Carl Ludwig Siegel
Encyclopedia
Carl Ludwig Siegel was a mathematician
specialising in number theory
and celestial mechanics
. He was one of the most important mathematicians of the 20th century.
, where he enrolled at the Humboldt University in Berlin in 1915 as a student in mathematics
, astronomy
, and physics
. Amongst his teachers were Max Planck
and Ferdinand Georg Frobenius
, whose influence made the young Siegel abandon astronomy and turn towards number theory instead. His best student was Jürgen Moser
, one of the founders of KAM theory (Kolmogorov
-Arnold
-Moser), which lies at the foundations of chaos theory
.
In 1917 Siegel was drafted into the German Army
. Since he refused military service, he was committed to a psychiatric institute. According to his own words, he withstood the experience only because of his support from Edmund Landau
, whose father had a clinic in the neighborhood. After the end of World War I
, he enrolled at the Georg-August University of Göttingen
, studying under Edmund Landau
, who was his doctoral thesis supervisor (Ph.D.
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 of Frankfurt am Main as the successor of Arthur Moritz Schönflies
. Siegel, who was deeply opposed to Nazism, was a close friend of the docents
Ernst Hellinger
and Max Dehn
and used his influence to help them. This attitude prevented Siegel's appointment as a successor to the chair of Constantin Carathéodory
in Munich. In Frankfurt he took part in a seminar 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 WWII are in an essay in his collected works.
In 1938, he returned to Göttingen
before emigrating in 1940 via Norway
to the United States
, where he joined the Institute for Advanced Study
in Princeton
, where he had already spent a sabbatical in 1935. He returned to Göttingen only after World War II
, when he accepted a post as professor
in 1951, which he kept until his retirement in 1959.
, diophantine equation
s, and celestial mechanics
in particular won him numerous honours. In 1978, he was awarded the Wolf Prize in Mathematics
, one of the most prestigious in the field.
Siegel's work spans analytic number theory
; and his theorem
on the finiteness of the integer points of curves, for genus
> 1, is historically important as a major general result on diophantine equations, when the field was essentially undeveloped. He worked on L-function
s, discovering the (presumed illusory) Siegel zero
phenomenon. His work derived from the Hardy-Littlewood circle method
on quadratic form
s proved very influential on the later, adele group theories encompassing the use of theta-functions. The Siegel modular form
s are recognised as part of the moduli theory of abelian varieties. In all this work the structural implications of analytic methods show through.
André Weil
, without hesitation, named Siegel as the greatest mathematician of the first half of the 20th century. 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 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.
about Siegel:
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
specialising in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and celestial mechanics
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
. He was one of the most important mathematicians of the 20th century.
Biography
Siegel was born in BerlinBerlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
, where he enrolled at the Humboldt University in Berlin in 1915 as a student in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. Amongst his teachers were Max Planck
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...
and Ferdinand Georg Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
, whose influence made the young Siegel abandon astronomy and turn towards number theory instead. His best student was Jürgen Moser
Jürgen Moser
Jürgen Kurt Moser or Juergen Kurt Moser was a German-American mathematician.-Professional biography:...
, one of the founders of KAM theory (Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...
-Arnold
Vladimir Arnold
Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...
-Moser), which lies at the foundations of chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
.
In 1917 Siegel was drafted into the German Army
German Army
The German Army is the land component of the armed forces of the Federal Republic of Germany. Following the disbanding of the Wehrmacht after World War II, it was re-established in 1955 as the Bundesheer, part of the newly formed West German Bundeswehr along with the Navy and the Air Force...
. Since he refused military service, he was committed to a psychiatric institute. According to his own words, he withstood the experience only because of his support from Edmund Landau
Edmund Landau
Edmund Georg Hermann Landau was a German Jewish mathematician who worked in the fields of number theory and complex analysis.-Biography:...
, whose father had a clinic in the neighborhood. After the end of World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
, he enrolled at the Georg-August University of Göttingen
Georg-August University of Göttingen
The University of Göttingen , known informally as Georgia Augusta, is a university in the city of Göttingen, Germany.Founded in 1734 by King George II of Great Britain and the Elector of Hanover, it opened for classes in 1737. The University of Göttingen soon grew in size and popularity...
, studying under Edmund Landau
Edmund Landau
Edmund Georg Hermann Landau was a German Jewish mathematician who worked in the fields of number theory and complex analysis.-Biography:...
, who was his doctoral thesis supervisor (Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
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 of Frankfurt am Main as the successor of Arthur Moritz Schönflies
Arthur Moritz Schönflies
Arthur Moritz Schoenflies , 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....
. Siegel, who was deeply opposed to Nazism, was a close friend of the docents
Docent
Docent is a title at some European universities to denote a specific academic appointment within a set structure of academic ranks below professor . Docent is also used at some universities generically for a person who has the right to teach...
Ernst Hellinger
Ernst Hellinger
Ernst David Hellinger was a German mathematician.-Early years:Ernst Hellinger was born on September 30, 1883 in Striegau, Silesia, Germany to Emil and Julie Hellinger. He grew up in Breslau, attended school and graduated from the Gymnasium there in 1902...
and Max Dehn
Max Dehn
Max Dehn was a German American mathematician and a student of David Hilbert. He is most famous for his work in geometry, topology and geometric group theory...
and used his influence to help them. This attitude prevented Siegel's appointment as a successor to the chair of Constantin Carathéodory
Constantin Carathéodory
Constantin Carathéodory was a Greek mathematician. He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory...
in Munich. In Frankfurt he took part in a seminar with Dehn, Hellinger, Paul Epstein
Paul Epstein
Paul Epstein was a German mathematician. He was known for his contributions to number theory, in particular the Epstein zeta function....
, 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 WWII are in an essay in his collected works.
In 1938, he returned to Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
before emigrating in 1940 via Norway
Norway
Norway , officially the Kingdom of Norway, is a Nordic unitary constitutional monarchy whose territory comprises the western portion of the Scandinavian Peninsula, Jan Mayen, and the Arctic archipelago of Svalbard and Bouvet Island. Norway has a total area of and a population of about 4.9 million...
to the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
, where he joined the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
in Princeton
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
, where he had already spent a sabbatical in 1935. He returned to Göttingen only after World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
, when he accepted a post as professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
in 1951, which he kept until his retirement in 1959.
Career
Siegel's work on number theoryNumber theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
s, and celestial mechanics
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
in particular won him numerous honours. In 1978, he was awarded the Wolf Prize in Mathematics
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.
Siegel's work spans analytic number theory
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 Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
; and his theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
on the finiteness of the integer points of curves, for genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
> 1, is historically important as a major general result on diophantine equations, when the field was essentially undeveloped. He worked on L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s, discovering the (presumed illusory) Siegel zero
Siegel zero
In mathematics, more specifically in the field of analytic number theory, a Siegel zero, named after Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeroes of Dirichlet L-function....
phenomenon. His work derived from the Hardy-Littlewood circle method
Hardy-Littlewood circle method
In mathematics, the Hardy–Littlewood circle method is one of the most frequently used techniques of analytic number theory. It is named for G. H. Hardy and J. E...
on quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s proved very influential on the later, adele group theories encompassing the use of theta-functions. The Siegel modular form
Siegel modular form
In mathematics, Siegel modular forms are a major type of automorphic form. These stand in relation to the conventional elliptic modular forms as abelian varieties do in relation to elliptic curves; the complex manifolds constructed as in the theory are basic models for what a moduli space for...
s are recognised as part of the moduli theory of abelian varieties. In all this work the structural implications of analytic methods show through.
André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
, without hesitation, named Siegel as the greatest mathematician of the first half of the 20th century. 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 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.
Works
by Siegel:- Gesammelte Werke, 3 Bände, Springer 1966
- with Jürgen MoserJürgen MoserJürgen Kurt Moser or Juergen Kurt Moser was a German-American mathematician.-Professional biography:...
Lectures on Celestial mechanics, based upon the older work Vorlesungen über Himmelsmechanik, Springer - 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)
- Letter to Louis J. Mordell, March 3, 1964.
about Siegel:
- Harold DavenportHarold DavenportHarold Davenport FRS was an English mathematician, known for his extensive work in number theory.-Early life:...
: 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 DieudonnéJean DieudonnéJean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
: Article in Dictionary of Scientific Biography - Eberhard Freitag: Siegelsche Modulfunktionen, Jahresbericht DMV, Bd.79, 1977, S.79-86
- Hel Braun: Eine Frau und die Mathematik 1933 - 1940, Springer 1990 (Reminiscence)
- Constance Reid: Hilbert, as well as Courant, Springer (The two biographies contain some information on Siegel.)
- Max Deuring: Carl Ludwig Siegel, 31. Dezember 1896 - 4. April 1981, Acta ArithmeticaActa ArithmeticaActa 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 Sciences....
, Vol.45, 1985, pp.93-113, online and Publications list - Goro ShimuraGoro Shimurais a Japanese mathematician, and currently a professor emeritus of mathematics at Princeton University.Shimura was a colleague and a friend of Yutaka Taniyama...
: "1996 Steele Prizes" (with Shimura's reminiscences concerning C. L. Siegel), Notices of the AMS, Vol. 43, 1996, pp. 1343-7, pdf - Serge LangSerge LangSerge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra...
: Mordell's Review, Siegel's letter to Mordell, diophantine geometry and 20th century mathematics, Notices American Mathematical Society 1995, Heft 3, auch in Gazette des Mathematiciens 1995,
See also
- Siegel disk
- Siegel's lemmaSiegel's lemmaIn 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 was proven by Axel Thue; Thue's proof used Dirichlet's box principle. Carl...
- Thue-Siegel-Roth theorem
- Brauer-Siegel theoremBrauer-Siegel theoremIn mathematics, the Brauer–Siegel theorem, named after Richard Brauer and Carl Ludwig Siegel, is an asymptotic result on the behaviour of algebraic number fields, obtained by Richard Brauer and Carl Ludwig Siegel...
- Siegel upper half-space
- Siegel-Weil formula
- Siegel modular formSiegel modular formIn mathematics, Siegel modular forms are a major type of automorphic form. These stand in relation to the conventional elliptic modular forms as abelian varieties do in relation to elliptic curves; the complex manifolds constructed as in the theory are basic models for what a moduli space for...
- Smith–Minkowski–Siegel mass formulaSmith–Minkowski–Siegel mass formulaIn mathematics, the Smith–Minkowski–Siegel mass formula is a formula for the sum of the weights of the lattices in a genus, weighted by the reciprocals of the orders of their automorphism groups...
- Riemann-Siegel theta function
- Riemann–Siegel formulaRiemann–Siegel formulaIn 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 in unpublished manuscripts of Bernhard Riemann...
External links
- Freddy Litten Die Carathéodory-Nachfolge in München 1938-1944
- 85. Band Heft 4 der DMV (mit 3 Arbeiten über Siegels Leben und Werk) (PDF-Datei; 6,77 MB)
- Siegel Approximation algebraischer Zahlen, Mathematische Zeitschrift, Bd.10, 1921, Dissertation
- Siegel „Additive Zahlentheorie in Zahlkörpern“, 1921, Jahresbericht DMV
- Webseite Uni Göttingen mit Biographie und Erläuterungen z. B. zur Klassenzahlformel