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81-583: Grigori Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман , IPA: [ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman] ; born 13 June 1966) is a Russian mathematician and geometer who is known for his contributions to the fields of geometric analysis , Riemannian geometry , and geometric topology . In 2005, Perelman resigned from his research post in Steklov Institute of Mathematics and in 2006 stated that he had quit professional mathematics, owing to feeling disappointed over

162-750: A Fields Medal for his work on the Ricci flow. However, Perelman declined to accept the prize. Sir John Ball , president of the International Mathematical Union , approached Perelman in Saint Petersburg in June 2006 to persuade him to accept the prize. After 10 hours of attempted persuasion over two days, Ball gave up. Two weeks later, Perelman summed up the conversation as follows: " He proposed to me three alternatives: accept and come; accept and don't come, and we will send you

243-421: A Ricci flow with surgery in three dimensions, systematically excising singular regions as they develop. As an immediate corollary of his construction, Perelman resolved a major conjecture on the topological classification in three dimensions of closed manifolds which admit metrics of positive scalar curvature . His third preprint (or alternatively Colding and Minicozzi's work) showed that on any space satisfying

324-465: A closed three-dimensional manifold has the property that any loop can be contracted into a point, then it must be topologically equivalent to a 3-sphere. Stephen Smale proved a high-dimensional analogue of Poincaré's conjecture in 1961, and Michael Freedman proved the four-dimensional version in 1982. Despite their work, the case of three-dimensional spaces remained completely unresolved. Moreover, Smale and Freedman's methods have had no impact on

405-404: A Riemannian metric, in certain geometric settings. As a byproduct, he was able to prove some new and striking theorems in the field of Riemannian geometry . Despite formal similarities, Hamilton's equations are significantly more complex and nonlinear than the heat equation, and it is impossible that such uniformization is achieved without contextual assumptions. In completely general settings, it

486-410: A compact nonnegatively curved submanifold, called a soul , whose normal bundle is diffeomorphic to the original space. From the perspective of homotopy theory , this says in particular that every complete Riemannian metric of nonnegative sectional curvature may be taken to be closed . Cheeger and Gromoll conjectured that if the curvature is strictly positive somewhere, then the soul can be taken to be

567-409: A consequence, Hamilton's compactness and the corresponding existence of subsequential limits could be applied somewhat freely. The "canonical neighborhoods theorem" is the second main result of Perelman's first preprint. In this theorem, Perelman achieved the quantitative understanding of singularities of three-dimensional Ricci flow which had eluded Hamilton. Roughly speaking, Perelman showed that on

648-477: A financial economist might study the structural reasons why a company may have a certain share price , a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock ( see: Valuation of options ; Financial modeling ). According to the Dictionary of Occupational Titles occupations in mathematics include

729-461: A followup unpublished paper, Perelman proved his "stability theorem," asserting that in the collection of all Alexandrov spaces with a fixed curvature bound, all elements of any sufficiently small metric ball around a compact space are mutually homeomorphic . Vitali Kapovitch, who described Perelman's article as being "very hard to read," later wrote a detailed version of Perelman's proof, making use of some further simplifications. Perelman developed

810-400: A manifold. The heat equation, such as when applied in the sciences to physical phenomena such as temperature , models how concentrations of extreme temperatures will spread out until a uniform temperature is achieved throughout an object. In three seminal articles published in the 1980s, Hamilton proved that his equation achieved analogous phenomena, spreading extreme curvatures and uniformizing

891-400: A manner which will help ensure that the plans are maintained on a sound financial basis. As another example, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while

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972-528: A microscopic level, every singularity looks either like a cylinder collapsing to its axis, or a sphere collapsing to its center. Perelman's proof of his canonical neighborhoods theorem is a highly technical achievement, based upon extensive arguments by contradiction in which Hamilton's compactness theorem (as facilitated by Perelman's noncollapsing theorem) is applied to construct self-contradictory manifolds. Other results in Perelman's first preprint include

1053-428: A new theorem characterizing manifolds in which collapsing is only assumed on a local level. In his preprint, he said the proof of his theorem would be established in another paper, but he did not then release any further details. Proofs were later published by Takashi Shioya and Takao Yamaguchi, John Morgan and Gang Tian , Jianguo Cao and Jian Ge, and Bruce Kleiner and John Lott . Perelman's preprints quickly gained

1134-549: A number of textbooks and expository articles. " Perelman's proofs are concise and, at times, sketchy. The purpose of these notes is to provide the details that are missing in [Perelman's first two preprints]... Regarding the proofs, [Perelman's papers] contain some incorrect statements and incomplete arguments, which we have attempted to point out to the reader. (Some of the mistakes in [Perelman's first paper] were corrected in [Perelman's second paper].) We did not find any serious problems, meaning problems that cannot be corrected using

1215-788: A political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells (some say roofing tiles). Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based primarily on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages

1296-424: A single point, and hence that the original space must be diffeomorphic to Euclidean space . In 1994, Perelman gave a short proof of Cheeger and Gromoll's conjecture by establishing that, under the condition of nonnegative sectional curvature, Sharafutdinov's retraction is a submersion . Perelman's theorem is significant in establishing a topological obstruction to deforming a nonnegatively curved metric to one which

1377-621: A third paper posted in July 2003, Perelman outlined an additional argument, sufficient for proving the Poincaré conjecture (but not the Thurston conjecture), the point being to avoid the most technical work in his second preprint. Perelman's first preprint contained two primary results, both to do with Ricci flow. The first, valid in any dimension, was based on a novel adaptation of Peter Li and Shing-Tung Yau 's differential Harnack inequalities to

1458-463: A three-dimensional version of his surgery techniques could be developed, and if a certain conjecture on the long-time behavior of Ricci flow could be established, then Thurston's conjecture would be resolved. This became known as the Hamilton program. In November 2002 and March 2003, Perelman posted two preprints to arXiv , in which he claimed to have outlined a proof of Thurston's conjecture. In

1539-612: A version of Morse theory on Alexandrov spaces. Despite the lack of smoothness in Alexandrov spaces, Perelman and Anton Petrunin were able to consider the gradient flow of certain functions, in unpublished work. They also introduced the notion of an "extremal subset" of Alexandrov spaces, and showed that the interiors of certain extremal subsets define a stratification of the space by topological manifolds . In further unpublished work, Perelman studied DC functions (difference of concave functions) on Alexandrov spaces and established that

1620-420: Is mathematics that studies entirely abstract concepts . From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics , and at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering , and other applications. Another insightful view put forth is that pure mathematics

1701-451: Is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract. As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science , engineering , business , and other areas of mathematical practice. Pure mathematics

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1782-475: Is inevitable that "singularities" occur, meaning that curvature accumulates to infinite levels after a finite amount of "time" has elapsed. Following Shing-Tung Yau 's suggestion that a detailed understanding of these singularities could be topologically meaningful, and in particular that their locations might identify the spheres and tori in Thurston's conjecture , Hamilton began a systematic analysis. Throughout

1863-574: Is most likely a fake, pointing to contradictions in statements supposedly made by Perelman. The writer Brett Forrest briefly interacted with Perelman in 2012. A reporter who had called him was told: " You are disturbing me. I am picking mushrooms. " Dissertation Research papers 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 . One of

1944-400: Is not necessarily applied mathematics : it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing

2025-515: Is positively curved, even at a single point. Some of Perelman's work dealt with the construction of various interesting Riemannian manifolds with positive Ricci curvature . He found Riemannian metrics on the connected sum of arbitrarily many complex projective planes with positive Ricci curvature, bounded diameter, and volume bounded away from zero. Also, he found an explicit complete metric on four-dimensional Euclidean space with positive Ricci curvature and Euclidean volume growth, and such that

2106-612: Is why I don't want to have everybody looking at me. " Nevertheless, on 22 August 2006, at the International Congress of Mathematicians in Madrid , Perelman was offered the Fields Medal " for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow ". He did not attend the ceremony and the presenter informed the congress that Perelman declined to accept

2187-653: The Pythagorean school , whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria ( c.  AD 350 – 415). She succeeded her father as librarian at the Great Library and wrote many works on applied mathematics. Because of

2268-676: The Schock Prize , and the Nevanlinna Prize . The American Mathematical Society , Association for Women in Mathematics , and other mathematical societies offer several prizes aimed at increasing the representation of women and minorities in the future of mathematics. Several well known mathematicians have written autobiographies in part to explain to a general audience what it is about mathematics that has made them want to devote their lives to its study. These provide some of

2349-476: The Steklov Institute in Saint Petersburg in the summer of 1995 for a research-only position. In his undergraduate studies, Perelman dealt with issues in the field of convex geometry . His first published article studied the combinatorial structures arising from intersections of convex polyhedra . With I. V. Polikanova, he established a measure-theoretic formulation of Helly's theorem . In 1987,

2430-475: The Steklov Institute of Mathematics . The institute was established in 1940 as a department of the Steklov Institute and is named after Vladimir Andreevich Steklov , a Soviet / Russian mathematician , mechanician and physicist . 59°56′03″N 30°20′34″E  /  59.9341°N 30.3429°E  / 59.9341; 30.3429 This article about a mathematics organization

2511-418: The Thurston geometrization conjecture , posited that given any closed three-dimensional manifold whatsoever, there is some collection of two-dimensional spheres and tori inside of the manifold which disconnect the space into separate pieces, each of which can be endowed with a uniform geometric structure. Thurston was able to prove his conjecture under some provisional assumptions. In John Morgan 's view, it

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2592-760: The USSR Academy of Sciences , where his advisors were Aleksandr Aleksandrov and Yuri Burago . In the late 1980s and early 1990s, with a strong recommendation from the geometer Mikhail Gromov , Perelman obtained research positions at several universities in the United States. In 1991, Perelman won the Young Mathematician Prize of the Saint Petersburg Mathematical Society for his work on Aleksandrov's spaces of curvature bounded from below. In 1992, he

2673-414: The asymptotic cone is non-uniquely defined. The Poincaré conjecture , proposed by mathematician Henri Poincaré in 1904, was throughout the 20th century regarded as a key problem in topology . On the 3-sphere , defined as the set of points at unit length from the origin in four-dimensional Euclidean space , any loop can be contracted into a point. Poincaré suggested that a converse might be true: if

2754-624: The geometrization conjecture or the Poincaré conjecture . In April 2003, Perelman visited the Massachusetts Institute of Technology , Princeton University , Stony Brook University , Columbia University , and New York University to give short series of lectures on his work, and to clarify some details for experts in the relevant fields. In the years afterwards, three detailed expositions appeared, discussed below. Since then, various parts of Perelman's work have also appeared in

2835-478: The graduate level . In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics; the students who pass are permitted to work on a doctoral dissertation . Mathematicians involved with solving problems with applications in real life are called applied mathematicians . Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of

2916-510: The 1990s, he found a number of new technical results and methods, culminating in a 1997 publication constructing a "Ricci flow with surgery" for four-dimensional spaces . As an application of his construction, Hamilton was able to settle a four-dimensional curvature-based analogue of the Poincaré conjecture. Yau has identified this article as one of the most important in the field of geometric analysis , saying that with its publication it became clear that Ricci flow could be powerful enough to settle

2997-586: The Italian and German universities, but as they already enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment , the same influences that inspired Humboldt. The Universities of Oxford and Cambridge emphasized the importance of research , arguably more authentically implementing Humboldt's idea of a university than even German universities, which were subject to state authority. Overall, science (including mathematics) became

3078-579: The Millennium Prize in July 2010. He considered the decision of the Clay Institute unfair for not sharing the prize with Richard S. Hamilton , and stated that " the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unjust. " The Clay Institute subsequently used Perelman's prize money to fund the "Poincaré Chair", a temporary position for young promising mathematicians at

3159-764: The Paris Institut Henri Poincaré . Perelman quit his job at the Steklov Institute in December 2005. His friends are said to have stated that he currently finds mathematics a painful topic to discuss; by 2010, some even said that he had entirely abandoned mathematics. Perelman is quoted in a 2006 article in The New Yorker saying that he was disappointed with the ethical standards of the field of mathematics. The article implies that Perelman refers particularly to alleged efforts of Fields medalist Shing-Tung Yau to downplay Perelman's role in

3240-713: The Ricci flow partly with the aim of attacking the conjecture. He had previously rejected the prestigious prize of the European Mathematical Society in 1996. Grigori Yakovlevich Perelman was born in Leningrad , Soviet Union (now Saint Petersburg, Russia) on June 13, 1966, to Jewish parents, Yakov (who now lives in Israel) and Lyubov (who still lives in Saint Petersburg with Perelman). Perelman's mother Lyubov gave up graduate work in mathematics to raise him. Perelman's mathematical talent became apparent at

3321-765: The Soviet team at the International Mathematical Olympiad hosted in Budapest, achieving a perfect score. He continued as a student of the School of Mathematics and Mechanics (the so-called "матмех" i.e. "math-mech") at Leningrad State University , without admission examinations, and enrolled at the university. After completing his PhD in 1990, Perelman began work at the Leningrad Department of Steklov Institute of Mathematics of

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3402-404: The Thurston conjecture. The key of Hamilton's analysis was a quantitative understanding of how singularities occur in his four-dimensional setting; the most outstanding difficulty was the quantitative understanding of how singularities occur in three-dimensional settings. Although Hamilton was unable to resolve this issue, in 1999 he published work on Ricci flow in three dimensions, showing that if

3483-545: The age of 10, and his mother enrolled him in Sergei Rukshin's after-school mathematics training program. His mathematical education continued at the Leningrad Secondary School 239 , a specialized school with advanced mathematics and physics programs. Perelman excelled in all subjects except physical education . In 1982, not long after his sixteenth birthday, he won a gold medal as a member of

3564-507: The analysis of Ricci flow , and proved the Poincaré conjecture and Thurston's geometrization conjecture , the former of which had been a famous open problem in mathematics for the past century. The full details of Perelman's work were filled in and explained by various authors over the following several years. In August 2006, Perelman was offered the Fields Medal for "his contributions to geometry and his revolutionary insights into

3645-476: The analytical and geometric structure of the Ricci flow", but he declined the award, stating: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo." On 22 December 2006, the scientific journal Science recognized Perelman's proof of the Poincaré conjecture as the scientific " Breakthrough of the Year ", the first such recognition in the area of mathematics. On 18 March 2010, it

3726-449: The assumptions of the Poincaré conjecture , the Ricci flow with surgery exists only for finite time, so that the infinite-time analysis of Ricci flow is irrelevant. The construction of Ricci flow with surgery has the Poincaré conjecture as a corollary. In order to settle the Thurston conjecture , the second half of Perelman's second preprint is devoted to an analysis of Ricci flows with surgery, which may exist for infinite time. Perelman

3807-465: The attention of the mathematical community, although they were widely seen as hard to understand since they had been written somewhat tersely. Against the usual style in academic mathematical publications, many technical details had been omitted. It was soon apparent that Perelman had made major contributions to the foundations of Ricci flow , although it was not immediately clear to the mathematical community that these contributions were sufficient to prove

3888-419: The best glimpses into what it means to be a mathematician. The following list contains some works that are not autobiographies, but rather essays on mathematics and mathematicians with strong autobiographical elements. St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences The name of the institution is a historical tradition and since 1995 it has no subordination to

3969-500: The earliest known mathematicians was Thales of Miletus ( c.  624  – c.  546 BC ); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry , by deriving four corollaries to Thales's theorem . The number of known mathematicians grew when Pythagoras of Samos ( c.  582  – c.  507 BC ) established

4050-472: The ethical breaches he perceived.) " It was unclear whether along with his resignation from Steklov and subsequent seclusion Perelman stopped his mathematics research. Yakov Eliashberg , another Russian mathematician, said that in 2007 Perelman confided to him that he was working on other things, but that it was too premature to discuss them. Perelman has shown interest in the Navier–Stokes equations and

4131-518: The ethical standards in the field. He lives in seclusion in Saint Petersburg and has declined requests for interviews since 2006. In the 1990s, partly in collaboration with Yuri Burago , Mikhael Gromov , and Anton Petrunin, he made contributions to the study of Alexandrov spaces . In 1994, he proved the soul conjecture in Riemannian geometry, which had been an open problem for the previous 20 years. In 2002 and 2003, he developed new techniques in

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4212-465: The extension of a well-known theorem of Nikolai Efimov to higher dimensions. Perelman's first works to have a major impact on the mathematical literature were in the field of Alexandrov spaces , the concept of which dates back to the 1950s. In a very well-known paper coauthored with Yuri Burago and Mikhael Gromov , Perelman established the modern foundations of this field, with the notion of Gromov–Hausdorff convergence as an organizing principle. In

4293-500: The focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. The German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of

4374-1000: The following. There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics or physics. Prominent prizes in mathematics include the Abel Prize , the Chern Medal , the Fields Medal , the Gauss Prize , the Nemmers Prize , the Balzan Prize , the Crafoord Prize , the Shaw Prize , the Steele Prize , the Wolf Prize ,

4455-633: The imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, and localized constructs, applied mathematicians work regularly in the study and formulation of mathematical models . Mathematicians and applied mathematicians are considered to be two of the STEM (science, technology, engineering, and mathematics) careers. The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics"

4536-436: The introduction of certain monotonic quantities and a "pseudolocality theorem" which relates curvature control and isoperimetry . However, despite being major results in the theory of Ricci flow, these results were not used in the rest of his work. The first half of Perelman's second preprint, in addition to fixing some incorrect statements and arguments from the first paper, used his canonical neighborhoods theorem to construct

4617-580: The kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of "freedom of scientific research, teaching and study." Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education , and then proceed to specialize in topics of their own choice at

4698-474: The king of Prussia , Fredrick William III , to build a university in Berlin based on Friedrich Schleiermacher 's liberal ideas; the goal was to demonstrate the process of the discovery of knowledge and to teach students to "take account of fundamental laws of science in all their thinking." Thus, seminars and laboratories started to evolve. British universities of this period adopted some approaches familiar to

4779-441: The medal later; third, I don't accept the prize. From the very beginning, I told him I have chosen the third one ... [the prize] was completely irrelevant for me. Everybody understood that if the proof is correct, then no other recognition is needed. " He was quoted as saying: " I'm not interested in money or fame, I don't want to be on display like an animal in a zoo. I'm not a hero of mathematics. I'm not even that successful; that

4860-507: The medal, which made him the only person to have ever declined the prize. He has also rejected a prestigious prize from the European Mathematical Society . On 18 March 2010, Perelman was awarded a Millennium Prize for solving the problem. On 8 June 2010, he did not attend a ceremony in his honor at the Institut Océanographique de Paris to accept his $ 1 million prize. According to Interfax , Perelman refused to accept

4941-537: The media. Masha Gessen , author of a biography about Perelman, "Perfect Rigour: A Genius and the Mathematical Breakthrough of the Century ", was unable to meet him. A Russian documentary about Perelman in which his work is discussed by several leading mathematicians, including Mikhail Gromov , Ludwig Faddeev , Anatoly Vershik , Gang Tian , John Morgan and others, was released in 2011 under

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5022-621: The methods introduced by Perelman. " " In this paper, we shall present the Hamilton-Perelman theory of Ricci flow. Based on it, we shall give the first written account of a complete proof of the Poincaré conjecture and the geometrization conjecture of Thurston. While the complete work is an accumulated efforts of many geometric analysts, the major contributors are unquestionably Hamilton and Perelman. [...] In this paper, we shall give complete and detailed proofs [...] especially of Perelman's work in his second paper in which many key ideas of

5103-488: The plane which is complete can be continuously immersed as a polyhedral surface. Later, he constructed an example of a smooth hypersurface of four-dimensional Euclidean space which is complete and has Gaussian curvature negative and bounded away from zero. Previous examples of such surfaces were known, but Perelman's was the first to exhibit the saddle property on nonexistence of locally strictly supporting hyperplanes. As such, his construction provided further obstruction to

5184-504: The possibility of being awarded a Fields medal , led him to state that he had quit professional mathematics by 2006. He said: " As long as I was not conspicuous, I had a choice. Either to make some ugly thing or, if I didn't do this kind of thing, to be treated as a pet. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit." (''The New Yorker'' authors explained Perelman's reference to "some ugly thing" as "a fuss" on Perelman's part about

5265-531: The probability and likely cost of the occurrence of an event such as death, sickness, injury, disability, or loss of property. Actuaries also address financial questions, including those involving the level of pension contributions required to produce a certain retirement income and the way in which a company should invest resources to maximize its return on investments in light of potential risk. Using their broad knowledge, actuaries help design and price insurance policies, pension plans, and other financial strategies in

5346-600: The problem of their solutions' existence and smoothness , according to Le Point . In 2014, Russian media reported that Perelman was working in the field of nanotechnology in Sweden . Shortly thereafter, however, he was spotted again in his native hometown of Saint Petersburg . Russian media speculated he is periodically visiting his sister in Sweden, while living in Saint Petersburg and taking care of his elderly mother. Perelman has avoided journalists and other members of

5427-441: The proof and play up the work of Cao and Zhu . Perelman added: "I can't say I'm outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest...It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated." This, combined with

5508-427: The proofs are sketched or outlined but complete details of the proofs are often missing. As we pointed out before, we have to substitute several key arguments of Perelman by new approaches based on our study, because we were unable to comprehend these original arguments of Perelman which are essential to the completion of the geometrization program. " In May 2006, a committee of nine mathematicians voted to award Perelman

5589-484: The real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. Many professional mathematicians also engage in the teaching of mathematics. Duties may include: Many careers in mathematics outside of universities involve consulting. For instance, actuaries assemble and analyze data to estimate

5670-408: The set of regular points has the structure of a manifold modeled on DC functions. For his work on Alexandrov spaces, Perelman was recognized with an invited lecture at the 1994 International Congress of Mathematicians . In 1972, Jeff Cheeger and Detlef Gromoll established their important soul theorem . It asserts that every complete Riemannian metric of nonnegative sectional curvature has

5751-484: The setting of Ricci flow. By carrying out the proof of the Bishop–Gromov inequality for the resulting Li−Yau length functional, Perelman established his celebrated "noncollapsing theorem" for Ricci flow, asserting that local control of the size of the curvature implies control of volumes. The significance of the noncollapsing theorem is that volume control is one of the preconditions of Hamilton's compactness theorem . As

5832-403: The seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle , and at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics . Moving into the 19th century, the objective of universities all across Europe evolved from teaching the "regurgitation of knowledge" to "encourag[ing] productive thinking." In 1810, Alexander von Humboldt convinced

5913-419: The three-dimensional case, as their topological manipulations, moving "problematic regions" out of the way without interfering with other regions, seem to require high dimensions in order to work. In 1982, William Thurston developed a novel viewpoint, making the Poincaré conjecture into a small special case of a hypothetical systematic structure theory of topology in three dimensions. His proposal, known as

5994-506: The title "Иноходец. Урок Перельмана" ("Maverick: Perelman's Lesson"). In April 2011, Aleksandr Zabrovsky, producer of "President-Film" studio, claimed to have held an interview with Perelman and agreed to shoot a film about him, under the tentative title The Formula of the Universe . Zabrovsky says that in the interview, Perelman explained why he rejected the one million dollar prize. A number of journalists believe that Zabrovsky's interview

6075-413: The year he began graduate studies, he published an article controlling the size of circumscribed cylinders by that of inscribed spheres . Surfaces of negative curvature were the subject of Perelman's graduate studies. His first result was on the possibility of prescribing the structure of negatively-curved polyhedral surfaces in three-dimensional Euclidean space . He proved that any such metric on

6156-943: Was Al-Khawarizmi . A notable feature of many scholars working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics , maths and astronomy of Ibn al-Haytham . The Renaissance brought an increased emphasis on mathematics and science to Europe. During this period of transition from a mainly feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli (founder of accounting ); Niccolò Fontana Tartaglia (notable engineer and bookkeeper); Gerolamo Cardano (earliest founder of probability and binomial expansion); Robert Recorde (physician) and François Viète (lawyer). As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in

6237-487: Was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture. On 1 July 2010, he rejected the prize of one million dollars, saying that he considered the decision of the board of the Clay Institute to be unfair, in that his contribution to solving the Poincaré conjecture was no greater than that of Richard S. Hamilton , the mathematician who pioneered

6318-650: Was invited to spend a semester each at the Courant Institute in New York University , where he began work on manifolds with lower bounds on Ricci curvature . From there, he accepted a two-year Miller Research Fellowship at the University of California, Berkeley , in 1993. After proving the soul conjecture in 1994, he was offered jobs at several top universities in the US, including Princeton and Stanford , but he rejected them all and returned to

6399-431: Was ongoing throughout the reign of certain caliphs, and it turned out that certain scholars became experts in the works they translated, and in turn received further support for continuing to develop certain sciences. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support

6480-471: Was only with Thurston's systematic viewpoint that most topologists came to believe that the Poincaré conjecture would be true. At the same time that Thurston published his conjecture, Richard Hamilton introduced his theory of the Ricci flow . Hamilton's Ricci flow is a prescription, defined by a partial differential equation formally analogous to the heat equation , for how to deform a Riemannian metric on

6561-469: Was unable to resolve Hamilton's 1999 conjecture on long-time behavior, which would make Thurston's conjecture another corollary of the existence of Ricci flow with surgery. Nonetheless, Perelman was able to adapt Hamilton's arguments to the precise conditions of his new Ricci flow with surgery. The end of Hamilton's argument made use of Jeff Cheeger and Mikhael Gromov 's theorem characterizing collapsing manifolds . In Perelman's adaptation, he required use of

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