, where he had already spent a
sabbatical
A sabbatical (from the Hebrew: (i.e., Sabbath); in Latin ; Greek: ) is a rest or break from work.
The concept of the sabbatical is based on the Biblical practice of ''shmita'' (sabbatical year), which is related to agriculture. According to ...
in 1935. He returned to Göttingen only after
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...
, when he accepted a post as
professor
Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "person who pr ...
in 1951, which he kept until his retirement in 1959. In 1968 he was elected a foreign associate of the U.S. National Academy of Sciences.
Career
Siegel's work on
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
,
diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
s, and
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
in particular won him numerous honours. In 1978, he was awarded the first
Wolf Prize in Mathematics
The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. ...
, one of the most prestigious in the field. When the prize committee decided to select the greatest living mathematician, the discussion centered around Siegel and
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел ...
as the leading candidates. The prize was ultimately split between them.
Siegel's work spans
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
; and his
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
on the
finiteness of the integer points of curves, for
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
> 1, is historically important as a major general result on diophantine equations, when the field was essentially undeveloped. He worked on
L-function
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ris ...
s, discovering the (presumed illusory)
Siegel zero
Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). A ...
phenomenon. His work, derived from the
Hardy–Littlewood circle method
In mathematics, the Hardy–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring's problem.
History
The initial idea is usually at ...
on
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s, appeared in the later,
adele group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the ...
theories encompassing the use of
theta-functions. The
Siegel modular varieties, which describe
Siegel modular form
In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular form ...
s, are recognised as part of the
moduli theory
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
of
abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular function ...
. In all this work the structural implications of analytic methods show through.
In the early 1970s Weil gave a series of seminars on the history of number theory prior to the 20th century and he remarked that Siegel once told him that when the first person discovered the simplest case of
Faulhaber's formula
In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the ''p''-th powers of the first ''n'' positive integers
:\sum_^n k^p = 1^p + 2^p + 3^p + \cdots + n^p
as a (''p''&nb ...
then, in Siegel's words, "Es gefiel dem lieben Gott." (It pleased the dear Lord.) Siegel was a profound student of the history of mathematics and put his studies to good use in such works as the
Riemann–Siegel formula In mathematics, the Riemann–Siegel formula is an asymptotic formula for the error of the approximate functional equation of the Riemann zeta function, an approximation of the zeta function by a sum of two finite Dirichlet series. It was found by ...
.
Works
by Siegel:
*''Transcendental numbers'', 1949
*''Analytic functions of several complex variables, Stevens 1949; 2008 pbk edition
*''Gesammelte Werke'', 3 Bände, Springer 1966
*with
Jürgen Moser
Jürgen Kurt Moser (July 4, 1928 – December 17, 1999) was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations.
Life
Moser's mother Ilse Strehl ...
''Lectures on Celestial mechanics'' 1971, based upon the older work ''Vorlesungen über Himmelsmechanik'', Springer 1956
*''On the history of the Frankfurt Mathematics Seminar'', Mathematical Intelligencer Vol.1, 1978/9, No. 4
*''Über einige Anwendungen diophantischer Approximationen'', Sitzungsberichte der Preussischen Akademie der Wissenschaften 1929 (sein Satz über Endlichkeit Lösungen ganzzahliger Gleichungen)
*''Transzendente Zahlen'', BI Hochschultaschenbuch 1967
*''Vorlesungen über Funktionentheorie'', 3 Bde. (auch in Bd.3 zu seinen Modulfunktionen, English translation "Topics in complex function theory",
3 vols., Wiley)
Letterto
Louis J. Mordell, March 3, 1964.
about Siegel:
*
Harold Davenport
Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory.
Early life
Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar Scho ...
: ''Reminiscences on conversations with Carl Ludwig Siegel'', Mathematical Intelligencer 1985, Nr.2
*Helmut Klingen, Helmut Rüssmann, Theodor Schneider: ''Carl Ludwig Siegel'', Jahresbericht DMV, Bd.85, 1983(Zahlentheorie, Himmelsmechanik, Funktionentheorie)
*
Jean Dieudonné
Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymo ...
: Article in Dictionary of Scientific Biography
*Eberhard Freitag: ''Siegelsche Modulfunktionen'', Jahresbericht DMV, vol. 79, 1977, pp. 79–86
*
Hel Braun
Helene (Hel) Braun (June 3, 1914 – May 15, 1986) was a German mathematician who specialized in number theory and modular forms. Her autobiography, ''The Beginning of A Scientific Career,'' described her experience as a female scientist working ...
: ''Eine Frau und die Mathematik 1933–1940'', Springer 1990 (Reminiscence)
*
Constance Reid
Constance Bowman Reid (January 3, 1918 – October 14, 2010)
was the author of several biographies of mathematicians and popular books about mathematics. She received several awards for mathematical exposition. She was not a mathematician but ...
:
Hilbert', as well as
Courant', Springer (The two biographies contain some information on Siegel.)
*
Max Deuring
Max Deuring (9 December 1907 – 20 December 1984) was a German mathematician. He is known for his work in arithmetic geometry, in particular on elliptic curves in characteristic p. He worked also in analytic number theory.
Deuring graduated fr ...
: ''Carl Ludwig Siegel, 31. Dezember 1896 – 4. April 1981'',
Acta Arithmetica
''Acta Arithmetica'' is a scientific journal of mathematics publishing papers on number theory. It was established in 1935 by Salomon Lubelski and Arnold Walfisz. The journal is published by the Institute of Mathematics of the Polish Academy of Sc ...
, Vol. 45, 1985, pp. 93–113
onlinean
Publications list*
Goro Shimura
was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multipli ...
"1996 Steele Prizes" (with Shimura's reminiscences concerning C. L. Siegel), Notices of the AMS, Vol. 43, 1996, pp. 1343–7, pdf*
Serge Lang: ''Mordell's Review, Siegel's letter to Mordell, diophantine geometry and 20th century mathematics'', Notices American Mathematical Society 1995, in Gazette des Mathematiciens 1995
See also
*
Bourget's hypothesis
*
Siegel's conjecture
*
Siegel's number
*
Siegel disk
*
Siegel's lemma In mathematics, specifically in transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence of these polynomials ...
*
Siegel upper half-space
In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive definite. It ...
*
Siegel–Weil formula
In mathematics, the Siegel–Weil formula, introduced by as an extension of the results of , expresses an Eisenstein series as a weighted average of theta series of lattices in a genus, where the weights are proportional to the inverse of the o ...
*
Siegel parabolic subgroup
*
Smith–Minkowski–Siegel mass formula In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices ( quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism g ...
*
Riemann–Siegel formula In mathematics, the Riemann–Siegel formula is an asymptotic formula for the error of the approximate functional equation of the Riemann zeta function, an approximation of the zeta function by a sum of two finite Dirichlet series. It was found by ...
*
Riemann–Siegel theta function In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as
:\theta(t) = \arg \left( \Gamma\left(\frac+\frac\right) \right) - \frac t
for real values of ''t''. Here the argument is chosen in such a way th ...
*
Siegel–Shidlovsky theorem
*
Siegel–Walfisz theorem In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel to primes in arithmetic progressions. It is a refinement both of the prime number theorem and of Dirichlet' ...
*
Siegel's theorem
In mathematics, Siegel's theorem on integral points states that for a smooth algebraic curve ''C'' of genus ''g'' defined over a number field ''K'', presented in affine space in a given coordinate system, there are only finitely many points on ''C' ...
(Minkowski–Hlawka theorem)
References
*
External links
*
Freddy Litten Die Carathéodory-Nachfolge in München 1938–194485. Vol. Heft 4 der DMV (with 3 articles about Siegel's life and works)(PDF; 6,77 MB)
Siegel ''Approximation algebraischer Zahlen'', Mathematische Zeitschrift, vol.10, 1921, DissertationSiegel „Additive Zahlentheorie in Zahlkörpern“, 1921, Jahresbericht DMV
{{DEFAULTSORT:Siegel, Carl Ludwig
1896 births
1981 deaths
20th-century German mathematicians
University of Göttingen alumni
University of Göttingen faculty
Goethe University Frankfurt faculty
Institute for Advanced Study faculty
Number theorists
Wolf Prize in Mathematics laureates
Knights Commander of the Order of Merit of the Federal Republic of Germany
Recipients of the Pour le Mérite (civil class)
Foreign associates of the National Academy of Sciences
Members of the Royal Swedish Academy of Sciences