1.
Andrey Kolmogorov
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Andrey Kolmogorov was born in Tambov, about 500 kilometers south-southeast of Moscow, in 1903. Kolmogorova, died giving birth to him, Andrey was raised by two of his aunts in Tunoshna at the estate of his grandfather, a well-to-do nobleman. Little is known about Andreys father and he was supposedly named Nikolai Matveevich Kataev and had been an agronomist. Nikolai had been exiled from St. Petersburg to the Yaroslavl province after his participation in the movement against the czars. He disappeared in 1919 and he was presumed to have killed in the Russian Civil War. Andrey Kolmogorov was educated in his aunt Veras village school, and his earliest literary efforts, Andrey was the editor of the mathematical section of this journal. In 1910, his aunt adopted him, and they moved to Moscow, later that same year, Kolmogorov began to study at the Moscow State University and at the same time Mendeleev Moscow Institute of Chemistry and Technology. Kolmogorov writes about this time, I arrived at Moscow University with a knowledge of mathematics. I knew in particular the beginning of set theory, I studied many questions in articles in the Encyclopedia of Brockhaus and Efron, filling out for myself what was presented too concisely in these articles. Kolmogorov gained a reputation for his wide-ranging erudition, during the same period, Kolmogorov worked out and proved several results in set theory and in the theory of Fourier series. In 1922, Kolmogorov gained international recognition for constructing a Fourier series that diverges almost everywhere, around this time, he decided to devote his life to mathematics. In 1925, Kolmogorov graduated from the Moscow State University and began to study under the supervision of Nikolai Luzin, Kolmogorov became interested in probability theory. In 1929, Kolmogorov earned his Doctor of Philosophy degree, from Moscow State University, in 1930, Kolmogorov went on his first long trip abroad, traveling to Göttingen and Munich, and then to Paris. He had various contacts in Göttingen. His pioneering work, About the Analytical Methods of Probability Theory, was published in 1931, also in 1931, he became a professor at the Moscow State University. In 1935, Kolmogorov became the first chairman of the department of probability theory at the Moscow State University, around the same years Kolmogorov contributed to the field of ecology and generalized the Lotka–Volterra model of predator-prey systems. In 1936, Kolmogorov and Alexandrov were involved in the persecution of their common teacher Nikolai Luzin, in the so-called Luzin affair. In a 1938 paper, Kolmogorov established the basic theorems for smoothing and predicting stationary stochastic processes—a paper that had military applications during the Cold War

2.
Andrew Wiles
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Sir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is most notable for proving Fermats Last Theorem, for which he received the 2016 Abel Prize, Wiles has received numerous other honours. Wiles was born in 1953 in Cambridge, England, the son of Maurice Frank Wiles, the Regius Professor of Divinity at the University of Oxford and his father worked as the Chaplain at Ridley Hall, Cambridge, for the years 1952–55. Wiles attended Kings College School, Cambridge, and The Leys School, Wiles states that he came across Fermats Last Theorem on his way home from school when he was 10 years old. He stopped by his local library where he found a book about the theorem. Fascinated by the existence of a theorem that was so easy to state that he, a ten-year-old, could understand it, Wiles earned his bachelors degree in mathematics in 1974 at Merton College, Oxford, and a PhD in 1980 at Clare College, Cambridge. After a stay at the Institute for Advanced Study in New Jersey in 1981, in 1985–86, Wiles was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques near Paris and at the École Normale Supérieure. From 1988 to 1990, Wiles was a Royal Society Research Professor at the University of Oxford and he rejoined Oxford in 2011 as Royal Society Research Professor. Wiless graduate research was guided by John Coates beginning in the summer of 1975, together these colleagues worked on the arithmetic of elliptic curves with complex multiplication by the methods of Iwasawa theory. He further worked with Barry Mazur on the conjecture of Iwasawa theory over the rational numbers. The modularity theorem involved elliptic curves, which was also Wiless own specialist area, the conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or perhaps impossible to prove. Despite this, Wiles, with his fascination with Fermats Last Theorem, decided to undertake the challenge of proving the conjecture. In June 1993, he presented his proof to the public for the first time at a conference in Cambridge and he gave a lecture a day on Monday, Tuesday and Wednesday with the title Modular Forms, Elliptic Curves and Galois Representations. There was no hint in the title that Fermats last theorem would be discussed, finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture. Then, seemingly as an afterthought, he noted that that meant that Fermats last theorem was true, in August 1993, it was discovered that the proof contained a flaw in one area. Wiles tried and failed for over a year to repair his proof, according to Wiles, the crucial idea for circumventing, rather than closing this area, came to him on 19 September 1994, when he was on the verge of giving up. Together with his former student Richard Taylor, he published a paper which circumvented the problem. Both papers were published in May 1995 in a volume of the Annals of Mathematics

3.
Jean-Pierre Serre
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Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000, born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951, from 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France and his wife, Professor Josiane Heulot-Serre, was a chemist, she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil, the French mathematician Denis Serre is his nephew. Serres thesis concerned the Leray–Serre spectral sequence associated to a fibration, together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, Serre subsequently changed his research focus. However, Weyls perception that the place of classical analysis had been challenged has subsequently been justified. In the 1950s and 1960s, a collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents, on coherent cohomology, even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field couldnt capture as much topology as singular cohomology with integer coefficients, amongst Serres early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties — those that become trivial after pullback by a finite étale map — are important and this acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique 4 and SGA5, from 1959 onward Serres interests turned towards group theory, number theory, in particular Galois representations and modular forms. In his paper FAC, Serre asked whether a finitely generated module over a polynomial ring is free. This question led to a deal of activity in commutative algebra. This result is now known as the Quillen-Suslin theorem, Serre, at twenty-seven in 1954, is the youngest ever to be awarded the Fields Medal. He went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000 and he has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre. He is a member of several scientific Academies and has received many honorary degrees

4.
David Mumford
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David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow, in 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University, Mumford was born in Worth, West Sussex in England, of an English father and American mother. His father William started a school in Tanzania and worked for the then newly created United Nations. In high school, he was a finalist in the prestigious Westinghouse Science Talent Search, after attending the Phillips Exeter Academy, Mumford went to Harvard, where he became a student of Oscar Zariski. At Harvard, he became a Putnam Fellow in 1955 and 1956 and he completed his Ph. D. in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. He met his first wife, Erika Jentsch, at Radcliffe College, after Erika died in 1988, he married his second wife, Jenifer Gordon. He and Erika had four children, Mumfords work in geometry combined traditional geometric insights with the latest algebraic techniques. He published on moduli spaces, with a theory summed up in his book Geometric Invariant Theory, on the equations defining an abelian variety and his books Abelian Varieties and Curves on an Algebraic Surface combined the old and new theories. His lecture notes on scheme theory circulated for years in unpublished form, at a time when they were, beside the treatise Éléments de géométrie algébrique and they are now available as The Red Book of Varieties and Schemes. Other work that was less thoroughly written up were lectures on varieties defined by quadrics, and this work on the equations defining abelian varieties appeared in 1966–7. He published some books of lectures on the theory. He also was one of the founders of the toroidal embedding theory and these pathologies fall into two types, bad behavior in characteristic p and bad behavior in moduli spaces. This second example is developed further in Mumfords third paper on classification of surfaces in characteristic p, worse pathologies related to p-torsion in crystalline cohomology were explored by Luc Illusie. Further such examples arise in Zariski surface theory and he also conjectures that the Kodaira vanishing theorem is false for surfaces in characteristic p. In the third paper, he gives an example of a surface for which Kodaira vanishing fails. The first example of a surface for which Kodaira vanishing fails was given by Michel Raynaud in 1978. In the second Pathologies paper, Mumford finds that the Hilbert scheme parametrizing space curves of degree 14, in the fourth Pathologies paper, he finds reduced and irreducible complete curves which are not specializations of non-singular curves

5.
Israel Gelfand
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Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gelfand, or Izrail M. Gelfand was a prominent Soviet mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and his legacy continues through his students, who include Endre Szemerédi, Alexandre Kirillov, Edward Frenkel, Joseph Bernstein, as well as his own son, Sergei Gelfand. A native of Kherson Governorate of the Russian Empire, Gelfand was born into a Jewish family in the small southern Ukrainian town of Okny, according to his own account, Gelfand was expelled from high school because his father had been a mill owner. Bypassing both high school and college, he proceeded to study at Moscow State University, where his advisor was the preeminent mathematician Andrei Kolmogorov. He nevertheless managed to attend lectures at the University and began study at the age of 19. The Gelfand–Tsetlin basis is a widely used tool in theoretical physics, Gelfand also published works on biology and medicine. For a long time he took an interest in cell biology and he worked extensively in mathematics education, particularly with correspondence education. In 1994, he was awarded a MacArthur Fellowship for this work, Gelfand was married to Zorya Shapiro, and their two sons, Sergei and Vladimir both live in the United States. A third son, Aleksandr, died of leukemia, following the divorce from his first wife, Gelfand married his second wife, Tatiana, together they had a daughter, Tatiana. The family also includes four grandchildren and three great-grandchildren, the memories about I. Gelfand are collected at the special site handled by his family. Gelfand held several degrees and was awarded the Order of Lenin three times for his research. In 1977 he was elected a Foreign Member of the Royal Society and he won the Wolf Prize in 1978, Kyoto Prize in 1989 and MacArthur Foundation Fellowship in 1994. Israel Gelfand died at the Robert Wood Johnson University Hospital near his home in Highland Park and he was less than five weeks past his 96th birthday. His death was first reported on the blog of his former collaborator Andrei Zelevinsky and confirmed a few hours later by an obituary in the Russian online newspaper Polit. ru. Gelfand, I. M. Lectures on linear algebra, Courier Dover Publications, ISBN 978-0-486-66082-0 Gelfand, I. M. Fomin, Sergei V. Silverman, Richard A. ed. Calculus of variations, Englewood Cliffs, ISBN 978-0-486-41448-5, MR0160139 Gelfand, I. Raikov, D. Shilov, G. Commutative normed rings, Translated from the Russian, with a chapter, New York. ISBN 978-0-8218-2022-3, MR0205105 Gelfand, I. M. Shilov, G. E. Generalized functions. Vol. I, Properties and operations, Translated by Eugene Saletan, Boston, MA, Academic Press, ISBN 978-0-12-279501-5, MR0166596 Gelfand, I. M. Shilov, G. E. Generalized functions

6.
Carl Ludwig Siegel
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Carl Ludwig Siegel was a German mathematician specialising in number theory and celestial mechanics. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation and he was named as one of the most important mathematicians of the 20th century. André Weil, without hesitation, named Siegel as the greatest mathematician of the first half of the 20th century. Atle Selberg said of Siegel and his work, Siegel was born in Berlin, 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 his best student was Jürgen Moser, one of the founders of KAM theory, which lies at the foundations of chaos theory. Another notable student was Kurt Mahler, the number theorist, Siegel was an antimilitarist, and in 1917, during World War I he was committed to a psychiatric institute as a conscientious objector. According to his own words, he withstood the experience only because of his support from Edmund Landau, after the end of World War I, he enrolled at the Georg-August University of Göttingen, studying under Landau, who was his doctoral thesis supervisor. He stayed in Göttingen as a teaching and research assistant, many of his 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 opposed to Nazism, was a close friend of the docents Ernst Hellinger and Max Dehn. This attitude prevented Siegels appointment as a successor to the chair of Constantin Carathéodory 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, Siegels 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 and 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. Siegels work on theory, diophantine equations, and celestial mechanics in particular won him numerous honours. In 1978, he was awarded the first Wolf Prize in Mathematics, when the prize committee decided to select the greatest living mathematician, the discussion centered around Siegel and Israel Gelfand as the leading candidates. The prize was split between them. He worked on L-functions, discovering the Siegel zero phenomenon and his work, derived from the Hardy–Littlewood circle method on quadratic forms, appeared in the later, adele group theories encompassing the use of theta-functions. The Siegel modular forms are recognised as part of the theory of abelian varieties

7.
Jean Leray
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Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology. He studied at École Normale Supérieure from 1926 to 1929 and he received his Ph. D. in 1933. Leray wrote an important paper that founded the study of solutions of the Navier–Stokes equations. From 1938 to 1939 he was professor at the University of Nancy and he did not join the Bourbaki group, although he was close with its founders. His main work in topology was carried out while he was in a prisoner of war camp in Edelbach and he concealed his expertise on differential equations, fearing that its connections with applied mathematics could lead him to be asked to do war work. Lerays work of this period proved seminal to the development of spectral sequences and sheaves and these were subsequently developed by many others, each separately becoming an important tool in homological algebra. He returned to work on differential equations from about 1950. He was professor at the University of Paris from 1945 to 1947 and he was awarded the Malaxa Prize, the Grand Prix in mathematical sciences, the Feltrinelli Prize, the Wolf Prize in Mathematics, and the Lomonosov Gold Medal. Leray spectral sequence Leray cover Lerays theorem Leray–Hirsch theorem OConnor, John J. Robertson, Jean Leray, MacTutor History of Mathematics archive, University of St Andrews. Jean Leray at the Mathematics Genealogy Project Jean Leray, by Armand Borel, Gennadi M. Henkin, and Peter D. Lax, Notices of the American Mathematical Society, vol

8.
Henri Cartan
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Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan and the brother of composer Jean Cartan, Cartan studied at the Lycée Hoche in Versailles, then at the École Normale Supérieure in Paris, receiving his doctorate in mathematics. Cartan is known for work in topology, in particular on cohomology operations, the method of killing homotopy groups. The number of his students was small, but includes Adrien Douady, Roger Godement, Max Karoubi, Jean-Louis Koszul, Jean-Pierre Serre. Cartan also was a member of the Bourbaki group and one of its most active participants. His book with Samuel Eilenberg Homological Algebra was an important text, Cartan used his influence to help obtain the release of some dissident mathematicians, including Leonid Plyushch and Jose Luis Massera. For his humanitarian efforts, he received the Pagels Award from the New York Academy of Sciences, the Cartan model in algebra is named after Cartan. Cartan died on 13 August 2008 at the age of 104 and his funeral took place the following Wednesday on 20 August in Die, Drome. Cartan received numerous honours and awards including the Wolf Prize in 1980 and he was an Invited Speaker at the ICM in 1932 in Zurich and a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts and in 1958 in Edinburgh. From 1974 until his death he had been a member of the French Academy of Sciences, Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications, thèse,1928 Sur les groupes de transformations analytiques,1935. Sur les classes de fonctions définies par des inégalités portant sur leurs dérivées successives,1940, cohomologie des groupes, suite spectrale, faisceaux, 1950-1951. Algèbres dEilenberg - Mac Lane et homotopie, 1954-1955, Homological Algebra, Princeton Univ Press,1956 ISBN 978-0-69104991-5 Séminaires de lÉcole normale supérieure, Secr. IHP, 1948-1964, New York, W. A. Benjamin ed.1967, théorie élémentaire des fonctions analytiques, Paris, Hermann,1961. Differential Forms, Dover 2006 Œuvres — Collected Works,3 vols, ed. Reinhold Remmert & Jean-Pierre Serre, Springer Verlag, Heidelberg,1967. Relations dordre en théorie des permutations des ensembles finis, Neuchâtel,1973, théorie élémentaire des fonctions analytiques dune ou plusieurs variables complexes, Paris, Hermann,1975. Elementary theory of functions of one or several complex variables, Dover 1995 Cours de calcul différentiel, Paris. Correspondance entre Henri Cartan et André Weil, Paris, SMF,2011, oConnor, John J. Robertson, Edmund F. Henri Cartan, MacTutor History of Mathematics archive, University of St Andrews. A17, retrieved 2008-08-25 Cartan, Henri, Eilenberg, Samuel, notices of the American Mathematical Society, Sept.2010, vol

9.
Lars Ahlfors
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Lars Valerian Ahlfors was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis. Ahlfors was born in Helsinki, Finland and his mother, Sievä Helander, died at his birth. His father, Axel Ahlfors, was a professor of engineering at the Helsinki University of Technology, the Ahlfors family was Swedish-speaking, so he first attended a private school where all classes were taught in Swedish. Ahlfors studied at University of Helsinki from 1924, graduating in 1928 having studied under Ernst Lindelöf and he assisted Nevanlinna in 1929 with his work on Denjoys conjecture on the number of asymptotic values of an entire function. In 1929 Ahlfors published the first proof of this conjecture, now known as the Denjoy–Carleman–Ahlfors theorem and it states that the number of asymptotic values approached by an entire function of order ρ along curves in the complex plane going toward infinity is less than or equal to 2ρ. He completed his doctorate from the University of Helsinki in 1930, Ahlfors worked as an associate professor at the University of Helsinki from 1933 to 1936. In 1936 he was one of the first two people to be awarded the Fields Medal, in 1935 Ahlfors visited Harvard University. He returned to Finland in 1938 to take up a professorship at the University of Helsinki, the outbreak of war led to problems although Ahlfors was unfit for military service. He was offered a post at the Swiss Federal Institute of Technology at Zurich in 1944, Ahlfors was a visiting scholar at the Institute for Advanced Study in 1962 and again in 1966. He was awarded the Wihuri Prize in 1968 and the Wolf Prize in Mathematics in 1981 and his book Complex Analysis is the classic text on the subject and is almost certainly referenced in any more recent text which makes heavy use of complex analysis. Ahlfors wrote several significant books, including Riemann surfaces and Conformal invariants. He made decisive contributions to meromorphic curves, value distribution theory, Riemann surfaces, conformal geometry, quasiconformal mappings, in 1933, he married Erna Lehnert, an Austrian who with her parents had first settled in Sweden and then in Finland. FUNDAMENTAL POLYHEDRONS AND LIMIT POINT SETS OF KLEINIAN GROUPS, proceedings of the National Academy of Sciences. Lars Ahlfors at the Mathematics Genealogy Project Ahlfors entry on Harvard University Mathematics department web site, lars Valerian Ahlfors, Notices of the American Mathematical Society, vol. Lars Valerian Ahlfors, a biographical memoir, National Academy of Sciences Biographical Memoir Author profile in the database zbMATH

10.
Oscar Zariski
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Oscar Zariski (born Oscher Zaritsky was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century. Zariski was born Oscher Zaritsky to a Jewish family and in 1918 studied at the University of Kiev, Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, which was proposed to him by Castelnuovo. At the time of his publication, he changed his name to Oscar Zariski. Zariski emigrated to the United States in 1927 supported by Solomon Lefschetz and he had a position at Johns Hopkins University where he became professor in 1937. During this period, he wrote Algebraic Surfaces as a summation of the work of the Italian school, the book was published in 1935 and reissued 36 years later, with detailed notes by Zariskis students that illustrated how the field of algebraic geometry had changed. It is still an important reference and it seems to have been this work that set the seal of Zariskis discontent with the approach of the Italians to birational geometry. He addressed the question of rigour by recourse to commutative algebra, the Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions. That theory is too limited for algebraic surfaces, and even for curves with singular points, a rational map is to a regular map as a rational function is to a polynomial, it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, the description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the case, to use also valuation theory to describe the phenomena such as blowing up. After spending a year 1946–1947 at the University of Illinois, Zariski became professor at Harvard University in 1947 where he remained until his retirement in 1969, in 1945, he fruitfully discussed foundational matters for algebraic geometry with André Weil. The two sets of foundations werent reconciled at that point, Zariski himself worked on equisingularity theory. Zariski proposed the first example of a Zariski surface in 1958, Zariski was awarded the Steele Prize in 1981, and in the same year the Wolf Prize in Mathematics with Lars Ahlfors. He wrote also Commutative Algebra in two volumes, with Pierre Samuel and his papers have been published by MIT Press, in four volumes. Zariski, Oscar, Abhyankar, Shreeram S. Lipman, Joseph, Mumford, David, an introduction to the theory of algebraic surfaces, Lecture notes in mathematics,83, Berlin, New York, Springer-Verlag, doi,10. Vol. II, Berlin, New York, Springer-Verlag, ISBN 978-0-387-90171-8, MR0389876 Zariski, Oscar, Kmety, François, Merle, Michel, Lichtin, Ben, the moduli problem for plane branches, University Lecture Series,39, Providence, R. I. American Mathematical Society, ISBN 978-0-8218-2983-7, MR0414561, Le problème des modules pour les branches planes Zariski, Oscar, Collected papers. Vol. I, Foundations of algebraic geometry and resolution of singularities, The MIT Press, Cambridge, Mass. -London, ISBN 978-0-262-08049-1, MR0505100 Zariski, Oscar, robertson, Edmund F. Oscar Zariski, MacTutor History of Mathematics archive, University of St Andrews