1.
Japan
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Japan is a sovereign island nation in Eastern Asia. Located in the Pacific Ocean, it lies off the eastern coast of the Asia Mainland and stretches from the Sea of Okhotsk in the north to the East China Sea, the kanji that make up Japans name mean sun origin. 日 can be read as ni and means sun while 本 can be read as hon, or pon, Japan is often referred to by the famous epithet Land of the Rising Sun in reference to its Japanese name. Japan is an archipelago consisting of about 6,852 islands. The four largest are Honshu, Hokkaido, Kyushu and Shikoku, the country is divided into 47 prefectures in eight regions. Hokkaido being the northernmost prefecture and Okinawa being the southernmost one, the population of 127 million is the worlds tenth largest. Japanese people make up 98. 5% of Japans total population, approximately 9.1 million people live in the city of Tokyo, the capital of Japan. Archaeological research indicates that Japan was inhabited as early as the Upper Paleolithic period, the first written mention of Japan is in Chinese history texts from the 1st century AD. Influence from other regions, mainly China, followed by periods of isolation, from the 12th century until 1868, Japan was ruled by successive feudal military shoguns who ruled in the name of the Emperor. Japan entered into a period of isolation in the early 17th century. The Second Sino-Japanese War of 1937 expanded into part of World War II in 1941, which came to an end in 1945 following the bombings of Hiroshima and Nagasaki. Japan is a member of the UN, the OECD, the G7, the G8, the country has the worlds third-largest economy by nominal GDP and the worlds fourth-largest economy by purchasing power parity. It is also the worlds fourth-largest exporter and fourth-largest importer, although Japan has officially renounced its right to declare war, it maintains a modern military with the worlds eighth-largest military budget, used for self-defense and peacekeeping roles. Japan is a country with a very high standard of living. Its population enjoys the highest life expectancy and the third lowest infant mortality rate in the world, in ancient China, Japan was called Wo 倭. It was mentioned in the third century Chinese historical text Records of the Three Kingdoms in the section for the Wei kingdom, Wa became disliked because it has the connotation of the character 矮, meaning dwarf. The 倭 kanji has been replaced with the homophone Wa, meaning harmony, the Japanese word for Japan is 日本, which is pronounced Nippon or Nihon and literally means the origin of the sun. The earliest record of the name Nihon appears in the Chinese historical records of the Tang dynasty, at the start of the seventh century, a delegation from Japan introduced their country as Nihon

2.
University of Tokyo
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The University of Tokyo, abbreviated as Todai, is a research university located in Bunkyo, Tokyo, Japan. The university has 10 faculties with a total of around 30,000 students,2,100 of whom are foreign and its five campuses are in Hongō, Komaba, Kashiwa, Shirokane and Nakano. It is the first of Japans National Seven Universities, the university was chartered by the Meiji government in 1877 under its current name by amalgamating older government schools for medicine and Western learning. It was renamed the Imperial University in 1886, and then Tokyo Imperial University in 1897 when the Imperial University system was created, in September 1923, an earthquake and the following fires destroyed about 700,000 volumes of the Imperial University Library. The books lost included the Hoshino Library, a collection of about 10,000 books, the books were the former possessions of Hoshino Hisashi before becoming part of the library of the university and were mainly about Chinese philosophy and history. In 1947, after Japans defeat in World War II, it re-assumed its original name, although the university was founded during the Meiji period, it has earlier roots in the Astronomy Agency, Shoheizaka Study Office, and the Western Books Translation Agency. These institutions were government offices established by the 徳川幕府 Tokugawa shogunate, kikuchi Dairoku, an important figure in Japanese education, served as president of Tokyo Imperial University. For the 1964 Summer Olympics, the university hosted the running portion of the pentathlon event. On 20 January 2012, Todai announced that it would shift the beginning of its academic year from April to September to align its calendar with the international standard, the shift would be phased in over five years. But this unilateral announcement by the president was received badly and the university abandoned the plans, according to the Japan Times, the university had 1,282 professors in February 2012. In 2014, the School of Science at the University of Tokyo introduced an undergraduate transfer program called Global Science Course. Academic Ranking of World Universities ranked the University of Tokyo 1st in Asia, Times Higher Education World University Rankings ranked the University of Tokyo 27th in the world in 2013 and 1st in the Asia University ranking in 2013. In 2015, Times Higher Education World University Rankings ranked the institution 23rd in the world and it ranks 12th in the world according to the Times Higher Education World Reputation Rankings 2016. QS World University Rankings in 2011 ranked the University of Tokyo 25th in the world, in the 2011 QS Asian University Rankings, which employs a different methodology, the University of Tokyo came 4th. Currently, University of Tokyo holds ranks 9th & 11th respectively for Natural Sciences & Engineering, Times Higher Education World Reputation Rankings ranked the University of Tokyo 12th in the world also 1st in Asia in 2016. Global University Ranking ranked the University of Tokyo 3rd in the world, Human Resources & Labor Review, a human competitiveness index & analysis published in Chasecareer Network, ranked the university 21st internationally and 1st in Asia in 2010. Mines ParisTech, Professional Ranking World Universities ranked the University of Tokyo 2nd in the world on the basis of the number of alumni listed among CEOs in the 500 largest worldwide companies, nature Publishing Index ranked the University of Tokyo 5th in the world in 2011. The main Hongo campus occupies the estate of the Maeda family

3.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

4.
Stony Brook University
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The State University of New York at Stony Brook is a public sea-grant and space-grant research university located in Stony Brook, New York in the United States. It is part of the State University of New York system, the institution was founded in 1957 in Oyster Bay as State University College on Long Island, and would evolve into the present university after a move to Stony Brook in 1962. Since its establishment in Stony Brook, the university has expanded to more than 200 major buildings with a combined area of more than 11 million gross square feet across 1,454 acres of land. In 2001, SUNY Stony Brook was elected to the Association of American Universities, joining four private universities and it is also a member of the larger Universities Research Association for which its president Samuel Stanley is a council president. Stony Brook is the largest single-site employer on Long Island, more than 24,500 students are enrolled at the university, which has over 14,500 employees and over 2,400 faculty. Stony Brook has a number of athletics teams, the Stony Brook Seawolves are members of the America East Conference and the Colonial Athletic Association competing at the Division I level of the NCAA since 1994. The State University of New York at Stony Brook was established in Oyster Bay in 1957 as the State University College on Long Island, by the governor and state of New York. Established almost a decade after the creation of New York’s public higher education system, leonard K. Olson was appointed as the first dean of the institution and was instrumental in the recruitment of faculty staff and planning of the later Stony Brook campus. SUCOLI opened with an class of 148 students, on the grounds of the William Robertson Coe Planting Fields estate. These first students were admitted on a tuition-free basis,1961 was a year of firsts as thirty students were conferred degrees in the first commencement and the University was appointed its first president, John Francis Lee. Lee left later that year due to political and bureaucratic matters regarding the future of the University, more recently, it has adopted the short-form name Stony Brook University. In 1963, only three years after the release of the Heald Report, the Governor commissioned the “Education of Health Professions” report, the report outlined the need for expansion of the university system to prepare medical professionals for the future needs of the state. In 1965 the State University appointed John S. Toll, a renowned physicist from the University of Maryland as the president of Stony Brook. In 1966 the University set forth initial timetables for the development of the Health Science Center which would house the University’s health programs, despite the budgetary concerns and challenges from Albany the University released a formalized plan early in 1968 and funding for recruitment of faculty was provided. At the same time, residential housing was expanded to 3,000, the Stony Brook Union opened in 1970, and in 1971, but the University lagged significantly in undergraduate education, prioritizing graduate education and research over undergraduate studies and student life. By 1975, enrollment had reached 16,000 and expansion crossed over Nicolls Road with the construction of the Health Science Center which would be completed in 1980, in 1981 John Marburger was inaugurated as the third president of the University and would continue the expansion of the institution. By the late 1980s the administration affirmed the need to other areas of the institution which included undergraduate education, student and residential life. The University approved a decision to transition athletics to the Division I of the NCAA and followed with the construction of the Stony Brook Arena, the 1990s affirmed Stony Brook’s success at building a research university with a strong undergraduate education

5.
Doctor of Philosophy
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A Doctor of Philosophy is a type of doctoral degree awarded by universities in many countries. Ph. D. s are awarded for a range of programs in the sciences, engineering. The Ph. D. is a degree in many fields. The completion of a Ph. D. is often a requirement for employment as a university professor, researcher, individuals with an earned doctorate can use the title of Doctor with their name and use the post-nominal letters Ph. D. The requirements to earn a Ph. D. degree vary considerably according to the country, institution, a person who attains a doctorate of philosophy is automatically awarded the academic title of doctor. A student attaining this level may be granted a Candidate of Philosophy degree at some institutions. A Ph. D. candidate must submit a project, thesis or dissertation often consisting of a body of academic research. In many countries, a candidate must defend this work before a panel of examiners appointed by the university. Universities award other types of doctorates besides the Ph. D. such as the Doctor of Musical Arts, a degree for music performers and the Doctor of Education, in 2016, ELIA launched The Florence Principles on the Doctorate in the Arts. The Florence Principles have been endorsed are supported also by AEC, CILECT, CUMULUS, the degree is abbreviated PhD, from the Latin Philosophiae Doctor, pronounced as three separate letters. In the universities of Medieval Europe, study was organized in four faculties, the faculty of arts. All of these faculties awarded intermediate degrees and final degrees, the doctorates in the higher faculties were quite different from the current Ph. D. degree in that they were awarded for advanced scholarship, not original research. No dissertation or original work was required, only lengthy residency requirements, besides these degrees, there was the licentiate. According to Keith Allan Noble, the first doctoral degree was awarded in medieval Paris around 1150, the doctorate of philosophy developed in Germany as the terminal Teachers credential in the 17th century. Typically, upon completion, the candidate undergoes an oral examination, always public, starting in 2016, in Ukraine Doctor of Philosophy is the highest education level and the first science degree. PhD is awarded in recognition of a contribution to scientific knowledge. A PhD degree is a prerequisite for heading a university department in Ukraine, upon completion of a PhD, a PhD holder can elect to continue his studies and get a post-doctoral degree called Doctor of Sciences, which is the second and the highest science degree in Ukraine. Scandinavian countries were among the early adopters of a known as a doctorate of philosophy

6.
National Diet Library
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The National Diet Library is the only national library in Japan. It was established in 1948 for the purpose of assisting members of the National Diet of Japan in researching matters of public policy, the library is similar in purpose and scope to the United States Library of Congress. The National Diet Library consists of two facilities in Tokyo and Kyoto, and several other branch libraries throughout Japan. The Diets power in prewar Japan was limited, and its need for information was correspondingly small, the original Diet libraries never developed either the collections or the services which might have made them vital adjuncts of genuinely responsible legislative activity. Until Japans defeat, moreover, the executive had controlled all political documents, depriving the people and the Diet of access to vital information. The U. S. occupation forces under General Douglas MacArthur deemed reform of the Diet library system to be an important part of the democratization of Japan after its defeat in World War II. In 1946, each house of the Diet formed its own National Diet Library Standing Committee, hani Gorō, a Marxist historian who had been imprisoned during the war for thought crimes and had been elected to the House of Councillors after the war, spearheaded the reform efforts. Hani envisioned the new body as both a citadel of popular sovereignty, and the means of realizing a peaceful revolution, the National Diet Library opened in June 1948 in the present-day State Guest-House with an initial collection of 100,000 volumes. The first Librarian of the Diet Library was the politician Tokujirō Kanamori, the philosopher Masakazu Nakai served as the first Vice Librarian. In 1949, the NDL merged with the National Library and became the national library in Japan. At this time the collection gained a million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, in 1986, the NDLs Annex was completed to accommodate a combined total of 12 million books and periodicals. The Kansai-kan, which opened in October 2002 in the Kansai Science City, has a collection of 6 million items, in May 2002, the NDL opened a new branch, the International Library of Childrens Literature, in the former building of the Imperial Library in Ueno. This branch contains some 400,000 items of literature from around the world. Though the NDLs original mandate was to be a library for the National Diet. In the fiscal year ending March 2004, for example, the library reported more than 250,000 reference inquiries, in contrast, as Japans national library, the NDL collects copies of all publications published in Japan. The NDL has an extensive collection of some 30 million pages of documents relating to the Occupation of Japan after World War II. This collection include the documents prepared by General Headquarters and the Supreme Commander of the Allied Powers, the Far Eastern Commission, the NDL maintains a collection of some 530,000 books and booklets and 2 million microform titles relating to the sciences

7.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory

8.
Galois theory
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In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, originally, Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, further abstraction of Galois theory is achieved by the theory of Galois connections. Further, it gives a clear, and often practical. Galois theory also gives an insight into questions concerning problems in compass. It gives an elegant characterisation of the ratios of lengths that can be constructed with this method, for instance, = x2 – x + ab, where 1, a + b and ab are the elementary polynomials of degree 0,1 and 2 in two variables. This was first formalized by the 16th-century French mathematician François Viète, in Viètes formulas, the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation, see Discriminant, Nature of the roots for details. This solution was then rediscovered independently in 1535 by Niccolò Fontana Tartaglia, Cardano then extended this to numerous other cases, using similar arguments, see more details at Cardanos method. After the discovery of Ferros work, he felt that Tartaglias method was no longer secret and his student Lodovico Ferrari solved the quartic polynomial, his solution was also included in Ars Magna. With the benefit of modern notation and complex numbers, the formulae in this book do work in the general case and it was Rafael Bombelli who managed to understand how to work with complex numbers in order to solve all forms of cubic equation. Crucially, however, he did not consider composition of permutations, lagranges method did not extend to quintic equations or higher, because the resolvent had higher degree. The quintic was almost proven to have no general solutions by radicals by Paolo Ruffini in 1799, whose key insight was to use permutation groups, not just a single permutation. This group was always solvable for polynomials of degree four or less, but not always so for polynomials of degree five and greater, prior to this publication, Liouville announced Galois result to the Academy in a speech he gave on 4 July 1843. According to Allan Clark, Galoiss characterization dramatically supersedes the work of Abel, Galois theory was notoriously difficult for his contemporaries to understand, especially to the level where they could expand on it. For example, in his 1846 commentary, Liouville completely missed the core of Galois method. Joseph Alfred Serret who attended some of Liouvilles talks, included Galois theory in his 1866 of his textbook Cours dalgèbre supérieure, serrets pupil, Camille Jordan had an even better understanding reflected in his 1870 book Traité des substitutions et des équations algébriques. Outside France Galois theory remained more obscure for a longer period, in Britain, Cayley failed to grasp its depth and popular British algebra textbooks didnt even mention Galois theory until well after the turn of the century

9.
Shokichi Iyanaga
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Shokichi Iyanaga was a Japanese mathematician. Iyanaga was born in Tokyo, Japan on April 2,1906 and he studied at the University of Tokyo from 1926 to 1929. As an undergraduate, he published two papers in the Japanese Journal of Mathematics and the Proceedings of the Imperial Academy of Tokyo, both of his papers appeared in print in 1928. After completing his degree in 1929, he stayed at Tokyo. He completed his Ph. D. in mathematics 1931, in 1931, Iyanaga obtained a scholarship from the French government. He also went to Hamburg, Germany where he studied with Austrian mathematician Emil Artin, in 1932, he attended the International Congress of Mathematicians in Zurich. During his time in Europe, he met with top mathematicians such as Claude Chevalley, Henri Cartan, Iyanaga returned to Tokyo in 1934 and was appointed Assistant Professor at the University of Tokyo. From 1935 to 1939, he didnt publish any research papers, according to Iyanaga, it was because of the pressure of teaching and other business to which he was not accustomed. He managed to solve a question of Artin on generalizing the principal ideal theorem, Iyanaga did publish many papers which arose through several courses such as algebraic topology, functional analysis, and geometry, which he taught. He became Professor at the University of Tokyo in 1942 and it was during World War II. Towards the end of the war, many Japanese cities were bombarded and he was busy in editing textbooks from primary and secondary schools and he continued to give courses and organise seminars. After the end of the war, he joined the Science Council of Japan in 1947 and he became a member of the Executive Committee of the International Mathematical Union in 1952. He was responsible for organizing the International Congress of Mathematicians in Amsterdam in 1954 and he was President of the International Commission on Mathematical Instruction from 1957 to 1978. Iyanaga spent the year 1961-62 at the University of Chicago and he became Dean of the faculty of Science at the University of Tokyo in 1965, a position he held until his retirement in 1967. After his retirement, he was a visiting professor during 1967-68 at the University of Nancy in France, from 1967 to 1977, he was a professor at Gakushuin University in Tokyo. Iyanaga received several honors and awards for his work and he received the Rising Sun from Japan in 1976. He was elected a member of the Japan Academy in 1978 and he received the Légion dhonneur in 1980. Shokichi Iyanaga at the Mathematics Genealogy Project

10.
Pierre Deligne
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Pierre René, Viscount Deligne is a Belgian mathematician. He is known for work on the Weil conjectures, leading to a proof in 1973. He is the winner of the 2013 Abel Prize,2008 Wolf Prize and he was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles. In 1968, he worked with Jean-Pierre Serre, their work led to important results on the l-adic representations attached to modular forms. Delignes also focused on topics in Hodge theory and he introduced weights and tested them on objects in complex geometry. He also collaborated with David Mumford on a new description of the spaces for curves. Their work came to be seen as an introduction to one form of the theory of algebraic stacks, perhaps Delignes most famous contribution was his proof of the third and last of the Weil conjectures. This proof completed a programme initiated and largely developed by Alexander Grothendieck, as a corollary he proved the celebrated Ramanujan–Petersson conjecture for modular forms of weight greater than one, weight one was proved in his work with Serre. From 1970 until 1984, when he moved to the Institute for Advanced Study in Princeton, during this time he did much important work outside of his work on algebraic geometry. He received a Fields Medal in 1978 and this idea allows one to get around the lack of knowledge of the Hodge conjecture, for some applications. All this is part of the yoga of weights, uniting Hodge theory, 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 – and more recent trends have used K-theory approaches and he was awarded the Fields Medal in 1978, the Crafoord Prize in 1988, the Balzan Prize in 2004, the Wolf Prize in 2008, and 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 Science and Letters, Quantum fields and strings, a course for mathematicians. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, edited by Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, american Mathematical Society, Providence, RI, Institute for Advanced Study, Princeton, NJ,1999. Vol.1, xxii+723 pp. Vol.2, pp. i--xxiv, Deligne wrote multiple hand-written letters to other mathematicians in the 1970s. These include Delignes letter to Piatetskii-Shapiro and it was proved by Kontsevich–Soibelman, McClure–Smith and others