Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, differential geometry. In the field of real analysis, he is known for the first rigorous formulation of the integral, the Riemann integral, his work on Fourier series, his contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity, he is considered by many to be one of the greatest mathematicians of all time. Riemann was born on September 17, 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover, his father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars.
His mother, Charlotte Ebell, died. Riemann was the second of six children and suffering from numerous nervous breakdowns. Riemann exhibited exceptional mathematical skills, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public. During 1840, Riemann went to Hanover to attend lyceum. After the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. In high school, Riemann studied the Bible intensively, but he was distracted by mathematics, his teachers were amazed by his adept ability to perform complicated mathematical operations, in which he outstripped his instructor's knowledge. In 1846, at the age of 19, he started studying philology and Christian theology in order to become a pastor and help with his family's finances. During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen, where he planned to study towards a degree in Theology. However, once there, he began studying mathematics under Carl Friedrich Gauss.
Gauss recommended that Riemann enter the mathematical field. During his time of study, Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, Gotthold Eisenstein were teaching, he stayed in Berlin for two years and returned to Göttingen in 1849. Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Albert Einstein's general theory of relativity. In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann being granted a regular salary. In 1859, following Dirichlet's death, he was promoted to head the mathematics department at the University of Göttingen, he was the first to suggest using dimensions higher than three or four in order to describe physical reality. In 1862 he married Elise Koch and they had a daughter. Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866, he died of tuberculosis during his third journey to Italy in Selasca where he was buried in the cemetery in Biganzolo.
Riemann was a dedicated Christian, the son of a Protestant minister, saw his life as a mathematician as another way to serve God. During his life, he held to his Christian faith and considered it to be the most important aspect of his life. At the time of his death, he was reciting the Lord’s Prayer with his wife and died before they finished saying the prayer. Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work, some deep insights may have been lost forever. Riemann's tombstone in Biganzolo refers to Romans 8:28: Here rests in God Georg Friedrich Bernhard Riemann Professor in Göttingen born in Breselenz, 17 September 1826 died in Selasca, 20 July 1866 For those who love God, all things must work together for the best. Riemann's published works opened up research areas combining analysis with geometry; these would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, complex manifold theory.
The theory of Riemann surfaces was elaborated by Felix Klein and Adolf Hurwitz. This area of mathematics is part of the foundation of topology and is still being applied in novel ways to mathematical physics. In 1853, Gauss asked his student Riemann to prepare a Habilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Göttingen in 1854 entitled "Ueber die Hypothesen welche der Geometrie zu Grunde liegen", it was only published twelve years in 1868 by Dedekind, two years after his death. Its early reception appears to have been slow but it is now recognized as one of the most important works in geometry; the subject founded by this work is Riemannian geometry. Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium; the fundamental object is called the Riemann curvature tensor. F
Barry Charles Mazur is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, the Mazur manifold in differential topology. Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate school and received his Ph. D. from Princeton University in 1959, becoming a Junior Fellow at Harvard from 1961 to 1964. He is the Gerhard Gade University Professor and a Senior Fellow at Harvard, his early work was in geometric topology. In an elementary fashion, he proved the generalized Schoenflies conjecture, around the same time as Morton Brown. Both Brown and Mazur received the Veblen Prize for this achievement.
He discovered the Mazur manifold and the Mazur swindle. His observations in the 1960s on analogies between primes and knots were taken up by others in the 1990s giving rise to the field of arithmetic topology. Coming under the influence of Alexander Grothendieck's approach to algebraic geometry, he moved into areas of diophantine geometry. Mazur's torsion theorem, which gives a complete list of the possible torsion subgroups of elliptic curves over the rational numbers, is a deep and important result in the arithmetic of elliptic curves. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves; this proof was carried in his seminal paper "Modular curves and the Eisenstein ideal". The ideas of this paper and Mazur's notion of Galois deformations, were among the key ingredients in Wiles's proof of Fermat's Last Theorem. Mazur and Wiles had earlier worked together on the main conjecture of Iwasawa theory. In an expository paper, Number Theory as Gadfly, Mazur describes number theory as a field which produces, without effort, innumerable problems which have a sweet, innocent air about them, tempting flowers.
He expanded his thoughts in the 2003 book Imagining Numbers and Circles Disturbed, a collection of essays on mathematics and narrative that he edited with writer Apostolos Doxiadis. In 1982 he was elected a member of the National Academy of Sciences, in 2012 he became a fellow of the American Mathematical Society. Mazur has received the Veblen Prize in geometry, the Cole Prize in number theory, the Chauvenet Prize for exposition, the Steele Prize for seminal contribution to research from the American Mathematical Society. In early 2013, he was presented with one of the 2011 National Medals of Science by President Barack Obama. Mazur, Barry. Prime numbers and the Riemann hypothesis. New York, NY: Cambridge University Press. ISBN 9781107499430. Mazur, Barry. Collected works of John Tate: parts i and ii. American Mathematical Society. ISBN 0821890913. Mazur, Barry. Imagining numbers:. New York: Farrar Straus Giroux. ISBN 0312421877. MR 1950850. Katz, Nicholas M.. Arithmetic moduli of elliptic curves. Annals of Mathematics Studies, 108.
Princeton University Press. ISBN 0-691-08349-5. MR 0772569. Fontaine–Mazur conjecture Mazur's control theorem Homepage of Barry Mazur O'Connor, John J.. Video of Mazur talking about his work, from the National Science & Technology Medals Foundation Barry Mazur on MathSciNet
Alexander Grothendieck was a French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics, he is considered by many to be the greatest mathematician of the 20th century. Born in Germany, Grothendieck was raised and lived in France. For much of his working life, however, he was, in effect, stateless; as he spelled his first name "Alexander" rather than "Alexandre" and his surname, taken from his mother, was the Dutch-like Low German "Grothendieck", he was sometimes mistakenly believed to be of Dutch origin. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des hautes études scientifiques and remained there until 1970, driven by personal and political convictions, he left following a dispute over military funding.
He became professor at the University of Montpellier and, while still producing relevant mathematical work, he withdrew from the mathematical community and devoted himself to political causes. Soon after his formal retirement in 1988, he moved to the Pyrenees, where he lived in isolation until his death in 2014. Grothendieck was born in Berlin to anarchist parents, his father, Alexander "Sascha" Schapiro, had Hasidic Jewish roots and had been imprisoned in Russia before moving to Germany in 1922, while his mother, Johanna "Hanka" Grothendieck, came from a Protestant family in Hamburg and worked as a journalist. Both had broken away from their early backgrounds in their teens. At the time of his birth, Grothendieck's mother was married to the journalist Johannes Raddatz and his birthname was recorded as "Alexander Raddatz." The marriage was dissolved in 1929 and Schapiro/Tanaroff acknowledged his paternity, but never married Hanka. Grothendieck lived with his parents in Berlin until the end of 1933, when his father moved to Paris to evade Nazism, followed soon thereafter by his mother.
They left Grothendieck in the care of a Lutheran pastor and teacher in Hamburg. During this time, his parents took part in the Spanish Civil War, according to Winfried Scharlau, as non-combatant auxiliaries, though others state that Sascha fought in the anarchist militia. In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterwards his father was interned in Le Vernet, he and his mother were interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners". The first was the Rieucros Camp, where his mother contracted the tuberculosis which caused her death and where Alexander managed to attend the local school, at Mende. Once Alexander managed to escape from the camp, intending to assassinate Hitler, his mother Hanka was transferred to the Gurs internment camp for the remainder of World War II. Alexander was permitted to live, separated from his mother, in the village of Le Chambon-sur-Lignon and hidden in local boarding houses or pensions, though he had to seek refuge in the woods during Nazis raids, surviving at times without food or water for several days.
His father was arrested under the Vichy anti-Jewish legislation, sent to the Drancy, handed over by the French Vichy government to the Germans to be sent to be murdered at the Auschwitz concentration camp in 1942. In Chambon, Grothendieck attended the Collège Cévenol, a unique secondary school founded in 1938 by local Protestant pacifists and anti-war activists. Many of the refugee children hidden in Chambon attended Cévenol, it was at this school that Grothendieck first became fascinated with mathematics. After the war, the young Grothendieck studied mathematics in France at the University of Montpellier where he did not perform well, failing such classes as astronomy. Working on his own, he rediscovered the Lebesgue measure. After three years of independent studies there, he went to continue his studies in Paris in 1948. Grothendieck attended Henri Cartan's Seminar at École Normale Supérieure, but he lacked the necessary background to follow the high-powered seminar. On the advice of Cartan and André Weil, he moved to the University of Nancy where he wrote his dissertation under Laurent Schwartz and Jean Dieudonné on functional analysis, from 1950 to 1953.
At this time he was a leading expert in the theory of topological vector spaces. From 1953 to 1955 he moved to the University of São Paulo in Brazil, where he immigrated by means of a Nansen passport, given that he refused to take French Nationality. By 1957, he set this subject aside in order to work in algebraic homological algebra; the same year he was invited to visit Harvard by Oscar Zariski, but the offer fell through when he refused to sign a pledge promising not to work to overthrow the United States government, a position that, he was warned, might have landed him in prison. The prospect did not worry him. Comparing Grothendieck during his Nancy years to the École Normale Supérieure trained students at that time: Pierre Samuel, Roger Godement, René Thom, Jacques Dixmier, Jean Cerf, Yvonne Bruhat, Jean-Pierre Serre, Bernard Malgrange, Leila Schneps says: He was so unknown to this group and to their professors, came from such a deprived and chaotic background, was, compared to them, so ignorant at the start of his research career
ArXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, astronomy, electrical engineering, computer science, quantitative biology, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month. ArXiv was made possible by the compact TeX file format, which allowed scientific papers to be transmitted over the Internet and rendered client-side. Around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Paul Ginsparg recognized the need for central storage, in August 1991 he created a central repository mailbox stored at the Los Alamos National Laboratory which could be accessed from any computer.
Additional modes of access were soon added: FTP in 1991, Gopher in 1992, the World Wide Web in 1993. The term e-print was adopted to describe the articles, it began as a physics archive, called the LANL preprint archive, but soon expanded to include astronomy, computer science, quantitative biology and, most statistics. Its original domain name was xxx.lanl.gov. Due to LANL's lack of interest in the expanding technology, in 2001 Ginsparg changed institutions to Cornell University and changed the name of the repository to arXiv.org. It is now hosted principally with eight mirrors around the world, its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv; the annual budget for arXiv is $826,000 for 2013 to 2017, funded jointly by Cornell University Library, the Simons Foundation and annual fee income from member institutions.
This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by 220 institutions. In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a life sentence". However, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. Although arXiv is not peer reviewed, a collection of moderators for each area review the submissions; the lists of moderators for many sections of arXiv are publicly available, but moderators for most of the physics sections remain unlisted.
Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, but to check whether the paper is appropriate for the intended subject area. New authors from recognized academic institutions receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for restricting scientific inquiry. A majority of the e-prints are submitted to journals for publication, but some work, including some influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof of Thurston's geometrization conjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman in November 2002.
Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there – let them go and read about it". Despite this non-traditional method of publication, other mathematicians recognized this work by offering the Fields Medal and Clay Mathematics Millennium Prizes to Perelman, both of which he refused. Papers can be submitted in any of several formats, including LaTeX, PDF printed from a word processor other than TeX or LaTeX; the submission is rejected by the arXiv software if generating the final PDF file fails, if any image file is too large, or if the total size of the submission is too large. ArXiv now allows one to store and modify an incomplete submission, only finalize the submission when ready; the time stamp on the article is set. The standard access route is through one of several mirrors. Sev
Pierre René, Viscount Deligne is a Belgian mathematician. He is known for work on the Weil conjectures, leading to a complete proof in 1973, he is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, 1978 Fields Medal. Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles, writing a dissertation titled Théorème de Lefschetz et critères de dégénérescence de suites spectrales, he completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled Théorie de Hodge. Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes Études Scientifiques near Paris on the generalization within scheme theory of Zariski's main theorem. In 1968, he worked with Jean-Pierre Serre. Deligne's focused on topics in Hodge theory, he tested them on objects in complex geometry. He collaborated with David Mumford on a new description of the moduli spaces for curves.
Their work came to be seen as an introduction to one form of the theory of algebraic stacks, has been applied to questions arising from string theory. Deligne's most famous contribution was his proof of the third and last of the Weil conjectures; this proof completed a programme initiated and developed by Alexander Grothendieck. As a corollary he proved the celebrated Ramanujan–Petersson conjecture for modular forms of weight greater than one. Deligne's 1974 paper contains the first proof of the Weil conjectures, Deligne's contribution being to supply the estimate of the eigenvalues of the Frobenius endomorphism, considered the geometric analogue of the Riemann hypothesis. Deligne's 1980 paper contains a much more general version of the Riemann hypothesis. From 1970 until 1984, Deligne was a permanent member of the IHÉS staff. During this time he did much important work outside of his work on algebraic geometry. In joint work with George Lusztig, Deligne applied étale cohomology to construct representations of finite groups of Lie type.
He received a Fields Medal in 1978. In 1984, Deligne moved to the Institute for Advanced Study in Princeton. In terms of the completion of some of the underlying Grothendieck program of research, he defined absolute Hodge cycles, as a surrogate for the missing and still conjectural theory of motives; this idea allows one to get around the lack of knowledge of the Hodge conjecture, for some applications. He reworked the Tannakian category theory in his 1990 paper for the Grothendieck Festschrift, employing Beck's theorem – the Tannakian category concept being the categorical expression of the linearity of the theory of motives as the ultimate Weil cohomology. All this is part of the yoga of uniting Hodge theory and the l-adic Galois representations; the Shimura variety theory is related, by the idea that such varieties should parametrize not just good families of Hodge structures, but actual motives. This theory is not yet a finished product, more recent trends have used K-theory approaches, he was awarded the Fields Medal in 1978, the Crafoord Prize in 1988, the Balzan Prize in 2004, the Wolf Prize in 2008, the Abel Prize in 2013.
In 2006 he was ennobled by the Belgian king as viscount. In 2009, Deligne was elected a foreign member of the Royal Swedish Academy of Sciences, he is a member of the Norwegian Academy of Letters. Deligne, Pierre. "La conjecture de Weil: I". Publications Mathématiques de l'IHÉS. 43: 273–307. Doi:10.1007/bf02684373. Deligne, Pierre. "La conjecture de Weil: II". Publications Mathématiques de l'IHÉS. 52: 137–252. Doi:10.1007/BF02684780. Deligne, Pierre. "Catégories tannakiennes". Grothendieck Festschrift vol II. Progress in Mathematics. 87: 111–195. Deligne, Pierre. "Real homotopy theory of Kähler manifolds". Inventiones Mathematicae. 29: 245–274. Doi:10.1007/BF01389853. MR 0382702. Deligne, Pierre. Commensurabilities among Lattices in PU. Princeton, N. J.: Princeton University Press. ISBN 0-691-00096-4. Quantum fields and strings: a course for mathematicians. Vols. 1, 2. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. Edited by Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C.
Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison and Edward Witten. American Mathematical Society, Providence, RI. Vol. 1: xxii+723 pp.. ISBN 0-8218-1198-3. Deligne wrote multiple hand-written letters to other mathematicians in the 1970s; these include "Deligne's letter to Piatetskii-Shapiro". Archived from the original on 7 December 2012. Retrieved 15 December 2012. "Deligne's letter to Jean-Pierre Serre". 2012-12-15. "Deligne's letter to Looijenga". Retrieved 15 December 2012; the following mathematical concepts are named after Deligne: Deligne–Lusztig theory Deligne–Mumford moduli space of curves Deligne–Mumford stacks Fourier–Deligne transform Deligne cohomology Deligne motive Deligne tensor product of abelian categories Langlands–Deligne local constantAdditionally, many different conjectures in mathematics have been called the De
Joe Harris (mathematician)
Joseph Daniel Harris, known nearly universally as Joe Harris, is a mathematician at Harvard University working in the field of algebraic geometry. He attended college at and received his Ph. D. from Harvard in 1978 under Phillip Griffiths. During the 1980s he was on the faculty of Brown University, moving to Harvard around 1988, he served as chair of the department at Harvard from 2002 to 2005. His work is characterized by its classical geometric flavor: he has claimed that nothing he thinks about could not have been imagined by the Italian geometers of the late 19th and early 20th centuries, that if he has had greater success than them, it is because he has access to better tools. Harris is well known for several of his books on algebraic geometry, notable for their informal presentations: Principles of Algebraic Geometry ISBN 978-0-471-05059-9, with Phillip Griffiths Geometry of Algebraic Curves, Vol. 1 ISBN 978-0-387-90997-4, with Enrico Arbarello, Maurizio Cornalba, Phillip Griffiths William Fulton, Joe Harris.
Representation Theory, A First Course, Graduate Texts in Mathematics, 129, New York: Springer-Verlag, doi:10.1007/978-1-4612-0979-9, ISBN 978-0-387-97495-8, MR 1153249, with William Fulton Joe Harris. Algebraic Geometry: A First Course, New York: Springer-Verlag, ISBN 978-0-387-97716-4 David Eisenbud, Joe Harris; the Geometry of Schemes, Graduate Texts in Mathematics, 197, New York: Springer-Verlag, ISBN 978-0-387-98638-8, MR 1730819, with David Eisenbud David Eisenbud, Joseph Harris. 3264 and All That: A Second Course in Algebraic Geometry. Cambridge University Press. ISBN 978-1107602724. Moduli of Curves ISBN 978-0-387-98438-4, with Ian Morrison. Harris has supervised 50 Ph. D. students, including Brendan Hassett, James McKernan, Rahul Pandharipande, Zvezdelina Stankova and Ravi Vakil
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques from commutative algebra, for solving geometrical problems about these sets of zeros; the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, parabolas, hyperbolas, cubic curves like elliptic curves, quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis and number theory. A study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, it becomes more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution. In the 20th century, algebraic geometry split into several subareas; the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more to the points with coordinates in an algebraically closed field. Real algebraic geometry is the study of the real points of an algebraic variety. Diophantine geometry and, more arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, p-adic fields.
A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers, it consists of algorithm design and software development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way, similar to its use in the study of differential and analytic manifolds; this is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring.
This means that a point of such a scheme may be either a subvariety. This approach enables a unification of the language and the tools of classical algebraic geometry concerned with complex points, of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach. In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 − 1 = 0. A "slanted" circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 − 1 = 0, x + y + z = 0. First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed.
We consider the affine space of dimension n over denoted An. When one fixes a coordinate system, one may identify An with kn; the purpose of not working with kn is to emphasize that one "forgets" the vector space structure that kn carries. A function f: An → A1 is said to be polynomial if it can be written as a polynomial, that is, if there is a polynomial p in k such that f = p for every point M with coordinates in An; the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the